Is Set Theory Misleading?
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I ask if set theory is misleading because it seems to contradict reality. Ya see with set theory the set of all non-negative integers has teh same cardinality, ie the same number of elements as the set of all positive even integers. Yet in reality the non-negative integers outnumber the positive even integers 2-to-1- forever (yes even out in infinity).
The reason is with set theory, once inside the {}, the numbers or whatever is put inside the {} is converted to e1,e2,e3,e4, and so on (e means elelment). Therefor all infinite and countable sets will ahve teh exact same thing {e1,e2,e3,e4,...}.
That is not so outside of the {}. And that tells me that set theory is misleading because it has you thinking that there are they same amount of numbers when in reality there is a difference.
And olegt chimes in with more bullshit:
Liar.
Yes and the number of non-negative integers will always be 2x the number of positive even integers- forever. No need to catch up and they cannot catch up.
I ask if set theory is misleading because it seems to contradict reality. Ya see with set theory the set of all non-negative integers has teh same cardinality, ie the same number of elements as the set of all positive even integers. Yet in reality the non-negative integers outnumber the positive even integers 2-to-1- forever (yes even out in infinity).
The reason is with set theory, once inside the {}, the numbers or whatever is put inside the {} is converted to e1,e2,e3,e4, and so on (e means elelment). Therefor all infinite and countable sets will ahve teh exact same thing {e1,e2,e3,e4,...}.
That is not so outside of the {}. And that tells me that set theory is misleading because it has you thinking that there are they same amount of numbers when in reality there is a difference.
And olegt chimes in with more bullshit:
Joe’s main difficulty with the concept of infinity is a failure to realize that infinity is a journey, not a destination.
Liar.
The even integers have no need to catch up to all integers. Both sequences keep going forever.
Yes and the number of non-negative integers will always be 2x the number of positive even integers- forever. No need to catch up and they cannot catch up.
posted by Joe G @ 8:55 AM
26 Comments:
At 9:44 AM,
Unknown said…
"I ask if set theory is misleading because it seems to contradict reality."
So? Mathematics is not limited to your experience of reality. Thankfully.
"Ya see with set theory the set of all non-negative integers has teh same cardinality, ie the same number of elements as the set of all positive even integers. Yet in reality the non-negative integers outnumber the positive even integers 2-to-1- forever (yes even out in infinity)."
Nope, the set of integers is the same size as the set of even integers.
"The reason is with set theory, once inside the {}, the numbers or whatever is put inside the {} is converted to e1,e2,e3,e4, and so on (e means elelment). Therefor all infinite and countable sets will ahve teh exact same thing {e1,e2,e3,e4,...}.
That is not so outside of the {}. And that tells me that set theory is misleading because it has you thinking that there are they same amount of numbers when in reality there is a difference."
In reality there is no difference in the size of those sets. Infinite cardinal numbers don't work the same as finite numbers. Cantor's work was quite controversial at the time.
At 10:05 AM,
Rich Hughes said…
Keep embarrassing yourself, Joe!
At 11:13 AM,
Joe G said…
Richie, you are an embarrassment to all humans.
At 11:15 AM,
Joe G said…
Nope, the set of integers is the same size as the set of even integers.
I am NOT talking about sets. I am talking about reality outside of the sets.
I am talking about the actual numbers, on a number line. There isn't any one-to-one correspondence between all non-negative integers and the positive even integers when you look at the number line.
At 11:24 AM,
Unknown said…
"I am NOT talking about sets. I am talking about reality outside of the sets."
Doesn't matter. Insdie or outside the {}s there are just as many integers are there are even integers.
{I am talking about the actual numbers, on a number line. There isn't any one-to-one correspondence between all non-negative integers and the positive even integers when you look at the number line."
Sure there is: match up the first integers with the first even, match up the second integer with the second even, etc.
At 11:31 AM,
Joe G said…
Insdie or outside the {}s there are just as many integers are there are even integers.
That's impossible
Sure there is: match up the first integers with the first even, match up the second integer with the second even, etc.
Nope, you don't get to use arbitray mapping with the number line. You have to match each number with its equal ON THE NUMBER LINE.
At 11:47 AM,
Unknown said…
" 'Insdie or outside the {}s there are just as many integers are there are even integers.'
That's impossible"
Nope, that's the cardinal numbers.
" 'Sure there is: match up the first integers with the first even, match up the second integer with the second even, etc.'
Nope, you don't get to use arbitray mapping with the number line. You have to match each number with its equal ON THE NUMBER LINE."
You can use any matching you like, there are no rules like that.
I think you're mixing up the values of the numbers with the number of numbers.
At 12:01 PM,
Joe G said…
Umm cardinal numbers refer to Set Theory and I am not talking about Set Theory.
You have to match each number with its equal ON THE NUMBER LINE.
You can use any matching you like, there are no rules like that.
Of course there are rules when comparing numbers on the number line.
I think you're mixing up the values of the numbers with the number of numbers.
Numbers mean something, Jerad. They hold a place on the number line and yes, they have a value.
As oleg said infinity is a journey. And as Einstein's train demonstrates the journey down the number line picking up all non-negative integers will pick up twice as many integers as a similar train traveling down the number line picking up only positive even integers.
The first train will always have twice the number of integers as teh second- always and forever on the journey to infinity.
At 4:31 PM,
Unknown said…
"Umm cardinal numbers refer to Set Theory and I am not talking about Set Theory."
You are if you're talking about sizes of infinite sets.
"You have to match each number with its equal ON THE NUMBER LINE."
Of course you don't!! When you're comparing two sets, item for item, you can match them up anyway you like!! And matching them up is how you see which is bigger.
"Of course there are rules when comparing numbers on the number line."
IF you're comparing their values, which I'm not doing. Just how many there are.
"Numbers mean something, Jerad. They hold a place on the number line and yes, they have a value."
Sure, if I'm using them that way. But when I'm trying to see HOW MANY NUMBERS THERE ARE in a couple of sets their values are not the point. Only how many there are.
At 5:01 PM,
Joe G said…
You are if you're talking about sizes of infinite sets.
Nope.
When you're comparing two sets, item for item, you can match them up anyway you like!!
I'm not comparing two sets.
At 5:15 PM,
Unknown said…
"I'm not comparing two sets."
You are if you're comparing the cardinality of the positive integers and the positive even integers.
Why do you think "set" means something other than "a collection"? And sets/collections have sizes. Finite ones, easy. To see which is bigger you line 'em up and see which set has more elements. Try that with infinite sets and you don't see that. Now . . . how to deal with that problem . . .
At 5:20 PM,
Joe G said…
You are if you're comparing the cardinality of the positive integers and the positive even integers.
Not comparing cardinalities. Not doing sets.
There isn't any collection. This isn't a Church.
At 1:58 AM,
Unknown said…
" 'You are if you're comparing the cardinality of the positive integers and the positive even integers.'
Not comparing cardinalities. Not doing sets.
There isn't any collection. This isn't a Church."
You brought up cardinalities. On your blog. Many times. You made fun of people who you disagreed with about cardinalities.
welcome to the party pal. - John McClane
Math is not a spectator sport.
At 7:16 AM,
Joe G said…
I am not talking about sets NOW, Jerad.
At 8:08 AM,
Unknown said…
"I am not talking about sets NOW, Jerad"
Oh, sorry, did I wander off topic from the OP which concludes with:
"Yes and the number of non-negative integers will always be 2x the number of positive even integers- forever. No need to catch up and they cannot catch up.|
At 8:31 AM,
Joe G said…
Right- no sets.
There isn't anything in Set Theory that sez everything has to go into a set before it can be considered.
At 9:08 AM,
Unknown said…
"There isn't anything in Set Theory that sez everything has to go into a set before it can be considered."
A set is just a collection of things! It doesn't change those things!
You seem really conflicted about what sets are. Set Theory is about a lot more than cardinality.
You also seem confused about cardinality. Just because cardinality doesn't care about the individual set elements' individual properties, only about how many there are, doesn't mean they lose those properties.
Of course this has been pointed out many, many times and you still refuse or are incapable of grasping it.
Oh well. Guess your brain isn't wired that way.
At 9:51 AM,
Joe G said…
"There isn't anything in Set Theory that sez everything has to go into a set before it can be considered."
A set is just a collection of things!
Non-sequitur.
It doesn't change those things!
All evidence to teh contray, of course.
You seem really conflicted about what sets are.
Nope.
Set Theory is about a lot more than cardinality.
Yup and so what?
You also seem confused about cardinality.
Nope, you do however.
At 11:57 AM,
Unknown said…
" 'A set is just a collection of things!'
Non-sequitur.
'It doesn't change those things!'
All evidence to teh contray, of course."
Or, so now you're buying Cantor's idea that, as long as they're in sets, there are just as many positive integers as there are even integers? Or are you just being sarcastic and wanting to have the last word?
" 'Set Theory is about a lot more than cardinality.'
Yup and so what?"
Well, on another thread I bothered to go look at the Wikipedia entry on Set Theory and I tried to post that material to that thread. It hadn't appeared last time I looked. Anyway, if you care you can find out. If you don't care, and you're not a mathematician so I could hardly blame you, then you really should stop telling people who do care and who have studied this stuff that they are idiots and assholes and such. Seriously.
Suppose I told you I had invented a perpetual motion matchine. And you tried to tell me I was wrong. And I kept telling you that you were stupid and wrong and twisting what you said in weird ways. You'd think I was a complete nutter.
Well, you're not a mathematician. You've only taken a few math courses. You haven't really studied Set Theory at any great depth. Maybe you should just accept that others might know more than you.
Just a thought.
" 'You also seem confused about cardinality.'
Nope, you do however."
Well, if you're not confused then you can tell me if there is a smallest infinite cardinal number. And tell me what it is or why there isn't one.
And you should be able to tell me how the cardinality of the primes compares to the cardinality of {4, 8, 12, 16, 20, 24 . . . . }
At 7:23 AM,
Joe G said…
Hey Jerad-
One train with two counters going down the number line. One counter counting the primes and the otehr counting {4,8,12,16,...}.
Which counter would count more?
And if you told me that you invented a pertual motion machine I would want to see it, duh.
And AGAIN:
What advantage is there to saying all countable and infinite sets have the same cardinality? What does that give us?
Ya see, Jerad, if YOU can't answer that ten you ain't no mathematician and you are nothing more than a blind follower.
At 10:24 AM,
Unknown said…
"Hey Jerad-"
Hey Joe, missed you yesterday. Did you know my name is in the book of Genesis? Pretty funny considering.
"One train with two counters going down the number line. One counter counting the primes and the otehr counting {4,8,12,16,...}.
Which counter would count more?"
What do you think? You know what I think. I think that the set of all primes has the same cardinality as the set of positive multiples of 4.
"And if you told me that you invented a pertual motion machine I would want to see it, duh."
Oh . . . yeah. I guess I would too!!
"And AGAIN:
What advantage is there to saying all countable and infinite sets have the same cardinality? What does that give us?"
Obviously it's of virtually no use in every day life. But it helps to put other areas of mathematics on a firmer ground since Set Theory can realistically be considered part of the foundation of mathematics.
"Ya see, Jerad, if YOU can't answer that ten you ain't no mathematician and you are nothing more than a blind follower."
I can't say I studied in great depth all the areas of mathematics that can count Set Theory as foundational. But I've looked at some.
Chaos theory in the guise of Mandelbrot Sets and that sort of thing. Measure theory, very important to measure theory.
On another thread I tried to post a list of math disciplines which are at least partially built on Set Theory but it never got through. So, I'll just say check out the Wikipedia entry on Set Theory and look through the descriptions of some of the 'uses'.
At 10:29 AM,
Joe G said…
I think that the set of all primes has the same cardinality as the set of positive multiples of 4.
That isn't what I asked. Try again.
What advantage is there to saying all countable and infinite sets have the same cardinality? What does that give us?"
Obviously it's of virtually no use in every day life.
I doubt it's of any use, period.
But it helps to put other areas of mathematics on a firmer ground since Set Theory can realistically be considered part of the foundation of mathematics.
LoL! I am not talking about Set Theory- just that one part of it.
I can't say I studied in great depth all the areas of mathematics that can count Set Theory as foundational.
Yes, I understand that you have problems in following along.
At 10:55 AM,
Unknown said…
" 'I think that the set of all primes has the same cardinality as the set of positive multiples of 4.'
That isn't what I asked. Try again."
You asked which would count more. Well if you go on that infinitely long journey then that's asking which set is bigger.
" 'Obviously it's of virtually no use in every day life.'
I doubt it's of any use, period."
Well, if it fails then Set Theory fails to a large extent.
" 'But it helps to put other areas of mathematics on a firmer ground since Set Theory can realistically be considered part of the foundation of mathematics.'
LoL! I am not talking about Set Theory- just that one part of it."
Like most axiomatic systems, if the basic building blocks fail then the whole ediface is in danger of collapsing. Which is why Cantor was looking at such things. There was this notion that mathematics might be a house built on the sand and might get washed away.
" 'I can't say I studied in great depth all the areas of mathematics that can count Set Theory as foundational. '
Yes, I understand that you have problems in following along."
Oh come on now. That's just being snotty and impolite. I'm trying to answer your questions and you're just rude.
At 11:22 AM,
Joe G said…
You asked which would count more. Well if you go on that infinitely long journey then that's asking which set is bigger.
True, and your formula doesn't jibe with reality. And that is why the journey is important.
Like most axiomatic systems, if the basic building blocks fail then the whole ediface is in danger of collapsing.
I doubt anything that deals with infinity is a basic building block. The basic building blocks would be laid down with finite sets.
At 1:58 AM,
Unknown said…
"True, and your formula doesn't jibe with reality. And that is why the journey is important."
So, in the infinite case do you agree with Cantor?
"I doubt anything that deals with infinity is a basic building block. The basic building blocks would be laid down with finite sets."
Doubt isn't the same thing as knowing is it? It's easy enough to find out if you wish.
At 7:10 AM,
Joe G said…
I have demonstrated that Cantor is wrong, so why would I agree with him?
And wrt infinity, no one knows...
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