Zachriel, Nested Hierarchies and Set Theory The Ignorance Exposed, Again

I had challenged Zachriel
Perhaps Zachriel can produce a nested hierarchy that is not based on characteristics.
Zachriel said
However the Power Set of an empty set {} is {{}}.
That's it. That is as far as it goes.
He made up {{{},{}},{},{{},{{},{},{}}}} because he thought he could slide one by.
Too bad he is just another exposed internet poseur .
And now he is choking on it, trying anything to distract from that fact.
Life is good.
Thanks Zachriel...
I had challenged Zachriel
Perhaps Zachriel can produce a nested hierarchy that is not based on characteristics.
Zachriel said
{{{},{}},{},{{},{{},{},{}}}}
However the Power Set of an empty set {} is {{}}.
That's it. That is as far as it goes.
He made up {{{},{}},{},{{},{{},{},{}}}} because he thought he could slide one by.
Too bad he is just another exposed internet poseur .
And now he is choking on it, trying anything to distract from that fact.
Life is good.
Thanks Zachriel...
875 Comments:
At 7:11 PM, Sheikh said…
Hey Joe, I just saw the background for this post over at Cornelius' blog, in which you said:
Joe G: {{ }} = {}
Because {} broken down is {{}}
That is what a power set is a set broken down.
LOL! Keep on keepin' on, Joe.
At 8:51 PM, Joe G said…
Geez another meaningless fartndart by an anonymous loser.
power set:
In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself.
IOW it is a set, broken down.
Go figure...
At 8:53 PM, Joe G said…
Hey, that was all explained in the OP.
LoL, indeed, ye Shiekh of willful ignorance...
At 9:08 PM, Sheikh said…
But Joe, are you still claiming that the power set of the empty set is the same as the empty set? The cardinality of {} is zero, while the cardinality of {{}} is one, so they can't be the same. Therefore, {{}} ≠ {}.
The excerpt from the wikipedia article that you posted contains the suggestive notation 2^S for the power set of a set S. It's useful because if S has n elements, its power set has 2^n elements.
For example, the power set of {x, y, z} is 2^3 = 8, which I believe came up in the thread on Cornelius' blog.
The empty set {} has 0 elements, so its power set (which happens to be {{}}) has 2^0 = 1 element.
At 9:34 PM, Joe G said…
But Joe, are you still claiming that the power set of the empty set is the same as the empty set?
The power set is the set broken down.
What is the set represented by the power set {{}}?
At 9:39 PM, Joe G said…
The power set of the empty set P({}) = {{}}.
At 9:48 PM, Sheikh said…
First, a correction to the third paragraph of my last post:
For example, the cardinality of the power set of {x, y, z} is 2^3 = 8, which I believe came up in the thread on Cornelius' blog.
Joe G,
The power set is the set broken down.
What is the set represented by the power set {{}}?
I'm not sure what you mean by the "set represented by the power set {{}}". {{}} is the power set of {}, which means it is the set of all subsets of {}. There's just one in this case, namely {}.
One notable thing here is that {} is a subset of every set, so any power set will have cardinality at least 1. That's another way to see why {} and {{}} cannot be the same.
At 10:10 PM, Joe G said…
{{}} is the power set of {}, which means it is the set of all subsets of {}.
Exactly.
From wikipedia:
If S = {}, then P(S) = {{}} is returned.
Which is the same as the post you ignored.
Power sets contain only what the set contains.
And if the contents are the same, ie equal then....
At 10:11 PM, Joe G said…
Also you are a pathetic imp for trying to make something of what I said while ignoring Zachriel's continued errors.
Typically pathetic...
At 10:16 PM, Joe G said…
Operations on the empty set:
Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the identity element for addition, and one is the identity element for multiplication.
Not that you would understand that...
At 10:49 PM, Sheikh said…
Joe,
Power sets contain only what the set contains.
And if the contents are the same, ie equal then....
Ok, I think I see the problem here now, and it is definitely related to your objections, let us say, to nested hierarchies in general.
The two sets {} and {{}} do not have the same contents. {} contains nothing, while {{}} contains a set, {}. The empty set is a bit strange, though, so why not consider a slightly less confusing example:
S = {x, y}
Then the power set of S is:
2^S = { {}, {x}, {y}, {x, y} }
Would you still say S and 2^S are the same? If I understand your post above, you would conclude that the "contents" of both sets are the samejust x and y. However, that's not the case. The elements of S are x and y, while the elements of 2^X are sets containing x and/or y, or nothing. It could even be that x and y are sets in their own right, btw.
If you are working with a set of sets (such as { {}, {x}, {y}, {x, y} }) you can't just throw away the "inside" braces and still expect to have the same set.
So, one question: Do you claim that the sets {x, y} and { {}, {x}, {y}, {x, y} } are equal?
At 10:58 PM, Joe G said…
The two sets {} and {{}} do not have the same contents. {} contains nothing, while {{}} contains a set, {}.
In reality {} contains nothing and itself.
IOW the problem you think you see is, in reality, you.
The empty set is a bit strange, though
The empty set is what I was talking about.
And the fact you keep ignoring my posts tells me you are on some agenda.
At 10:59 PM, Joe G said…
Do you claim that the sets {x, y} and { {}, {x}, {y}, {x, y} } are equal?
I don't know anyone who doesn't:
In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself.
IOW the contents are the same. The way they are arranged is different.
At 11:03 PM, Joe G said…
Ok, I think I see the problem here now, and it is definitely related to your objections, let us say, to nested hierarchies in general.
I predict you will never produce a valid resource pertaining to nested hierarchies that both defines and describes the rules for constructing one that supports any of your claims nor refutes anything I have said.
At 11:07 PM, Joe G said…
I also predict that you will never support anything you say that you think refutes what I said.
You are just another pontificating bluffer.
At 11:11 PM, Joe G said…
If you are working with a set of sets (such as { {}, {x}, {y}, {x, y} }) you can't just throw away the "inside" braces and still expect to have the same set.
If x = your left hand and y = your right hand, you still only have one of each even when represented as a power set.
If x = $10 and y = $20 you still only have $30 even when broken down into a power set.
At 11:13 PM, Joe G said…
Do you claim that the sets {x, y} and { {}, {x}, {y}, {x, y} } are equal?
Let x = $10 and y = $20
The set {x,y} contains $30
the power set { {}, {x}, {y}, {x, y} } also contains $30.
$30 = $30
At 11:23 PM, Sheikh said…
So I'll take it that you agree that {x, y} and { {}, {x}, {y}, {x, y} } are not equal.
In reality {} contains nothing and itself.
Ok, let me use more precise language than the word "contents":
1. {} has no elements. It has one subset, {}.
2. {{}} has one element, {}. It has two subsets, {} and {{}}.
Do you still want to stick to your claim that {} = {{}}?
Here's another way to see this is impossible. If {} = {{}}, then {} ⊂{{}} and {{}} ⊂ {}. The first inclusion is obviously true, but {{}} ⊂ {} is false. That's because {} is not an element of {}.
The distinction between the statements "x is an element of S" and "x is a subset of S" is very important.
At 11:29 PM, Joe G said…
So I'll take it that you agree that {x, y} and { {}, {x}, {y}, {x, y} } are not equal.
Only if you are a complete dishonest moron.
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
What part of that don't you understand?
Do you still want to stick to your claim that {} = {{}}?
I have already supported the claim.
Again just because you can ignore my posts doesn't mean your ignorance refutes them.
At 11:30 PM, Joe G said…
BTW thank you for fulfilling my predictions.
At 11:30 PM, Sheikh said…
Joe,
Do you claim that the sets {x, y} and { {}, {x}, {y}, {x, y} } are equal?
I don't know anyone who doesn't:
I didn't see this before my last post. The two sets are definitely not the same, however.
What's your definition equality of sets?
BTW, here's a good link on set theory:
http://en.wikipedia.org/wiki/Naive_set_theory
Here's a relevant quote:
We also allow for an empty set, often denoted Ø and sometimes {}: a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.) Note that Ø ≠ {Ø}.
At 11:31 PM, Joe G said…
Until tomorrow...
At 11:33 PM, Joe G said…
I didn't see this before my last post. The two sets are definitely not the same, however.
The CONTENTS are the SAME they are EQUAL.
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
What part of that don't you understand?
At 11:58 PM, Sheikh said…
Joe G,
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
What rule are you using to calculate the total amount of money in { {}, {x}, {y}, {x, y} }? I guess you could make a case for getting $60, but I don't see how it adds to $30.
At 8:06 AM, oleg said…
Joe,
Here is a reallife example of a set and its power set.
Michael Denton wrote two books, Evolution: a Theory in Crisis, or E for short, and Nature's Destiny: How the Laws of Biology Reveal Purpose in the Universe or N. The set Books = {E, N} contains two members.
When you ask someone "Which of these books did you read?" you may hear one of four different answers: none, Evolution, Nature's Destiny, or both. Each of the outcomes is a set of books (not a book): an empty set, a set containing one book, or a set containing two books. Formally, Outcomes = {{}, {E}, {N}, {E, N}}.
Clearly, the sets Books and Outcomes are different beasts. Even the sizes are different: two books, four possible outcomes.
At 9:22 AM, Joe G said…
What rule are you using to calculate the total amount of money in { {}, {x}, {y}, {x, y} }? I guess you could make a case for getting $60, but I don't see how it adds to $30.
If x = $10 and y = $20 you only have $30 in both cases {x,y} and { {}, {x}, {y}, {x, y} }
You don't get more money just by rearranging what you already have.
This:
{ {}, {x}, {y}, {x, y} }
just shows how many ways you can arrange that money.
Do you really think your money doubles just because you can arrange it several different ways?
At 9:29 AM, Joe G said…
oleg:
Here is a reallife example of a set and its power set.
The example I provided is a real set and power set.
Michael Denton wrote two books, Evolution: a Theory in Crisis, or E for short, and Nature's Destiny: How the Laws of Biology Reveal Purpose in the Universe or N. The set Books = {E, N} contains two members.
{E,N} contains the SAME stuff as {{}, {E}, {N}, {E, N}}.
They are equal.
When you ask someone "Which of these books did you read?" you may hear one of four different answers: none, Evolution, Nature's Destiny, or both. Each of the outcomes is a set of books (not a book): an empty set, a set containing one book, or a set containing two books. Formally, Outcomes = {{}, {E}, {N}, {E, N}}.
So what?
What does that have to do with anything I have said?
Clearly, the sets Books and Outcomes are different beasts.
Again so what?
What does that have to do with anything I have said?
Even the sizes are different: two books, four possible outcomes.
So what?
The contents of {E,N} are the same as {{}, {E}, {N}, {E, N}}.
That is what I am saying.
Are you saying the contents are different?
If so prove it or admit that you are a bluffing pontificator.
At 9:32 AM, Joe G said…
Let x = $10 and y = $20
The set {x,y} contains $30
the power set { {}, {x}, {y}, {x, y} } also contains $30.
$30 = $30
If you could double your money just by constructing a power set I would think everyone in the world would be doing it.
IOW Shiekh you are one of the stupidest people, ever, in the history of mankind.
At 9:59 AM, Joe G said…
How many books are in the set {E,N}?
How many books are in its power set {{}, {E}, {N}, {E, N}}?
I say the answer to both questions is 2 there are two books in the set and two books in the power set.
At 10:02 AM, oleg said…
Joe, the contents are not the same. They are books in one case, sets of books in the other. In one case we have two members, in the other four. If two sets are equal their sizes are equal. Conversely, unequal sizes mean that the sets are not equal.
At 10:30 AM, Joe G said…
Joe, the contents are not the same.
They have to be.
They are books in one case, sets of books in the other.
They are a set of books in {E,N} and they are both a set of books and individual books in {{}, {E}, {N}, {E, N}}.
The books are the same.
In one case we have two members, in the other four.
You are counting the same two members twice.
There are still only two books.
If two sets are equal their sizes are equal.
They both contain the same thing.
The power set is just a different way of expressing the set.
At 10:30 AM, Joe G said…
How many books are in the set {E,N}?
How many books are in its power set {{}, {E}, {N}, {E, N}}?
I say the answer to both questions is 2 there are two books in the set and two books in the power set.
At 10:42 AM, Joe G said…
A set containing nothing {}, ie an empty set, has the same contents as a set containing only a set that contains nothing {{}}.
Nothing from nothing leaves nothing.
At 10:50 AM, Sheikh said…
Joe,
Did you check out that quote from the wikipedia page on naive set theory that I posted on your blog?
We also allow for an empty set, often denoted Ø and sometimes {}: a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.) Note that Ø ≠ {Ø}.
The last sentence, Ø ≠ {Ø}, also means {} ≠ {{}} if you use the notation {} instead of Ø.
Furthermore, you can see that {x, y} ≠ { {}, {x}, {y}, {x, y} } in several ways. For example, {x} ∉ {x, y}. There are elements of { {}, {x}, {y}, {x, y} } which are not in {x, y}, in other words.
Just think about it, thoughif taking the power set of a set always gave you the original set, what would the purpose of the power set be? There would be no point in doing this operation.
Your illustration of the two sets in money terms is actually quite helpful. {x, y} would stand for an envelope containing a $10 bill and a $20 bill.
On the other hand, { {}, {x}, {y}, {x, y} } would then stand for a large envelope containing four smaller envelopes, one being empty, one containing a $10 bill, one containing a $20 bill, and finally one containing a $10 bill and a $20 bill. Of course you need $60 to create this physical model of the set.
The power set of a set S is not simply a "rearrangement" of the original set S. You have to have enough copies of the original elements to form all possible subsets of S at the same time.
At 10:53 AM, oleg said…
Joe, I moved the discussion to AtBC. See you there.
At 11:01 AM, Joe G said…
On the other hand, { {}, {x}, {y}, {x, y} } would then stand for a large envelope containing four smaller envelopes, one being empty, one containing a $10 bill, one containing a $20 bill, and finally one containing a $10 bill and a $20 bill.
That is false.
{ {}, {x}, {y}, {x, y} } would be the different outcomes of the question "how much money should I take with me?"
You can take no money, $10, $20 or $30.
Just think about it, thoughif taking the power set of a set always gave you the original set, what would the purpose of the power set be? There would be no point in doing this operation.
Yes there is to dtermine the number of proper subsets and then listing them to confirm it.
That is the purpose of the operation.
The contents do not change.
The power set of a set S is not simply a "rearrangement" of the original set S.
It is a rearrangement of a set into ALL of its proper subsets.
At 11:03 AM, Joe G said…
Joe, I moved the discussion to AtBC. See you there.
Hold your breath while you wait.
If you want to discuss something with me this is the place to do so.
If you want to continue to be a pontificating asshole then run to atbc...
At 11:05 AM, Joe G said…
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
At 11:07 AM, oleg said…
Joe, you can continue talking to yourself here if you wish. Bye.
At 11:15 AM, Sheikh said…
Joe,
That is false.
{ {}, {x}, {y}, {x, y} } would be the different outcomes of the question "how much money should I take with me?"
You can take no money, $10, $20 or $30.
I see what you are saying, although this seems more complicated than necessary. The elements of my representation are in 11 correspondence with yours, though:
{} <> take no money
{x} <> take $10
{y} <> take $20
{x, y} <> take $30
That means we are talking about essentially the same set.
It is a rearrangement of a set into ALL of its proper subsets.
Nitpick: you don't want the word "proper" there. The power set of a set is the set of all subsets of the original set.
Did you see my reference to the fact that {} ≠ {{}}?
At 11:19 AM, Sheikh said…
One clarification on my last postThis is the correspondence I meant to specify:
empty envelope <> take no money
envelope with $10 <> take $10
envelope with $20 <> take $20
envelope with $10 and $20 <> take $30
At 12:11 PM, oleg said…
Joe wrote:
{E,N} contains the SAME stuff as {{}, {E}, {N}, {E, N}}.
They are equal.
That's not how things work in set theory. Two sets are equal if and only if they have precisely the same elements. The two sets above do not have the same elements. The elements of the former are books E and N. The elements of the latter are sets of books {}, {E}, {N}, and {E,N}.
Denton wrote two books. A person can give four different answers to the question "Which of those books did you read?" So you use two different sets to describe the books and the answers.
At 12:30 PM, Joe G said…
{ {}, {x}, {y}, {x, y} } would be the different outcomes of the question "how much money should I take with me?"
You can take no money, $10, $20 or $30.
Shiekh:
I see what you are saying, although this seems more complicated than necessary.
It's not more complicated.
That is the way it is.
subset:
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.
At 12:32 PM, Joe G said…
oleg:
That's not how things work in set theory. Two sets are equal if and only if they have precisely the same elements. The two sets above do not have the same elements. The elements of the former are books E and N. The elements of the latter are sets of books {}, {E}, {N}, and {E,N}.
We are not talking about two SETS.
We are talking about a set and its represenative power set.
Denton wrote two books. A person can give four different answers to the question "Which of those books did you read?"
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
At 12:35 PM, Joe G said…
describing sets:
Unlike a multiset, every element of a set must be unique; no two members may be identical.
At 12:36 PM, Joe G said…
How many books are in the set {E,N}?
How many books are in its power set {{}, {E}, {N}, {E, N}}?
I say the answer to both questions is 2 there are two books in the set and two books in the power set.
At 12:40 PM, Sheikh said…
Joe,
Again, from the wikipedia page on Naive Set Theory:
We also allow for an empty set, often denoted Ø and sometimes {}: a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.) Note that Ø ≠ {Ø}.
Do you have a response to the last sentence? It directly contradicts your claim that {} = {{}}.
Also, this whole discussion is getting bogged down on the issue of set equality. Please provide your definition of what it means for two sets to be equal, including a supporting reference (wikipedia will do).
At 12:41 PM, oleg said…
Joe wrote: We are not talking about two SETS.
We are. A power set is a set. By definition,
Given a set S, the power set (or powerset) of S is the set of all subsets of S, including the empty set and S itself.
So we use the standard definition of set equality to check whether the two sets (the set and its power set) are equal. They are not. They have different numbers of elements. A set of N elements will have a power set of 2^N elements. Even the sizes are different.
At 12:43 PM, oleg said…
Joe, there are no books in the power set. A book is E or N. The power set contains {}, {E}, {N}, and {E,N}. These are sets of books.
At 12:55 PM, Joe G said…
We are not talking about two SETS.
We are. A power set is a set. By definition,
Given a set S, the power set (or powerset) of S is the set of all subsets of S, including the empty set and S itself.
By DEFINITION a SET Unlike a multiset, every element of a set must be unique; no two members may be identical.
Joe, there are no books in the power set.
If "E" and "N" are in the power set then it contains those two books.
A book is E or N. The power set contains {}, {E}, {N}, and {E,N}. These are sets of books.
{E] is a set containing the book "E". {N} is the set containing the book "N". {E,N} is the set containing both books.
At 12:56 PM, Joe G said…
Shiekh,
If you have an empty set you have exactly nothing.
A power set containing and empty set and nothing else also contains nothing.
Nothing = nothing.
At 12:58 PM, Joe G said…
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
At 1:02 PM, oleg said…
Joe, here are two simple questions for you.
1. What are the elements of set {E,N}?
2. What are the elements of set {{}, {E}, {N}, {E,N}}?
Thank you in advance.
At 3:47 PM, Joe G said…
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
At 3:51 PM, Joe G said…
An empty set is {} it contains nothing.
The power set of an empty set is {{}}.
And although it has one element, tha element is an empty set, which is nothing.
Simple math demonstrates 1 x 0 = 0
and 0 + 0 = 0
At 3:53 PM, Joe G said…
The Animal Kingdom is a set.
And just because we can break it down into subsets phyla, classes, orders, etc., doesn't mean the former contains less animals than the latter.
The set "Animal Kingdom" contains the same number of animals as its power set.
At 3:56 PM, Sheikh said…
Joe,
I provided a quote from wikipedia saying:
Note that Ø ≠ {Ø}.
Here's another:
For example, the Powerset of the empty set is the set containing the empty set, {Ø}. Note that this set is different from the empty set: {Ø} has a member (Ø); Ø has none.
(from http://en.wikibooks.org/wiki/Set_Theory/Naive_Set_Theory)
In view of your prediction above:
I also predict that you will never support anything you say that you think refutes what I said.
you should be able to find a reference which supports your contention that Ø = {Ø}. Had any luck finding one yet?
Regarding your assertion that "A power set containing and empty set and nothing else also contains nothing": Where did you read that {} and {{}} are "nothing"? If you want to continue to assert this, prove it. The following two pages would be helpful:
http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
At 4:04 PM, Sheikh said…
Joe,
Simple math demonstrates 1 x 0 = 0
and 0 + 0 = 0
At least we can agree on something!
The link I just gave to the wikipedia page on ZermeloFraenkel set theory shows a construction of the empty set. It also includes an axiom which asserts the existence of a power set of any set. Finally, the first axiom on the page defines set equality.
That should give you a good start on your proof that {} = {{}}.
At 4:08 PM, Joe G said…
Regarding your assertion that "A power set containing and empty set and nothing else also contains nothing": Where did you read that {} and {{}} are "nothing"?
What the fuck do you think an empty set is?
An empty set = nothing.
At 4:09 PM, Joe G said…
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
Agree or disagree?
At 4:10 PM, Joe G said…
The Animal Kingdom is a set.
And just because we can break it down into subsets phyla, classes, orders, etc., doesn't mean the former contains less animals than the latter.
The set "Animal Kingdom" contains the same number of animals as its power set.
At 4:15 PM, Sheikh said…
Joe,
What the fuck do you think an empty set is?
An empty set = nothing.
Is this on the ZermeloFraenkel wikipedia page somewhere? I searched for the word "nothing" there and found 2 instances, neither of which appear in this context.
From the wikipedia page on the empty set:
The empty set is not the same thing as nothing; rather, it is a set with nothing inside it and a set is always something. This issue can be overcome by viewing a set as a bag – an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."
At 4:17 PM, oleg said…
Joe,
Why don't you answer a simple question or two?
At 4:20 PM, Joe G said…
oleg,
Why don't you answer my questions?
At 4:23 PM, Joe G said…
From the wikipedia page on the empty set:
Philosophy?
Give me a break...
At 4:23 PM, Joe G said…
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
Agree or disagree?
At 4:24 PM, Joe G said…
The Animal Kingdom is a set.
And just because we can break it down into subsets phyla, classes, orders, etc., doesn't mean the former contains less animals than the latter.
The set "Animal Kingdom" contains the same number of animals as its power set.
Agree or disagree?
At 4:26 PM, Joe G said…
Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."
"the set of all triangles with four sides = 0
the set of all numbers that are bigger than nine but smaller than eight, = none
the set of all opening moves in chess that involve a king. = none
At 4:27 PM, Joe G said…
none:
1 : not any
2 : not one : nobody
3 : not any such thing or person
4 : no part : nothing
At 4:45 PM, oleg said…
I will answer questions that are relevant to the point we are discussing. Your later questions are irrelevant to it.
Here, here and here you said that a set is equal its power set. I am saying that this is against the standard notion of set equality. For two finite sets it is trivial to check whether they are equal: all one needs to do is compare their elements. And since you are unwilling to list the elements of the two sets, I will do that for you.
So here are the elements:
1. E, N.
2. {}, {E}, {N}, {E,N}.
They are not the same. Therefore the two sets are not equal.
Joe, this is basic set theory. The notions are abstract, but they are not complicated.
At 4:49 PM, Sheikh said…
Joe,
Philosophy?
Give me a break...
If you have any references that assert that the empty set is "nothing", please post away.
The Animal Kingdom is a set.
Agree
And just because we can break it down into subsets phyla, classes, orders, etc., doesn't mean the former contains less animals than the latter.
There are several ways to interpret this I think. I take it what you mean here is that if you take the union of all organisms in all taxa in the animal kingdom, you don't get more animals than were in the entire animal kingdom to begin with. I would agree if that's what you mean. This operation is taking the union of all subsets of a set, however, which is different than taking the power set of a set.
The set "Animal Kingdom" contains the same number of animals as its power set.
Disagree. Suppose there were only two animals, named Adam and Eve, let's say. The power set of this animal kingdom would be { {}, {Adam}, {Eve}, {Adam, Eve} }, which has 4 elements.
At 4:54 PM, Sheikh said…
"the set of all triangles with four sides = 0
the set of all numbers that are bigger than nine but smaller than eight, = none
the set of all opening moves in chess that involve a king. = none
Yes, the empty set has zero elements. However, "The empty set is not the same thing as nothing".
Have you found a proof that {} = {{}} yet, btw?
At 8:30 PM, Joe G said…
"The empty set is not the same thing as nothing".
My claims are all about the CONTENTS an empty set contains nothing.
At 8:33 PM, Joe G said…
oleg:
I will answer questions that are relevant to the point we are discussing. Your later questions are irrelevant to it.
All my questions are very relevant to my claims.
Here, here and here you said that a set is equal its power set.
The contents are the same.
I have been very clear on that point.
That you refuse to even understand what I am saying proves that you are on some evotard agenda.
At 8:37 PM, Joe G said…
The set "Animal Kingdom" contains the same number of animals as its power set.
Disagree. Suppose there were only two animals, named Adam and Eve, let's say. The power set of this animal kingdom would be { {}, {Adam}, {Eve}, {Adam, Eve} }, which has 4 elements.
I said ANIMALS not elements.
There are only TWO ANIMALS Adam and Eve in your power set.
But thank you for proving you are just an obtuse jerk.
If you have any references that assert that the empty set is "nothing", please post away.
An empty set CONTAINS nothing.
At 8:41 PM, Joe G said…
A power set is a representation of a set.
As I said it is a set broken down into all of its subsets.
A set of two books {E,N} does not magically turn into four books (two copies of each) just by breaking it down into its represenative power set {{},{E},{N},{E,N}}.
If oleg is holding two books in his hand and asks a student "which of these books have you read?", even though the student can answer one of 4 different ways, oleg is still only holding two books.
Agree or disagree?
At 8:48 PM, Joe G said…
The empty set {} has 0 elements,
Nothing.
so its power set (which happens to be {{}}) has 2^0 = 1 element.
The one element that happens to be more of nothing.
operations on an empty set:
Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the identity element for addition, and one is the identity element for multiplication.
At 8:58 PM, oleg said…
Now I'm confused, Joe. You seemed to argue that {x,y} equals {{}, {x}, {y}, {x,y}} at one point. Now you don't?
At 9:01 PM, Sheikh said…
I said ANIMALS not elements.
There are only TWO ANIMALS Adam and Eve in your power set.
But thank you for proving you are just an obtuse jerk.
Ok, I did read that part too quickly. The elements of the power set are not animals. They are sets of animals. I still disagree with your statement. These things are true, however:
Adam ∈ {Adam, Eve}
Eve ∈ {Adam, Eve}
Adam ∉ { {}, {Adam}, {Eve}, {Adam, Eve} }
Eve ∉ { {}, {Adam}, {Eve}, {Adam, Eve} }
This example animal kingdom contains two animals as elements. The power set contains no animals as elements.
An empty set CONTAINS nothing.
That's true (if by that you mean it has no members). The same is not true of {{}}, since it has a member, {}.
Do you now agree that {} ≠ {{}}? Remember, we are talking about set theory, not "common sense" notions of what sets "contain".
At 9:04 PM, Sheikh said…
Operations performed on the empty set (as a set of things to be operated upon) are unusual. For example, the sum of the elements of the empty set is zero, but the product of the elements of the empty set is one (see empty product). Ultimately, the results of these operations say more about the operation in question than about the empty set. For instance, zero is the identity element for addition, and one is the identity element for multiplication.
I've noticed you posted this a couple of times already. What's the relevance to this discussion?
At 9:42 PM, Joe G said…
Now I'm confused, Joe.
No, you're just an asshole.
Ya see you even linked to me saying it is about contents and now you are pleading ignorance.
You seemed to argue that {x,y} equals {{}, {x}, {y}, {x,y}} at one point.
I said the CONTENTS are the same.
I also said that the power set is a representation of the set.
At 9:48 PM, Joe G said…
Ok, I did read that part too quickly. The elements of the power set are not animals. They are sets of animals. I still disagree with your statement. These things are true, however:
Adam ∈ {Adam, Eve}
Eve ∈ {Adam, Eve}
Adam ∉ { {}, {Adam}, {Eve}, {Adam, Eve} }
Eve ∉ { {}, {Adam}, {Eve}, {Adam, Eve} }
This example animal kingdom contains two animals as elements. The power set contains no animals as elements.
The power set contains two animals.
An empty set CONTAINS nothing.
That's true (if by that you mean it has no members).
It contains nothing.
The same is not true of {{}}, since it has a member, {}.
More nothing.
I keep explaining it to you and you just keep ignoring what I say.
Adding nothing to nothing gives you what?
At 10:14 PM, Sheikh said…
Joe,
The power set contains two animals.
You're going to have to explain what you mean by "contains" here then. You don't appear to be using it in the standard way. No elements or subsets of the power set in the Adam/Eve example are animals.
Of course the elements of the power set do have animals as members/elements. That's a separate question though.
Adding nothing to nothing gives you what?
What do you mean by "adding"? Are you adding numbers? Adding sets? Write it out in mathematical symbols and I'll try to carry out the operation.
At 10:54 PM, Joe G said…
You're going to have to explain what you mean by "contains" here then.
I already have.
Both Adam and Eve are contained in sets sets are a collection Set theory is the branch of mathematics that studies sets, which are collections of objects.
The only set that doesn't contain something is the empty set.
You don't appear to be using it in the standard way.
Obviously I am sets consist of and contain objects, with the exception of the empty set which contains nothing.
Of course the elements of the power set do have animals as members/elements. That's a separate question though.
Nope that is the question  how many animals are in the power set of the set {Adam, Eve}?
Adding nothing to nothing gives you what?
What do you mean by "adding"?
The nothing from one empty set added to the nothing of it's empty subset  0 + 0 = 0
Are you adding numbers? Adding sets?
The CONTENTS My argument is all about the CONTENTS.
Sets contain something a set that doesn't contain anything is called an empty set.
At 10:59 PM, Joe G said…
x = $10 bill
y = $20 bill
oleg has the set {x,y}ie, oleg has $30.
oleg figures out the different ways to configure his funds:
{ {}, {x}, {y}, {x, y} }
How much money does oleg have now?
Look below for my answer


















$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
What part of that don't you understand?
At 11:05 PM, Sheikh said…
Joe,
Why not just use the standard definitions and notation? That way there will be no ambiguity. Let's see if we're on the same page. Which of these statements are true and which are false?
1. Adam ⊂ {Adam, Eve}
2. Adam ∈ {Adam, Eve}
3. Adam ⊂ { {}, {Adam}, {Eve}, {Adam, Eve} }
4. Adam ∈ { {}, {Adam}, {Eve}, {Adam, Eve} }
5. {} ⊂ {{}}
6. {} = {{}}
At 11:11 PM, Sheikh said…
Joe,
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
This is probably for olegt, but let me ask you, is it true in your example that (x, y) ∈ {x, y}?
At 11:12 PM, Joe G said…
Why not just use the standard definitions and notation?
I am using standard definitions and notations.
Do you have any specifics in mind?
Sets are collections of objects.
That means sets contain things they have contents (except the empty set which doesn't have any contents)
Let's see if we're on the same page.
We aren't on the same page.
You refuse to understand what I am saying even though I have made it perfectly clear.
You think if you act like an asshole long enough that you may "win" something.
And I find that sort of behaviour to be common amongst evotards.
At 11:22 PM, Sheikh said…
Ok, by contents, it seems you mean elements or members. The reason I am being so particular about this point is that you have a tendency to equivocate between the contents of a set and the contents of the contents. That leads very quickly to erroneous results.
So which of the statements I wrote out are true and which are false?
At 11:23 PM, Joe G said…
This is probably for olegt,
No, it's not just for oleg.
but let me ask you, is it true in your example that (x, y) ∈ {x, y}?
You are asking me if (x,y) belongs to/ are members of {x,y}?
And I did find this on power sets:
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 23 = 8. This relationship is one of the reasons for the terminology power set.
IOW oleg is exposed as an asshole with an agenda for continuing to deny the difference and insist a power set is a set.
At 11:26 PM, Joe G said…
Ok, by contents, it seems you mean elements or members.
Whatever Zachriel.
The reason I am being so particular about this point is that you have a tendency to equivocate between the contents of a set and the contents of the contents.
You have no idea what you are talking about.
So which of the statements I wrote out are true and which are false?
I am not playing your games.
You were caught misrepresenting me so fuck off...
At 11:32 PM, Sheikh said…
You are asking me if (x,y) belongs to/ are members of {x,y}?
Yes, specifically, is (x, y) a member of {x, y}?
IOW oleg is exposed as an asshole with an agenda for continuing to deny the difference and insist a power set is a set.
Eh? Every power set is a set. Not all sets are power sets of some other set though. For example, {x, y, z} isn't the power set of any set because its cardinality isn't a power of 2.
At 7:40 AM, oleg said…
So, Joe, do you agree that {x,y} does not equal { {}, {x}, {y}, {x, y} }?
At 10:30 AM, Joe G said…
Shiekh:
Every power set is a set.
No, every POWER set is a POWER set and must be called such.
At 10:32 AM, Joe G said…
oleg:
So, Joe, do you agree that {x,y} does not equal { {}, {x}, {y}, {x, y} }?
The CONTENTS are the same, ie the CONTENTS are EQUAL.
The POWER set is a representation of the set.
The CONTENTS do not change when going from a set to a power set.
At 10:34 AM, Joe G said…
power set:
In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.
At 10:59 AM, Sheikh said…
Joe,
No, every POWER set is a POWER set and must be called such.
There's no requirement to always call a power set a power set rather than just a set. Like you said, a set is just a collection of objects, and a power set fits this description.
In the same way, I could call my 4x4 vehicle either a truck or a Ford truckeither term is correct. Of course where I live people will look at you funny if you call it anything other than a "rig".
But that's a minor issue. What about my previous question: Is (x, y) a member of {x, y}?
At 11:17 AM, oleg said…
Joe,
You have to define what the term contents mean with respect to sets. If by contents you mean the set elements then you're wrong. If you don't then you should define the term.
At 1:03 PM, Joe G said…
oleg:
You have to define what the term contents mean with respect to sets.
The contents are what they contain.
Sets are a collection of objects ie they contain those collected objects.
I have been over this already.
At 1:04 PM, Joe G said…
No, every POWER set is a POWER set and must be called such.
Zachriel:
There's no requirement to always call a power set a power set rather than just a set.
Yes there is and I pointed it out.
At 1:09 PM, Joe G said…
Set theory is the branch of mathematics that studies sets, which are collections of objects.
IOW sets CONTAIN objects.
Those objects can be animals, books, money, etc.
And the object(s) contained in a set is (are) the same as the objects contained in its power set.
agree or disagree?
At 1:40 PM, Joe G said…
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
At 1:44 PM, Sheikh said…
Zachriel:
There's no requirement to always call a power set a power set rather than just a set.
Yes there is and I pointed it out.
What you posted states clearly that 2^S is a set:
In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself.
Why don't we deal with a specific example. Is this statement true or false:
{ {}, {x}, {y}, {x, y} } is a set.
While you're at it, tell me whether this statement is true or false as well:
{} ∈ {}.
I'm not Zachriel, btw.
At 2:59 PM, Joe G said…
Zachriel:
What you posted states clearly that 2^S is a set:
In mathematics, given a set S, the power set (or powerset) of S, written , P(S), ℘(S) or 2S, is the set of all subsets of S, including the empty set and S itself.
No it says it is a set of all subsets, including the empty set and S (the set) itself, ie a POWER set.
Then we have this part which you ignored as if your ignorance means something:
power sets:
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 23 = 8. This relationship is one of the reasons for the terminology power set. (bold added)
What part of taht don't you understand?
Why don't we deal with a specific example. Is this statement true or false:
{ {}, {x}, {y}, {x, y} } is a set.
No, it's a POWER set. It is the POWER set of the SET {x,y}.
That you refuse to understand that just exposes your agenda of dishonesty and willfull ignorance.
Now why don't you deal with all my posts that you have been ignoring?
They explain everything so it would be in your best interest to read them.
At 3:01 PM, Joe G said…
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
Set theory is the branch of mathematics that studies sets, which are collections of objects.
IOW sets CONTAIN objects.
Those objects can be animals, books, money, etc.
And the object(s) contained in a set is (are) the same as the objects contained in its power set. (references already provided)
agree or disagree?
At 3:06 PM, oleg said…
Sheikh is right: { {}, {x}, {y}, {x, y} } is a set.
Joe, read the definition of a power set. 2^S is a set whose elements are all possible subsets of S. So it is a set. Not every set is a power set, but every power set is a set.
At 3:36 PM, Joe G said…
oleg,
Stop ignoring all the relevant qualifying text:
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 23 = 8. This relationship is one of the reasons for the terminology power set.
A power set is a set in the same way a multiset is a set.
Ya see sets have unique members power sets and multisets are not so constrained.
A set is a collection of distinct objects,
At 3:37 PM, Joe G said…
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
Set theory is the branch of mathematics that studies sets, which are collections of objects.
IOW sets CONTAIN objects.
Those objects can be animals, books, money, etc.
And the object(s) contained in a set is (are) the same as the objects contained in its power set. (references already provided)
agree or disagree?
At 3:44 PM, Sheikh said…
No, it's a POWER set. It is the POWER set of the SET {x,y}.
By "No", I take it you are saying that the statement "{ {}, {x}, {y}, {x, y} } is a set" is false. Thanks for answering.
Suppose the subject of power sets had never come up here, though, and that we were just talking about set theory in general. How would you have known that { {}, {x}, {y}, {x, y} } is not a set? You said:
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
How does { {}, {x}, {y}, {x, y} } fail to satisfy the definition of a set which you provided? It's a collection of other sets. Looks like a set to me.
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
Set theory is the branch of mathematics that studies sets, which are collections of objects.
IOW sets CONTAIN objects.
Those objects can be animals, books, money, etc.
And the object(s) contained in a set is (are) the same as the objects contained in its power set. (references already provided)
If by "object(s) contained in a set" you mean the set's members or elements, I disagree. If you mean something different by this phrase, then you'll have to define it.
This quote from wikipedia explains the issue:
Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
http://en.wikipedia.org/wiki/Element_%28mathematics%29
At 4:32 PM, Joe G said…
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
How does { {}, {x}, {y}, {x, y} } fail to satisfy the definition of a set which you provided?
You have two of the same thing two x's and two y's.
The owrd "distinct" in the definition of "set" doesn't allow for duplications.
A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on.
Set theory is the branch of mathematics that studies sets, which are collections of objects.
IOW sets CONTAIN objects.
Those objects can be animals, books, money, etc.
And the object(s) contained in a set is (are) the same as the objects contained in its power set. (references already provided)
If by "object(s) contained in a set" you mean the set's members or elements, I disagree.
Really? How do the contents change?
How are the contents any different {x,y} to { {}, {x}, {y}, {x, y} }?
This quote from wikipedia explains the issue:
It supports what I have been saying.
At 4:37 PM, Joe G said…
A power set is a representation of a set.
Whenever a good mathematician sees a set the power set is also observed.
The contents of the set do not change just because it is expanded into its power set.
The contents are just rearranged into all possible subsets.
If oleg is holding two books and asks a student which of the two books she/ he has read, even though the student can answer one of 4 different ways oleg is still only holding two books.
At 4:40 PM, Joe G said…
Shiekh/ Zachriel must think a power set is some sort of way to get around the law of conservation of energy.
The contents of a set magically and mysteriously change, double!
Either that or Shiekh/ Zacherial is a total buffoon.
You decide...
At 4:50 PM, Sheikh said…
You have two of the same thing two x's and two y's.
The elements of { {}, {x}, {y}, {x, y} } are {}, {x}, {y}, and {x, y}. All four are distinct.
How are the contents any different {x,y} to { {}, {x}, {y}, {x, y} }?
That's impossible for me to answeryou haven't defined "contents".
A power set is a representation of a set.
What does that mean? The word "representation" has various meanings in math (e.g., group representations). In what way is a power set a representation of a set?
Whenever a good mathematician sees a set the power set is also observed.
Huh?
At 4:53 PM, Sheikh said…
The contents of a set magically and mysteriously change, double!
Either that or Shiekh/ Zacherial is a total buffoon.
You decide...
...or the power set of a set is different from the original set, so there is no reason to expect them to have the same number of elements.
At 4:57 PM, Joe G said…
Shiekh/ Zachriel:
The elements of { {}, {x}, {y}, {x, y} } are {}, {x}, {y}, and {x, y}. All four are distinct.
The xs are the same xs and the ys are the same ys.
This power set { {}, {x}, {y}, {x, y} } would not exist without the set {x,y}.
And you cannot provide one valid reference that says otherwise.
And if { {}, {x}, {y}, {x, y} } was a set it should be able to exist independendtly of {x,y}.
How are the contents any different {x,y} to { {}, {x}, {y}, {x, y} }?
That's impossible for me to answeryou haven't defined "contents".
Yes I have. The contents are what is inside of the {} ie the OBJECTS that make up the set.
A power set is a representation of a set.
What does that mean?
Exactly what I said a power set is a set broken down into all of its subsets.
Whenever a good mathematician sees a set the power set is also observed.
Huh?
Excatly the response I would expect from a mathematically illitertate evotard.
At 4:58 PM, Joe G said…
The contents of the set do not change just because it is expanded into its power set.
The contents are just rearranged into all possible subsets.
If oleg is holding two books and asks a student which of the two books she/ he has read, even though the student can answer one of 4 different ways oleg is still only holding two books.
Don't you assholes get sick of selectively quotemining my posts?
At 5:01 PM, Joe G said…
The contents of a set magically and mysteriously change, double!
Either that or Shiekh/ Zacherial is a total buffoon.
You decide...
...or the power set of a set is different from the original set, so there is no reason to expect them to have the same number of elements.
The elements, ie the CONTENTS of the sets, are duplicates.
That is why oleg is only holding TWO books even though there are 4 possible answers.
Are you that much of a fuckhead you refuse to understand that?
At 5:07 PM, Sheikh said…
Joe,
I know you post over at UD and some other places where ID is discussed. I also now there are a number of mathematicians and others who have strong mathematical backgrounds involved in ID. Have you thought about running some of this stuff by any of those folks just to be sure you haven't made a mistake? William Dembski, Granville Sewell, hell, even Sal Cordova probably could supply some helpful advice.
At 5:13 PM, Sheikh said…
That is why oleg is only holding TWO books even though there are 4 possible answers.
Nobody has disputed that. No one has said that additional books magically appear. Like you said, however, there are 4 possible answers. The power set is the set of answers (not books) in your interpretation.
At 5:34 PM, Sheikh said…
The elements, ie the CONTENTS of the sets, are duplicates.
At least you're using the standard term here, which I'll count as progress. Your statement is incorrect, unfortunately. The four elements {}, {x}, {y}, {x, y} of the power set are all distinct. If you believe otherwise, prove that two of these sets are equal using axiom 1 on the ZermeloFraenkel wikipedia page.
At 5:51 PM, Joe G said…
That is why oleg is only holding TWO books even though there are 4 possible answers.
Nobody has disputed that. No one has said that additional books magically appear.
MY CLAIM is about the number of books.
Like you said, however, there are 4 possible answers.
So what? I am talking about the books and I have made that very clear.
The power set is the set of answers (not books) in your interpretation.
Nope I was always talking about the books, ie the contents.
At 5:52 PM, Joe G said…
This power set { {}, {x}, {y}, {x, y} } would not exist without the set {x,y}.
And if { {}, {x}, {y}, {x, y} } was a set it should be able to exist independendtly of {x,y}.
And you cannot provide one valid reference that says otherwise.
At 5:54 PM, Joe G said…
Sheikh/ Zachriel,
What makes you think that I haven't run this by math experts?
And why is it you cannot produce a valid reference supporting your claim?
At 6:00 PM, Sheikh said…
This power set { {}, {x}, {y}, {x, y} } would not exist without the set {x,y}.
Why?
And you cannot provide one valid reference that says otherwise.
Speaking of valid references, have you found one supporting your claim that {} = {{}}? I've already given two that state otherwise.
At 6:10 PM, Joe G said…
This power set { {}, {x}, {y}, {x, y} } would not exist without the set {x,y}.
Zachriel:
Why?
Because power sets are derived from sets duh.
And you cannot provide one valid reference that says otherwise.
Speaking of valid references,
As I said.
have you found one supporting your claim that {} = {{}}?
Yes 0 + 0 = 0
Ya see as I said this {{}} is nothing more than two of nothing and this {} is nothing.
It's all about the contents...
At 6:11 PM, Joe G said…
And if { {}, {x}, {y}, {x, y} } was a set it should be able to exist independendtly of {x,y}.
And you cannot provide one valid reference that says otherwise.
At 6:12 PM, Sheikh said…
What makes you think that I haven't run this by math experts?
Well, I guess in a sense you have, in this thread. Not speaking of myself, of course, but of olegt. Do you know what a projective Hilbert space is, Joe? olegt certainly does, and you can't begin to understand those without knowing basic set theory.
At 6:24 PM, Sheikh said…
Because power sets are derived from sets duh.
But what if no one had ever formulated the idea of a power set. I could still write down S = { {}, {x}, {y}, {x, y} } and it would be a welldefined set, despite your objections about duplications. No reference to {x, y} is necessary.
Yes 0 + 0 = 0
Ya see as I said this {{}} is nothing more than two of nothing and this {} is nothing.
It's all about the contents...
Surely this important result must be published somewhere. Do you have an independent source?
And of course if {} = {{}} were true, it would throw a wrench into the wellknown construction of the natural numbers using this correspondence:
0 = { }
1 = {0} = {{ }}
2 = {0,1} = {0, {0}} = {{ }, {{ }}}
3 = {0,1,2} = {0, {0}, {0, {0}}} = {{ }, {{ }}, {{ }, {{ }}}}
etc.
This probably came up on Cornelius' blog as well. Why has no one discovered this error until now?
At 6:26 PM, Joe G said…
Not speaking of myself, of course, but of olegt.
And yet oleg has been shown to be wrong.
Go figure...
(oleg is not a trusted source.)
OTOH I have corresponded with trusted sources and you are being an obtuse jerk.
At 6:30 PM, Joe G said…
But what if no one had ever formulated the idea of a power set. I could still write down S = { {}, {x}, {y}, {x, y} } and it would be a welldefined set,
Unfortunately for you you cannot produce a valid resource to support your claim.
Surely this important result should be published somewhere?
Or are you just full of shit?
Put up or shut up...
At 6:32 PM, Joe G said…
According to you, oleg and Zachriel and most likely many more evotards this {{{},{}},{},{{},{{},{},{}}}} is a valid representations of sets.
Yet they are all empty and the best one can do with an empty set is get the power set of {{}}.
IOW not one of you has even the basic understanding of set theory and you think you can come here and lecture me.
At 6:44 PM, Sheikh said…
Unfortunately for you you cannot produce a valid resource to support your claim.
Surely this important result should be published somewhere?
"A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on."
{ {}, {x}, {y}, {x, y} } fits this description, so...
At 6:47 PM, Joe G said…
Unfortunately for you you cannot produce a valid resource to support your claim.
Surely this important result should be published somewhere?
"A set is a collection of distinct objects, considered as an object in its own right.
The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on."
{ {}, {x}, {y}, {x, y} } fits this description, so...
That is only YOUR OPINION and your opinion is meaningless.
You need a valid reference that says this { {}, {x}, {y}, {x, y} } is a valid set.
Without that you are just full of shit.
At 6:52 PM, Joe G said…
x is an element in { {}, {x}, {y}, {x, y} } and x is dupicated.
y is an element in { {}, {x}, {y}, {x, y} } and y is duplicated.
Anything else I can help you with?
At 6:53 PM, Sheikh said…
OTOH I have corresponded with trusted sources and you are being an obtuse jerk.
How about posting a link to this thread on one of the evolution/ID forums you frequent, and let's see what sort of reaction you get from ID proponents. It's a great opportunity to expose me (not Zachrieldo you have access to my IP so that you can verify I am not him?) as a fraud, etc.
Yet they are all empty and the best one can do with an empty set is get the power set of {{}}
Hm. What's the power set of {{}}? Can you write that out?
Again, why hasn't this very basic error ever been pointed out in the construction of the natural numbers which I mentioned above? Countless math majors and other students have seen this but have failed to notice the problem with step 1? How can it be?
At 6:57 PM, Sheikh said…
x is an element in { {}, {x}, {y}, {x, y} } and x is dupicated.
y is an element in { {}, {x}, {y}, {x, y} } and y is duplicated.
Remember this?
Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
At 6:59 PM, Joe G said…
What's the power set of {{}}?
{{}} is the power set thanks for exposing your ignorance I even told you it was the power set.
There isn't any such thing as a power set of a power set.
Again thanks for proving you don't know what you are talking about as if I needed more evidence...
At 7:00 PM, Joe G said…
It's a great opportunity to expose me (not Zachrieldo you have access to my IP so that you can verify I am not him?) as a fraud, etc.
You are an anonymous piece of shit.
It is a given that you are a fraud...
At 7:02 PM, Sheikh said…
There isn't any such thing as a power set of a power set.
http://mathforum.org/library/drmath/view/66741.html
At 7:03 PM, Joe G said…
x is an element in { {}, {x}, {y}, {x, y} } and x is dupicated.
y is an element in { {}, {x}, {y}, {x, y} } and y is duplicated.
Remember this?
Sets can themselves be elements.
Yes I do and the sets that are elements also contain elements.
Did you know that?
At 7:04 PM, Joe G said…
x is an element in { {}, {x}, {y}, {x, y} } and x is dupicated.
y is an element in { {}, {x}, {y}, {x, y} } and y is duplicated.
Remember this?
Yes and it doesn't help you.
If it had {1,2,{1,2}} you would be on to something that supports your stupidity.
At 7:07 PM, Sheikh said…
You might have overlooked this part:
For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
So 3 is not an element of {1, 2, {3, 4}}. Neither is 4. Likewise x and y are not elements of { {}, {x}, {y}, {x, y} }.
At 7:07 PM, Joe G said…
There isn't any such thing as a power set of a power set.
http://mathforum.org/library/drmath/view/66741.html
A forum is not a valid reference.
At 7:10 PM, Joe G said…
So 3 is not an element of {1, 2, {3, 4}}. Neither is 4.
Yes they are.
They are elements of the set {3,4} making them elements of the set {1,2,{3,4}}.
Are you that fucking stupid?
At 7:21 PM, Sheikh said…
Yes they are.
They are elements of the set {3,4} making them elements of the set {1,2,{3,4}}.
Are you that fucking stupid?
Again, from http://en.wikipedia.org/wiki/Element_%28mathematics%29 :
Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
Sorry, Joe. It says there are just three elements, 1, 2, and {3, 4}. 3 and 4 are not elements of {1, 2, {3, 4}}.
C'mon, this is getting ridiculous. Everyone makes mistakes, it's no big deal. Just admit the error and move on.
At 7:43 PM, Sheikh said…
Speaking of admitting error, I'll admit one right herethe construction of the natural numbers I referred to earlier, where the first few steps go:
0 = { }
1 = {0} = {{ }}
2 = {0,1} = {0, {0}} = {{ }, {{ }}}
3 = {0,1,2} = {0, {0}, {0, {0}}} = {{ }, {{ }}, {{ }, {{ }}}}
does not actually involve power sets. My bad.
However, if you look at the right hand sides of the first three lines, {}, {{}}, { {}, {{}} }, it is true that:
P({}) = {{}}
and
P({{}}) = { {}, {{}} }
so the first two steps amount to computing the power set of the set in the previous line. The third step screws up the pattern, however.
I think you still would have a problem with this construction though, and would argue that all the sets in that list equal {}. So a good question is, why has no one discovered the problem?
At 7:52 PM, Sheikh said…
There isn't any such thing as a power set of a power set.
See problem 7 on page 9 of this lecture on set theory. Actually the entire thing looks worth reading.
www.math.nccu.edu/~melikyan/mat_math/Lect/Sec21.pdf
The author is Dr. Hayk Melikyan of North Carolina Central University.
At 8:33 PM, oleg said…
Joe wrote: A power set is a set in the same way a multiset is a set.
That is incorrect, Joe. A power set is always a set, but a multiset is not necessarily a set. Look at how these terms are defined. Here is Wikipedia:
In mathematics, given a set S, the power set (or powerset) of S is the set of all subsets of S, including the empty set and S itself.
In mathematics, a multiset (or bag) is a generalization of a set. While each member of a set has only one membership, a member of a multiset can have more than one membership (meaning that there may be multiple instances of a member in a multiset, not that a single member instance may appear simultaneously in several multisets).
From the bolded sections it is clear that a power set is a particular type of a set, but a multiset is not. A generalization means that the narrow definition of something (a set) is broadened to include features not possessed by a set. In this case, a multiset can have multiple instances of the same element, whereas a set cannot.
So, every power set is a set, but not every set is a power set. In contrast, not every multiset is a set, but every set is a multiset.
Try to parse this.
At 8:49 PM, oleg said…
Joe G wrote: You need a valid reference that says this { {}, {x}, {y}, {x, y} } is a valid set.
Joe, mathematics is not done that way. If everyone had to produce "a valid reference", how would mathematicians derive new results and define new terms? There would be no references, right?
No, mathematicians define objects, postulate their properties, and then derive results from there. You can check whether { {}, {x}, {y}, {x, y} } is a set by comparing it against the definition: A set is a collection of distinct objects, considered as an object in its own right.
Let's see. Is { {}, {x}, {y}, {x, y} } a collection of objects? A set is considered an object in its own right, so the four sets {}, {x}, {y}, and {x,y} are objects. Check. Are they distinct? Clearly, the empty set is distinct from the other three. {x} is distinct from {y} (unless x=y), and both {x} and {y} are distinct from {x,y}. Check.
So we see that { {}, {x}, {y}, {x, y} } fits the definition of a set so long as x and y are distinct. If you still disagree with us you have to point out which part of the set definition is violated.
At 9:02 PM, oleg said…
Sheikh: So 3 is not an element of {1, 2, {3, 4}}. Neither is 4.
Joe: Yes they are.
They are elements of the set {3,4} making them elements of the set {1,2,{3,4}}.
No, Joe. 3 and 4 are elements of the set {3,4} and {3,4} is an element of {1, 2, {3, 4}}, but that does not make 3 and 4 elements of the larger set. Membership is not transitive.
At 9:44 PM, Sheikh said…
One more reference to the power set of a power set
You'll find several others if you go to http://books.google.com and search for "power set of a power set".
At 9:52 PM, Joe G said…
oleg:
No, Joe. 3 and 4 are elements of the set {3,4} and {3,4} is an element of {1, 2, {3, 4}}, but that does not make 3 and 4 elements of the larger set. Membership is not transitive.
Reference please.
Your word isn't good enough here.
Ya see anything and everything contained in a set is a member.
What you are saying is that grandchildren are not part of the family tree sure they are members of their imediate family but that doesn't mean they are members of the overall family tree.
And that is just plain stupid.
At 9:59 PM, Joe G said…
You need a valid reference that says this { {}, {x}, {y}, {x, y} } is a valid set.
oleg:
Joe, mathematics is not done that way.
Then fuck off as all you have is your warped opinion of the definition.
Let's see. Is { {}, {x}, {y}, {x, y} } a collection of objects?
They have to be DISTINCT objects.
In YOUR fucking stupid example of Denton's books you had "E" being one book ("Evolution" A Theory in Crisis") and "N" being the other "Nature's Destiny...").
That set = {E,N}
The power set = {{},{E},{N},{E,N}} both "E"s refer to the same book.
Both "N"s refer to the same book.
There is a duplication.
And it is very telling that you both keep ignoring the following:
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 23 = 8. This relationship is one of the reasons for the terminology power set. (reference already provided)
What part of that don't you understand?
At 10:01 PM, Joe G said…
oleg:
A generalization means that the narrow definition of something (a set) is broadened to include features not possessed by a set. In this case, a multiset can have multiple instances of the same element, whereas a set cannot.
This { {}, {x}, {y}, {x, y} } has multiple instances of the same element.
x is an element and it is there twice.
y is an element and it is there twice.
What the fuck is wrong with you?
At 10:03 PM, Sheikh said…
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 2^3 = 8. This relationship is one of the reasons for the terminology power set. (reference already provided)
No arguments with that, Joe. Not sure how you extract from that the erroneous conclusion that power sets are not sets, though.
At 10:04 PM, Joe G said…
Sorry, Joe. It says there are just three elements, 1, 2, and {3, 4}. 3 and 4 are not elements of {1, 2, {3, 4}}.
There are three elements in the set {1,2{3,4}} but 3 and 4 are elements and they are contained in the same fucking set as 1,2 they are part of that fucking collection of objects.
They are fucking objects.
What is your fucking problem?
At 10:07 PM, Joe G said…
As an example, the power set of {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The cardinality of the original set is 3, and the cardinality of the power set is 2^3 = 8. This relationship is one of the reasons for the terminology power set. (reference already provided)
No arguments with that, Joe.
Then you admit to using incorrect terminology.
It's about time...
Thank you
Or did you not understand the meaning of that last sentence:
This relationship is one of the reasons for the terminology POWER SET. (emphasis added)
At 10:08 PM, oleg said…
Joe wrote: They have to be DISTINCT objects.
In YOUR fucking stupid example of Denton's books you had "E" being one book ("Evolution" A Theory in Crisis") and "N" being the other "Nature's Destiny...").
That set = {E,N}
The power set = {{},{E},{N},{E,N}} both "E"s refer to the same book.
Both "N"s refer to the same book.
There is a duplication.
There is no duplication, Joe. I tried to convey to you the meaning of the power set by using a simple example with Denton's books. The power set is just the right mathematical object to describe the possible answers to the question "Which books by Denton did you read?" There can be four distinct answers to that question, Joe:
(1) none,
(2) Evolution,
(3) ,
(4) both.
The four answers are distinct objects, Joe. Therefore they form a set. It is not a set of books. It is a set of sets of books. All possible sets of two books.
Think about that.
At 10:10 PM, Sheikh said…
There are three elements in the set {1,2{3,4}} but 3 and 4 are elements and they are contained in the same fucking set as 1,2 they are part of that fucking collection of objects.
They are fucking objects.
What is your fucking problem?
Erm, if 1, 2, 3, and 4 were all elements of the set {1, 2, {3, 4}}, then there would be (at least) 4 elements. The wikipedia article says that {1, 2, {3, 4}} has only three elements. That's the problem.
At 10:13 PM, Sheikh said…
A power set is still a set, Joe. If it wasn't, how could you take the power set of a power set? I trust you did find several references to the concept by following my google books link.
At 10:18 PM, Sheikh said…
Just a thought, Joe. Would any of your trusted sources be interested in posting here? That might be helpful in resolving our disagreements.
OTOH I have corresponded with trusted sources and you are being an obtuse jerk.
At 10:18 PM, oleg said…
If Joe so desires to see a "valid reference" lets go to Wikipedia, entry Element (mathematics):
Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
Are we done with this?
At 10:33 PM, Sheikh said…
You know, Joe, one thing I find odd about your position is that you've made these two claims:
1. {x, y} = { {}, {x}, {y}, {x, y} }
2. Power sets are not sets.
But by definition P({x, y}) = { {}, {x}, {y}, {x, y} }. So if we accept your claim (1) from above, that means P({x, y}) = {x, y}. All of a sudden, P({x, y}) is a set after all, contradicting claim (2). What happened?
At 10:54 PM, Joe G said…
oleg:
There is no duplication, Joe. I tried to convey to you the meaning of the power set by using a simple example with Denton's books. The power set is just the right mathematical object to describe the possible answers to the question "Which books by Denton did you read?"
But THAT has NOTHING to do with what I am saying.
I have tried to convey that to you many times but you just refuse to accept it.
The BOOKS are the same and that means the contents of the set and power set are the same.
The only difference is a power set is a set broken down into all possible subsets.
At 10:55 PM, Joe G said…
Erm, if 1, 2, 3, and 4 were all elements of the set {1, 2, {3, 4}}, then there would be (at least) 4 elements.
So by your "logic" Homo sapiens are not elements, ie members, of the Animal Kingdom.
Are you really that stupid?
At 10:57 PM, Joe G said…
You know, Joe, one thing I find odd about your position is that you've made these two claims:
1. {x, y} = { {}, {x}, {y}, {x, y} }
The CONTENTS are equal, ie the same.
2. Power sets are not sets.
They are a special case, just as is a multiset.
At 10:59 PM, Joe G said…
oleg:
If Joe so desires to see a "valid reference" lets go to Wikipedia, entry Element (mathematics):
Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.
and 4 are still elements they are elements of the set {3,4}.
And just as Homo sapiens are members, ie elements of the Animal Kingdom, {3,4} is a member of {1,2,{3,4}} which means 3 and 4 are members of that larger set.
At 11:01 PM, Joe G said…
A power set is still a set, Joe.
That is incorrect terminology.
But I understand why you would want to push that shit and refuse to understand what "contents" means.
And no my resources will not post here. They think I am stupid for continuing this discussion.
At 11:02 PM, Sheikh said…
So by your "logic" Homo sapiens are not elements, ie members, of the Animal Kingdom.
No, how did you come up with that?
At 11:06 PM, Joe G said…
So by your "logic" Homo sapiens are not elements, ie members, of the Animal Kingdom.
No, how did you come up with that?
Then 3 and 4 are elements, ie members of {1,2,{3,4}} it's that simple asshole.
Do you know how many subsets it takes to get to Homo sapiens?
You just don't know what you are talking about and you got caught again.
At 11:08 PM, Sheikh said…
1. {x, y} = { {}, {x}, {y}, {x, y} }
The CONTENTS are equal, ie the same.
You've stated that {x, y} = { {}, {x}, {y}, {x, y} }, with no hedging about "contents" several times.
2. Power sets are not sets.
They are a special case, just as is a multiset.
Special case of what? What does that even mean? You've stated that power sets are not sets unequivocally.
You can't maintain both claims without contradicting yourself.
At 11:11 PM, Sheikh said…
Do you know how many subsets it takes to get to Homo sapiens?
You just don't know what you are talking about and you got caught again.
I have a guess as to one potential incorrect argument you have in mind, but please spell it out. Note that the set of mammals is not an element of the set of animals, in case you plan to go that route.
At 6:08 AM, oleg said…
Joe, we have provided you with a reference that clearly states that 3 and 4 are not elements of {1,2,{3,4}}. We have explained why that is so. If you can't get that after all this, you can't get it evar.
Give it up. You were not made for mathematics.
At 10:05 AM, Joe G said…
Note that the set of mammals is not an element of the set of animals, in case you plan to go that route.
Then Homo sapiens are not members, ie elements of the set Animal Kingdom.
At 10:09 AM, Joe G said…
olrg:
Joe, we have provided you with a reference that clearly states that 3 and 4 are not elements of {1,2,{3,4}}.
3 and 4 are in the larger set they are part of the collection ie that set.
The reference says that the set {3,4} is an element and 3 and 4 are elements of that element, therefor they are elements of the larger set.
That is what a subset is.
subset:
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B.
I know English isn't your strong suit but there it is in English.
Choke on it asshole...
At 10:11 AM, Joe G said…
You've stated that {x, y} = { {}, {x}, {y}, {x, y} }, with no hedging about "contents" several times.
No asshole my claim has ALWAYS been about the contents.
So fuck you.
You've stated that power sets are not sets unequivocally.
And I provided a reference that says power sets must be called power sets that is if one wants to use the CORRECT TERMINOLOGY.
Now if one wants to be a deceptive asshole then one will equivocate.
At 10:49 AM, Sheikh said…
Joe,
No asshole my claim has ALWAYS been about the contents.
And you claimed on the basis of those "contents" arguments that the sets are equal. Remember this?
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
What part of that don't you understand?
It's right there in black and white.
And I provided a reference that says power sets must be called power sets that is if one wants to use the CORRECT TERMINOLOGY.
Now if one wants to be a deceptive asshole then one will equivocate.
That's not the issue I raised. Once again:
1. {x, y} is a set (we both agree on this).
2. You claim that { {}, {x}, {y}, {x, y} } is not a set, since it's a power set and there are "duplications".
3. But if {x, y} = { {}, {x}, {y}, {x, y} }, then {x, y} is also not a set. This is what is known in mathematics as a contradiction, and you have yet to resolve it.
At 10:56 AM, Joe G said…
And you claimed on the basis of those "contents" arguments that the sets are equal.
If the contents are the same then the sets would be equal.
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
What part of that don't you understand?
It's right there in black and white.
True it is all about the CONTENTS $30 = $30
And I provided a reference that says power sets must be called power sets that is if one wants to use the CORRECT TERMINOLOGY.
Now if one wants to be a deceptive asshole then one will equivocate.
That's not the issue I raised.
THAT IS THE ISSUE.
Period, end of story.
Not only that you don't understand subsets.
So fuck off and take your ignorance with you.
At 10:58 AM, Sheikh said…
Joe,
Then Homo sapiens are not members, ie elements of the set Animal Kingdom.
No, that doesn't follow from what I said. Go ahead and spell out your argument and we'll go over it. I'll even give you more informationI'll write H for the set of humans and A for the set of animals. Then these three statements are true:
1. H ⊂ A
2. H ∉ A
3. If x ∈ H, then x ∈ A.
At 11:05 AM, Sheikh said…
Joe,
And I provided a reference that says power sets must be called power sets that is if one wants to use the CORRECT TERMINOLOGY.
So according to you, the correct terminology for both {x, y} and { {}, {x}, {y}, {x, y} } is "power set", not "set". Since they're equal (they have the same "contents"), we should use the same term to describe both. Ok.
At 12:37 PM, Joe G said…
And I provided a reference that says power sets must be called power sets that is if one wants to use the CORRECT TERMINOLOGY.
So according to you, the correct terminology for both {x, y} and { {}, {x}, {y}, {x, y} } is "power set", not "set".
Nope that is only according to your twisted version of what I said.
Since they're equal (they have the same "contents"), we should use the same term to describe both.
That is false.
Just because the contents are the same doesn't mean we call them the same thing.
At 12:50 PM, Sheikh said…
That is false.
Just because the contents are the same doesn't mean we call them the same thing.
Well, if the two sets are equal (which you have said over and over), how do we know which to call a power set and which to just call a set?
Now you seem to be changing your position and saying that although they have the same "contents", {x, y} ≠ { {}, {x}, {y}, {x, y} }. Could you state your position clearly?
At 12:51 PM, Joe G said…
Then Homo sapiens are not members, ie elements of the set Animal Kingdom.
No, that doesn't follow from what I said.
Yes it does.
You said 3 and 4 and not elements of {1,2, {3,4}}
The set (subset) {3,4} would be the same thing as the set (subset) Homo sapiens.
1. H ⊂ A
2. H ∉ A
3. If x ∈ H, then x ∈ A.
Except H ∈ A
Only a complete imbecile would think that Homo sapiens are not elements of, ie members of, the Animal Kingdom.
subsets:
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B.
Nice of you to keep ignoring that...
At 12:52 PM, Joe G said…
Hey Sheikh,
Take a look...
LoL!
At 12:55 PM, Joe G said…
Just because the contents are the same doesn't mean we call them the same thing.
Well, if the two sets are equal (which you have said over and over), how do we know which to call a power set and which to just call a set?
Well I said the CONTENTS are equal you lying son of a bitch.
And the difference is easily recognizable the power set contains dupicate elements and has the set as a subset.
Now you seem to be changing your position
Nope you are just being an obtuse fuckhead.
At 12:57 PM, Joe G said…
subsets:
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B.
That means 3 and 4 are elements of {1,2,{3,4}} because {3,4} are elements of the subset which belong to the larger superset.
At 1:08 PM, Sheikh said…
Yes it does.
You said 3 and 4 and not elements of {1,2, {3,4}}
The set (subset) {3,4} would be the same thing as the set (subset) Homo sapiens.
Heh, {3, 4} is not a subset of {1, 2, {3, 4}}. Please direct Dr. Dembski to your new thread and ask him what he thinks.
At 1:12 PM, Sheikh said…
Joe,
Well I said the CONTENTS are equal you lying son of a bitch.
Earlier Joe:
$30 = $30 so {x,y} has to equal { {}, {x}, {y}, {x, y} }
At 1:15 PM, Joe G said…
Heh, {3, 4} is not a subset of {1, 2, {3, 4}}.
BWAAAAAAAHAAAAHAAAAAAAAA
No it's not a subset just a set inside of another set, ie a subset...
At 1:15 PM, Sheikh said…
That means 3 and 4 are elements of {1,2,{3,4}} because {3,4} are elements of the subset which belong to the larger superset.
I know what a subset is, Joe. And this doesn't make sense:
because {3,4} are elements of the subset which belong to the larger superset.
At 1:17 PM, Sheikh said…
BWAAAAAAAHAAAAHAAAAAAAAA
No it's not a subset just a set inside of another set, ie a subset...
Is 2 a subset of {1, 2, {3, 4}}? Why or why not?
Again, please get some of your trusted sources to look at the thread. Perhaps ID Guy might show up?
At 1:18 PM, Joe G said…
Well I said the CONTENTS are equal you lying son of a bitch.
Earlier Joe:
$30 = $30
so {x,y} has to equal { {}, {x}, {y}, {x, y} }
Just the contents are equal.
At 1:19 PM, Sheikh said…
One more suggestion, Joethe Conservapedia pages on Set theory are very incomplete. How about you help them out by writing some pages?
http://www.conservapedia.com/Set_theory
At 1:21 PM, Joe G said…
That means 3 and 4 are elements of {1,2,{3,4}} because {3,4} are elements of the subset which belong to the larger superset.
I know what a subset is, Joe.
All evidence to the contrary.
And this doesn't make sense:
because {3,4} are elements of the subset which belong to the larger superset.
That means you do not know what a subset is.
I even provided the reference and you keep ignoring it.
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. Correspondingly, set B is a superset of A since all elements of A are also elements of B.
All elements of the subset are elements of the superset.
At 1:24 PM, Joe G said…
Is 2 a subset of {1, 2, {3, 4}}?
It could be.
Any member of a set can be a subset.
Again, please get some of your trusted sources to look at the thread.
They would just shake tehir heads and laugh at you and oleg.
At 1:24 PM, Sheikh said…
Just the contents are equal.
I think you're learning, Joe, despite yourself. Do you now agreee that {x, y} ≠ { {}, {x}, {y}, {x, y} }?
At 1:26 PM, Joe G said…
Just the contents are equal.
I think you're learning,
Hey asshole that has been my position all along.
IOW it appears that YOU are finally learning, despite yourself.
At 1:32 PM, Sheikh said…
All elements of the subset are elements of the superset.
No, and this is the critical point. If you disagree, formulate a proof using the ZF axioms I linked to earlier. (And you don't want to say superset there, just set).
Is 2 a subset of {1, 2, {3, 4}}?
It could be.
"It could be"? This is a yes or no question. Which is it?
Hey asshole that has been my position all along.
Ok, let's just accept that for the moment and move on. What about the statement {} = {{}}? True or False?
At 2:12 PM, Sheikh said…
From your new post:
That set contains 3 elements 1, 2 and {3,4} (They do not understand that the element {3,4} contains elements 3 and 4 so they say that 3 and 4 are not elements of the larger set.)
Obviously 3 ∈ {3, 4} and 4 ∈ {3, 4}. No one disputes that. That doesn't mean that 3 ∈ {1, 2, {3, 4}}. If it's true, prove it. I wouldn't spend too much time on it and waste a very pleasant summer day, because you won't succeed.
At 2:23 PM, oleg said…
Joe,
{3,4} is an element of {1,2,{3,4}}, but not a subset thereof. Likewise, 1 is an element but not a subset of {1,2,{3,4}}.
{1} is a subset of {1,2,{3,4}} and also {{3,4}} is a subset of {1,2,{3,4}}.
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