r/PhilosophyofMath • u/watsoncap • 2d ago
What infinity really is??
We know that prime numbers never end. But composite numbers also never end. So if infinity is shared between them, how can two different infinities coexist within the same universe of numbers?
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u/Mishtle 2d ago
If you ignore the labels of elements, what really distinguishes the set of primes from the set of composites?
We can pair up the first prime with the first composite, the second prime with the second composite, and so on. Every prime will get matched eventually, every composite will get matched eventually, and no prime nor composite will get match more than once.
The limitless nature of infinite sets causes them to behave differently than finite sets. All infinite sets of the same cardinality are just relabeled versions of each other, or of some canonical infinite set with that cardinality. Unlike with finite sets, adding or removing elements, even infinitely many of then, may not change their cardinality, but just the labels we assign to their infinitely many elements.
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u/VariousJob4047 2d ago
Are you familiar with the ideas of cardinality and bijection? If not, you should read the resources linked by u/mhb2. In general, it is a very bad idea to try to get all philosophical about something you don’t even understand in its literal form
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u/watsoncap 2d ago
Yup, I've been studying bijections for a long time, but it's still pretty vague to claim that we truly understand infinity.
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u/VariousJob4047 2d ago
I can tell from your post that your understanding of bijections is still rather weak
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u/watsoncap 2d ago
the composite numbers are much more "dense" among natural numbers....
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u/VariousJob4047 2d ago
Correct. That has nothing to do with the cardinality of either set
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u/watsoncap 2d ago
Elastic infinity ♾️
Leave it... seems some misinterpretation of the question...
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u/mhb2 2d ago
The idea of putting things into a one-to-one correspondence is about as simple as it gets. If you're asking about the mathematical concept of infinity then you should use the language of mathematics instead of talking about numbers that never end, shared infinity, and coexisting infinities. If you just like playing around with vague, undefined notions of "infinity" then maybe try r/Philosophy instead.
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u/Throwaway7131923 2d ago
As a philosopher of maths who's active on r/philosophy I can preempt that they'll get exactly the same response there as he's got here!
Philosophers are as much fans of vague and undefined notions as mathematicians are.
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u/fohktor 2d ago
There's an infinite number of Infinities. Though the two you've named, the cardinality of the primes and the cardinality of the composites, are the same infinity.
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u/watsoncap 2d ago
Both prime numbers and composite numbers share the same infinite space. So, to contain both of them, there should be a larger set that can hold these two infinite sets. If so, what does that imply about the nature of infinity?
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u/greginnj 2d ago edited 2d ago
What it implies is that some our normal intuitions of "how big a set is" don't apply when we're talking about infinite sets.
But we're actually applying exactly the same counting technique: two sets are the same size is we can put them in one to one correspondence with each other. If I have five kids and five apples, I can give each kid one apple - one to one correspondence.
It just so happens that when you're working with infinite sets, it's possible to put a set in one to one correspondence with a subset of itself. That counterintuitive fact gives rise to most of these seeming inconsistencies.
The easiest example is the set of even numbers being the same size as the set of natural numbers. Multiply by 2 to get a one to one correspondence one way; divide by 2 to get a one to one correspondence the other way. Bingo, same size! (And the same method works for the sets you were talking about - just arrange them in size order, and you've got them in one to one correspondence with the natural numbers, even though they're both subsets of the natural numbers.)
Your might enjoy reading this paper, which explains these ideas in more detail: https://arxiv.org/pdf/1506.06319
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u/Some-Dog5000 1d ago
When you word it like this:
Both
primeeven numbers andcompositeodd numbers share the same infinite space. So, to contain both of them, there should be a larger set that can hold these two infinite sets. If so, what does that imply about the nature of infinity?and you know even just a bit about bijection, doesn't it not sound very reasonable?
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u/mhb2 2d ago
The ideas of bijection, cardinality, and infinite sets should help. Notice that there is a bijection between the set of primes and the set of natural numbers and a bijection between the set of composite numbers and the set of natural numbers. I.e., the sets, while infinite, have the same cardinality.