[Monte314]

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Well it should be evident that if we define 2 sets as having infinite members they are going to be the same size by definition. I can also imagine the case of two infinite sets where no bijection exists to prove it, say integers and primes.

Nope.

[shytiger]

	Originally Posted by thod To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Well it should be evident that if we define 2 sets as having infinite members they are going to be the same size by definition. I can also imagine the case of two infinite sets where no bijection exists to prove it, say integers and primes.

The one-to-one correspondence is essential to counting infinite sets (or in the case of uncountable infinite sets, measuring them). There is a one-to-one correspondence between the integers and the primes because they are both countable and both infinite.

If no one-to-one correspondence exists, then the two sets are not the same "size". Cantor's brilliant
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was one of the first proofs that the real numbers are uncountable and hence the reals are fundamentally "bigger" than the integers.

[envirodude]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	... These ideas are very important in advanced mathematics, but they are worth knowing about just as aspects of popular culture. The most important consequence mentioned in the video is the fact that mathematics (and logic) are irremediably limited: there are statements in mathematics and logic that can neither be proven nor disproven. ...

I find this discussion cool and interesting. For me, as a non-mathematician, it's like watching little animated warriors pulling out different weapons and zapping each other. I don't understand much of the dialogue, and I don't know how any of the weapons work, but I can still follow the action (sort of). (hope this doesn't sound dismissive)

Monte (et al), I can see this concept of different-sized infinities being important philosophically, but is there any practical application? If you want people less geeky than me (say 99%) to care, you need a better hook than incompleteness. It never would have crossed my mind that perhaps mathematics and logic were not irremediably limited.

[Straylight]

I think the human brain isn't well resourced to deal with the concept of infinity. Seriously. How was there ever a survival value to understanding infinity?

[setsume]

TED - fucking up your brain with twisted semantics. Well yeah we can actually blame Cantor for that, really, but I kind of doubt TED would have picked it up otherwise.

[Monte314]

	Originally Posted by envirodude To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Monte (et al), I can see this concept of different-sized infinities being important philosophically, but is there any practical application?

Many mathematical results use the fact that there are infinities of different sizes in their proofs, often in combination with some variation of the Axiom of Choice, or the Pigeon-Hole Principle. Lots of important mathematical results would disappear without them.

The question of when this loss has practical import is a question I'll leave to others.

[shytiger]

One way to think of infinity of countable sets is that, when you say "please sir, may I have another?", you always get one. Many proofs that involve infinities work this way. It's the concept that you can always find a number bigger than the one you have.

With uncountable sets things just get creepy. I am convinced nature would never allow it, and it's a folly of the deranged human mind.

"God created the integers; all else is the work of man." Kronecker.

[ummon]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Many mathematical results use the fact that there are infinities of different sizes in their proofs, often in combination with some variation of the Axiom of Choice, or the Pigeon-Hole Principle. Lots of important mathematical results would disappear without them. The question of when this loss has practical import is a question I'll leave to others.

As a physicist, I'd like to chime in that a great many pure math results have practical import.

[setsume]

	Originally Posted by Thrushbeard To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I was taught that infinity was a concept not a number. Am I right?

You were taught correctly. Infinity is a limit, "Stop! math ends here, and philosophy begins here." The concept of uncountability is useful is because there is some common sense philosophy, "You are not God!"

[Latro]

	Originally Posted by setsume To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	You were taught correctly. Infinity is a limit, "Stop! math ends here, and philosophy begins here." The concept of uncountability is useful is because there is some common sense philosophy, "You are not God!"

I disagree with this. Reasoning about infinite cardinals and infinite ordinals (and other infinite objects, like hyperreal numbers, that are in neither of these categories) is not philosophy. I think the philosophy comes in when it comes time to decide on axioms to decide statements which the more intuitive axioms do not decide--statements like the continuum hypothesis or the axiom of choice. One might argue that any establishment of axioms is philosophy and only reasoning from given axioms is mathematics, but this is not a particularly practical perspective even for pure mathematicians.

[Vermillion]

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	But, doesn't this mean that logical reasoning will *never* provide comprehensive knowledge... even about mathematics and logic themselves? Yes.

As long as math continues to provide us with tangible results I'm not to concerned about it.

[setsume]

	Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I disagree with this. Reasoning about infinite cardinals and infinite ordinals (and other infinite objects, like hyperreal numbers, that are in neither of these categories) is not philosophy. I think the philosophy comes in when it comes time to decide on axioms to decide statements which the more intuitive axioms do not decide--statements like the continuum hypothesis or the axiom of choice.

I don't think there is anything intuitive about infinite objects. We cannot construct or imagine infinite objects. we have only discursively created a concept for it, but we have not made a infinite number of judgments subsuming an infinite number of objects under the concept. We are finite beings so we cannot do this.

[Latro]

	Originally Posted by setsume To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I don't think there is anything intuitive about infinite objects. We cannot construct or imagine infinite objects. we have only discursively created a concept for it, but we have not made a infinite number of judgments subsuming an infinite number of objects under the concept. We are finite beings so we cannot do this.

I disagree again, especially with regard to countably infinite objects, because of the power of induction, which is ultimately a very intuitive concept, since it proves a statement about each *given* natural number (or member of another countable set) using only finitely many implications. Larger objects like the reals tend to require more structure to be easy to reason about, though, I concede that.

[Eye on Earth]

To make infinity more complicated, the "space" between rational and irrational infinite sets can also be equated to the distance between dimensions. Let your mind ponder that one for a while.

[Hugh]

If there are an infinite number of points within 1inch, then how can i ever touch something? Wouldn't my finger just get infinitely closer?
Sorry for the silly question, this wonder has gone unanswered since I was a kid.

[K27]

	Originally Posted by Hugh To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	If there are an infinite number of points within 1inch, then how can i ever touch something? Wouldn't my finger just get infinitely closer? Sorry for the silly question, this wonder has gone unanswered since I was a kid.

This apparent paradox exists because a guy called Zeno forgot that when you move your fingers, you are moving across infinite points already. You cannot move a finite point in this real physical space which is a continuum -- finite points takes up literally no space at all.

As you move, however short the distance is, you move across infinite point. So how is that a paradox at all?

This is why I think Zeno is making a fuss out of nothing. I'm not impressed.

[Monte314]

	Originally Posted by setsume To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I don't think there is anything intuitive about infinite objects. We cannot construct or imagine infinite objects. we have only discursively created a concept for it, but we have not made a infinite number of judgments subsuming an infinite number of objects under the concept. We are finite beings so we cannot do this.

There is a school of mathematical thought, referred to as "Intuitionism", that rejects all but the countable infinity for the reason setsume suggests. For these folks, even the Real Numbers are a fiction.

Many of these people insist on constructive proofs of results, which must produce the posited objects using finitistic processes that terminate; this is called the "Constructivist" approach. In particular, "proof by contradiction" is not regarded by them as valid, nor are proofs using "transfinite" methods such as Zorn's Lemma.

These mathematicians are in the minority, but they cannot be discounted. For example, the Brouwer Fixed Point Theorem is a famous and important theorem in analysis. It was proven by Contradiction by Luitzen Brouwer... a constructivist! Even he is able to laugh about this.

[setsume]

Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

There is a school of mathematical thought, referred to as "Intuitionism", that rejects all but the countable infinity for the reason setsume suggests. For these folks, even the Real Numbers are a fiction. Many of these people insist on constructive proofs of results, which must produce the posited objects using finitistic processes that terminate; this is called the "Constructivist" approach. In particular, "proof by contradiction" is not regarded by them as valid, nor are proofs using "transfinite" methods such as Zorn's Lemma. These mathematicians are in the minority, but they cannot be discounted. For example, the Brouwer Fixed Point Theorem is a famous and important theorem in analysis. It was proven by Contradiction by Luitzen Brouwer... a constructivist! Even he is able to laugh about this.

Yes, but I thought it was the law of excluded middle that they don't consider valid. I'm not really willing to go in that direction really, I just don't think infinity should be considered a number but at most a limit. Real numbers are very fishy too and I've been for some time skeptical about their nature and use other than as algorithms for producing approximations to a limit.

---------- Post added 08-11-2012 at 05:28 PM ----------

Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

I disagree again, especially with regard to countably infinite objects, because of the power of induction, which is ultimately a very intuitive concept, since it proves a statement about each *given* natural number (or member of another countable set) using only finitely many implications. Larger objects like the reals tend to require more structure to be easy to reason about, though, I concede that.

I will have to think more about this. Induction seem to me to be just a consequence of the concept of natural numbers. The problem for me isn't that they cannot be reasoned about but that they are just discursive concepts without an intuitive connection to reality.

[Eye on Earth]

I am more impressed by the concept that if I have an infinite number of sheep I would need an infinite number of rocks to match to the sheep.

*EoE tries to count sheep and ends up throwing rocks at them*

I see a flaw in matching theory, but I'm going to need a few more days to test my idea out. Presently, I can state there are no sheep, therefore I need no rocks. I'm thinking this puts me at a state of mathematical equilibirum. In essence, the ideas of infinity also contribute to the concept/understanding of zero.

Would anyone care to expand on this idea?

[envirodude]

I'm wondering if any of this is related to indeterminate forms in calculus (
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), like 0/0 or, more relevant here, inf/inf. I see inf/inf as asking, of these two infinities, which one is bigger? The possible answers were: inf (numerator is infinitely larger), 0 (denominator is infinitely larger), or, some finite number. Is there any correspondence between the treatment of infinity in L'Hopital and in set theory?

@EOE, just remember to tell the sheep you love them
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[Latro]

Originally Posted by envirodude To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

I'm wondering if any of this is related to indeterminate forms in calculus ( To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. ), like 0/0 or, more relevant here, inf/inf. I see inf/inf as asking, of these two infinities, which one is bigger? The possible answers were: inf (numerator is infinitely larger), 0 (denominator is infinitely larger), or, some finite number. Is there any correspondence between the treatment of infinity in L'Hopital and in set theory? @EOE, just remember to tell the sheep you love them To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

Cardinals and (extended) real numbers are really completely different animals.

[Eye on Earth]

Great, I've gone and puzzled myself stupid.

Is zero infinitely zero to the extent that it cannot ever exceed itself. Doesn't that define infinity in a simple sense?

Please help me to clean this idea up. Links and proofs are encouraged.

[envirodude]

	Originally Posted by Latro To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Cardinals and (extended) real numbers are really completely different animals.

Thanks, and nice pun. I've hereby filed set theory under "special math for mathematicians" and posted a logo on the file.

( Show Spoiler )

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[Zodd]

Infinity is sooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooo...

Actually I don't think I can describe it in one post. So I'll just post a gif that explains the feeling I sometimes have when thinking about the size of the universe and the quantum levels :

( Show Spoiler )

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It's also what I imagine what goes on in Monte's head and what I feel when I think about Monte. But seriously that is quite how I feel when I think about quantum level and the universe being infinite.

[Eye on Earth]

Hey Monte - is zero infinite or absolute? Can one exist without the other?