Set Mathematics

While talking about sets with respect to mathematics on cannot help see the irony between the definition of a set and alleged infinite sets. The problem is even more exacerbated when thinking about the set of all real numbers.
So perhaps that is why there are issues when it comes to infinity and sets. They were never supposed to go together.
While talking about sets with respect to mathematics on cannot help see the irony between the definition of a set and alleged infinite sets. The problem is even more exacerbated when thinking about the set of all real numbers.
In mathematics, a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. The concept of a set is one of the most fundamental in mathematics.and
A set is a welldefined collection of distinct objects.No one can collect an infinite number of things. And with real numbers, if you are starting with the positive Reals, then you don't even know where it starts. There isn't a welldefined first positive Real number and there will never be a welldefined collection of them.
So perhaps that is why there are issues when it comes to infinity and sets. They were never supposed to go together.
26 Comments:
At 2:37 AM, Unknown said…
No one can collect an infinite number of things. And with real numbers, if you are starting with the positive Reals, then you don't even know where it starts. There isn't a welldefined first positive Real number and there will never be a welldefined collection of them.
Just because there isn't a smallest Real number doesn't mean you can't talk about them as a set. Give me a number and I can tell you whether or not it's Real. And I can give you a number that isn't Real.
A well defined set doesn't mean you can list them all. When a set is well defined you can say for sure whether or not something is in the set.
Some infinite sets can be matched up 1to1 with the positive integers. But even then that doesn't mean there is a smallest element of the set.
The set of Rational numbers is countably infinite (which means it has the same cardinality as the integers) so that means it is possible to put them in a list and be sure your infinite list will eventually get to any specified rational number. You can't do that with the Reals so they have a different cardinality.
Both the Rationals and the Reals are well defined, neither set has a smallest number, the Rationals are countably infinite while the Reals are not and if you give me a number I can tell you whether it is or is not in either set.
At 2:40 AM, Unknown said…
So perhaps that is why there are issues when it comes to infinity and sets. They were never supposed to go together.
There are no 'issues'. Cantor figured out how to deal with such things. And lots of mathematicians have built upon his work.
At 8:15 AM, Joe G said…
There are issues and no one uses the concept of equal cardinalities for countably infinite sets so no one has built upon it. No one can even collect the real numbers so clearly there is an issue
At 8:19 AM, Joe G said…
Reading isn't one of Jerad's skills.
Just because there isn't a smallest Real number doesn't mean you can't talk about them as a set.
You clearly have difficulties with the definition of a set.
A well defined set doesn't mean you can list them all.
Cuz you say so? Really?
When a set is well defined you can say for sure whether or not something is in the set.
If it isn't listed then how do you know? If you cannot collect it then how is it in the set? And if you never stop collecting then how are future items in the set?
Some infinite sets can be matched up 1to1 with the positive integers. But even then that doesn't mean there is a smallest element of the set.
That doesn't make any sense.
At 9:13 AM, Unknown said…
There are issues and no one uses the concept of equal cardinalities for countably infinite sets so no one has built upon it. No one can even collect the real numbers so clearly there is an issue
It is used by mathematicians. Cantor's work opened the door to a whole new area of mathematics. The Real numbers are one of the basic structures used in mathematics.
No issues.
Reading isn't one of Jerad's skills.
I'm just trying to explain the mathematics.
You clearly have difficulties with the definition of a set.
Where is it said that a set has to have a smallest element? Or a largest element?
Cuz you say so? Really?
No, I'm just telling you how the mathematics works. It's not my opinion, it's just the way it is.
When a set is well defined you can say for sure whether or not something is in the set.
If it isn't listed then how do you know? If you cannot collect it then how is it in the set? And if you never stop collecting then how are future items in the set?
It comes down to how the set is defined that enables you to determine if something is in it or not.
Some infinite sets can be matched up 1to1 with the positive integers. But even then that doesn't mean there is a smallest element of the set.
That doesn't make any sense.
That's the way it works. Cantor's work was revolutionary, one of the greatest discoveries/results of all of mathematics.
At 9:31 AM, Joe G said…
It is used by mathematicians.
For what? Reference please
Where is it said that a set has to have a smallest element?
It isn't well defined if you cannot say what the first element is.
That's the way it works.
Except it doesn't work if you don't know what the first element is. And if you disagree then we are right back to being able to match up the reals with the integers.
At 9:48 AM, Unknown said…
For what? Reference please
It's used in Set Theory. Obviously.
https://en.wikipedia.org/wiki/Cardinality_of_the_continuum
https://en.wikipedia.org/wiki/Continuum_hypothesis
It isn't well defined if you cannot say what the first element is.
That is incorrect.
From Wikipedia: In mathematics, an expression is called welldefined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not welldefined or ambiguous.[1] A function is welldefined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not welldefined (and thus: not a function).[2] The term welldefined is also used to indicate whether a logical statement is unambiguous.
And specifically in Set Theory: A set is welldefined if there is no ambiguity as to whether or not an object belongs to it, i.e., a set is defined so that we can always tell what is and what is not a member of the set. (From numerous sources)
Nothing to do with whether or not it has a smallest element.
Except it doesn't work if you don't know what the first element is. And if you disagree then we are right back to being able to match up the reals with the integers.
That is how it works. You can prove that the rationals can be matched up 1to1 with the integers even though there is not a smallest rational number. You do pick something to be 'first' but it's not the smallest.
There are lots of places you can find a scheme; search for cardinality of the rationals. This video seems a good general introduction and the part about the rationals starts partway through.
https://www.youtube.com/watch?v=IHqQomPjyI
At 2:47 PM, Joe G said…
OK so saying that all countably infinite sets have the same cardinality is not used by anyone for anything
In mathematics, an expression is called welldefined or unambiguous if its definition assigns it a unique interpretation or value.
The value is missing
A set is welldefined if there is no ambiguity as to whether or not an object belongs to it, i.e., a set is defined so that we can always tell what is and what is not a member of the set.
You don't know what the object it.
At 4:02 PM, Unknown said…
OK so saying that all countably infinite sets have the same cardinality is not used by anyone for anything
It is used by mathematicians.
In mathematics, an expression is called welldefined or unambiguous if its definition assigns it a unique interpretation or value.
The value is missing
The point is that if, for sets, you specify and value or an element I can tell you whether or not it is or is not in a well defined set.
A set is welldefined if there is no ambiguity as to whether or not an object belongs to it, i.e., a set is defined so that we can always tell what is and what is not a member of the set.
You don't know what the object it.
If you specify one then I can tell you whether or not it is or is not in a well defined set. Object is the same as element.
At 10:56 PM, Joe G said…
It is used by mathematicians.
For what?
The point is that if, for sets, you specify and value or an element I can tell you whether or not it is or is not in a well defined set.
So you'll know it when you see it. That's not well defined
At 1:44 AM, Unknown said…
For what?
For working with sets in set theory. It's a major line of mathematical research.
So you'll know it when you see it. That's not well defined
That's what mathematicians mean when they say a set is well defined. Essentially it means the definition is clear enough that it's easy to determine whether or not something is in the set.
Like any highly developed field, it has it's own specialised terms or use of terms. For example, in the mathematical field of graph theory the terms: path, root and tree have specific meanings separate from the ones used outside of mathematics.
At 9:20 AM, Joe G said…
For working with sets in set theory
That's too vague. Try again
That's what mathematicians mean when they say a set is well defined.
They will know it when they see it? Really?
Back to the OP there cannot be an infinite set as it falls outside of the definition of a set.
At 10:24 AM, Unknown said…
That's too vague. Try again
From Wikipedia: Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
And from the Large Cardinal article: In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philosophical schools (see Motivations and epistemic status below).
A large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.
Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC. These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).
There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in the list of large cardinal properties are large cardinal properties.
They will know it when they see it? Really?
That's the way the term is defined.
Back to the OP there cannot be an infinite set as it falls outside of the definition of a set.
A set is a collection of objects, where is it said it has to be finite?
Cantor figured out how to work with infinite sets. And that some infinite sets were bigger than others. Revolutionary stuff. Hard to grasp once past the basics though.
At 1:12 PM, Joe G said…
Wow, nothing in what you posted says that the claim of all countably infinite sets having the same cardinality is of any use.
A set is a collection of objects, where is it said it has to be finite?
That is the only set that can be deemed collectable, in theory.
An infinite collection is an oxymoron.
At 3:14 PM, Unknown said…
Wow, nothing in what you posted says that the claim of all countably infinite sets having the same cardinality is of any use.
I post an easily accessible reference to the fact that there is research regarding infinite cardinals and you insist that your particular bugaboo is addressed specifically.
Look up the research papers, look at the issues. Just skimming the abstracts does not give the whole story. You have to really understand the mathematical issues.
That is the only set that can be deemed collectable, in theory.
Cantor showed how to deal with infinite sets. If you can find a mistake in his work then you might have something.
An infinite collection is an oxymoron.
The mathematics community has dealt with this issue a century ago. If you wish to contest all that work then it's up to you to find fault in all that work.
Any basic introductory Calculus course introduces infinities. It's a basic part of advanced mathematics. Advanced being Freshman level.
At 3:37 PM, Joe G said…
I post an easily accessible reference to the fact that there is research regarding infinite cardinals and you insist that your particular bugaboo is addressed specifically.
And? Is it my fault that you cannot address what I am saying?
Cantor showed how to deal with infinite sets.
It is an oxymoron> there cannot be an infinite set.
The mathematics community has dealt with this issue a century ago
Bullshit
Any basic introductory Calculus course introduces infinities.
OK, but that is not the same as an infinite set/ a set that contains infinite elements.
Look, if all you can do is dance and not address the issues then clearly you are not the mathematician tat you think you are.
At 4:03 PM, Unknown said…
And? Is it my fault that you cannot address what I am saying?
Did you read through the references? Did you follow the arguments? How do you know?
It is an oxymoron> there cannot be an infinite set.
According you you. What mathematical research work have you published? The integers are clearly an infinite set, how does your approach deal with them?
Bullshit
You've been provided with references. And there clearly are well defined infinite sets.
OK, but that is not the same as an infinite set/ a set that contains infinite elements.
You never took a limit to infinity? Something done in introductory Calculus courses.
Look, if all you can do is dance and not address the issues then clearly you are not the mathematician tat you think you are.
I don't need to say anything else really. You clearly have never taken even a proper Calculus course. You argue against wellestablished and accepted mathematics and ask others to spell out all the work for you when you haven't even tried to do the work yourself?
Limits to infinity is a basic mathematical construct. Are you saying Calculus is not well founded?
At 7:24 PM, Joe G said…
Did you read through the references? Did you follow the arguments? How do you know?
Yes I did and if you could prove me wrong then you would. But you can't
It is an oxymoron> there cannot be an infinite set.
According you you.
According to logic, reasoning and most of all, reality.
The integers are clearly an infinite set, how does your approach deal with them?
Are they? Are they all collected? No
You never took a limit to infinity?
Are you fucking retarded? THAT does not have anything to do with a set of infinite elements
You argue against wellestablished and accepted mathematics and ask others to spell out all the work for you when you haven't even tried to do the work yourself?
Again, fuck you. You can't even stay focused. You think tat someone can actually collect infinite elements.
Limits to infinity is a basic mathematical construct. Are you saying Calculus is not well founded?
I never said anything about it. Clearly you are just a sore loser.
At 1:40 AM, Unknown said…
Yes I did and if you could prove me wrong then you would. But you can't
One of the quotes I gave from Wikipedia referenced ZFC . . . what is that and how does it affect the study of large cardinals?
According to logic, reasoning and most of all, reality.
We're talking about mathematics. In mathematics there are many, many, many infinite sets. And they are used even in first year Calculus courses.
Are they? Are they all collected? No
If the integers aren't infinite then there would be a largest one. What is the largest integer?
Are you fucking retarded? THAT does not have anything to do with a set of infinite elements
Of course it does. How can you take the limit as x approaches infinity if x is not an element of an infinite set?
Again, fuck you. You can't even stay focused. You think tat someone can actually collect infinite elements.
I said no such thing. But I can work with infinite sets and there is a whole area of mathematics dealing with that.
I never said anything about it. Clearly you are just a sore loser.
You can't do Calculus without infinite sets. Look af Fourier analysis, widely used in engineering.
https://en.wikipedia.org/wiki/Fourier_analysis
Scroll down the page and look at the integrals and the summations. Infinities everywhere. You have to be able to deal with those or you can't do the work.
At 10:55 AM, Joe G said…
Again you are ignoring the definition of a set. So yes, if you ignore the definition anything can be a set.
How can you take the limit as x approaches infinity if x is not an element of an infinite set?
Ignore the set because there isn't any such thing as a set with infinite elements BY DEFINITION.
You can't do Calculus without infinite sets.
Bullshit. Cantor came well after calculus was established
At 11:02 AM, Joe G said…
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
You think that someone can actually collect infinite elements.
I said no such thing.
Of course you did when you insisted there can be a set with infinite elements.
Wake up
At 11:10 AM, Unknown said…
Again you are ignoring the definition of a set. So yes, if you ignore the definition anything can be a set.
Who says 'collection' has to be finite? A set, as used in mathematics, is a collection of distinct objects. And they can be infinite. Which is where you get infinite cardinalities.
Ignore the set because there isn't any such thing as a set with infinite elements BY DEFINITION.
Show me a mathematical definition of set which restricts it to a finite concept.
So, how would you take the limit as x approaches infinity as is frequently done in calculus? Open up any calculus textbook, limits to infinity are everywhere.
Bullshit. Cantor came well after calculus was established
Yes, people did deal with infinite sets before Cantor but he made the process rigorous and consistent.
You cannot escape the fact that in mathematics sets can be infinite. Just because you think the normal, everyday definition of set means they can't doesn't cut it. We're talking specific meanings for mathematics. And if you're not familiar with the specific mathematical usage then you shouldn't argue about them.
Ask any mathematician and they will tell you that sets can be infinite. That's standard usage. If you want to pick a different word for something like all the integers then fine, go ahead. But mathematicians will still refer to them as a set.
At 11:24 AM, Joe G said…
Who says 'collection' has to be finite?
Anyone and everyone with any sense.
Show me a mathematical definition of set which restricts it to a finite concept.
Read the OP.
So, how would you take the limit as x approaches infinity as is frequently done in calculus?
It has nothing to do with sets.
Dealing with infinity is NOT the same as dealing with sets of infinite elements. You are fucking insane.
You cannot escape the fact that in mathematics sets can be infinite.
You cannot escape the fact that a set of infinite elements goes against the definition of a set.
No one can collect infinite objects. No one. That alone demonstrates that a set with infinite elements is an oxymoron.
And I don't care if mathematicians failed English. That isn't an excuse.
At 2:32 PM, Joe G said…
So, if In mathematics, a set is a collection of distinct objects, considered as an object in its own right;
And no one can collect infinite elements (ie the distinct objects of the set);
Then there cannot be a set of infinite elements
At 3:33 PM, Unknown said…
Anyone and everyone with any sense.
When mathematicians talk about a collection they mean something that could be infinite. And there are very clear in that. You object but they have been clear.
You cannot escape the fact that a set of infinite elements goes against the definition of a set.
It goes against what YOU think a set is. Mathematicians have been very, very clear about what they think a set can be. You can argue about the choice of terms but they have been very clear. That doesn't make them 'wrong', it just means you don't like the term they chose. That's a different thing.
No one can collect infinite objects. No one. That alone demonstrates that a set with infinite elements is an oxymoron.
Who said, recently:
Consider the following four sets:
A = {1,2,3,4,5…}
B = {2,4,6,8,10…}
C = {3.1, 3.2, 3.3, 3.4, 3.5…}
D = {3.2, 3.4, 3.6, 3.8, 3.10…}
Yes, that was you. You talked about infinite sets with infinite elements. If you're going back on that then you're going to have to rework quite a lot of your previous discussions.
And I don't care if mathematicians failed English. That isn't an excuse.
When mathematicians use a term they are very clear in how they are using it. You might not like the usage but you can't say they are wrong because they contradict common usage. This is frequently the case in mathematics. Consider the term integration. In mathematics that has a completely different meaning than that in common usage. But you would never say the mathematicians are 'wrong'.
And no one can collect infinite elements (ie the distinct objects of the set);
Your interpretation of the word 'collect' is not the same as the mathematicians' use of the term.
Then there cannot be a set of infinite elements
What collective noun would you use for the integers then? Go on. You don't like set so suggest an alternative. You've spent days bitching and moaning about 'set' so come up with an alternative.
At 3:41 PM, Joe G said…
When mathematicians talk about a collection they mean something that could be infinite.
So they talk about a collection that isn't a collection. Got it.
And I know what I said. I was using the current convention, duh. The point is in the OP. Did you read it?
What collective noun would you use for the integers then?
The integers; All of the integers; (All of) the positive integers whatever the context is
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