# Natural numbers and infinity

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#1
I have it on good authority that there are an infinite number of Natural numbers, and also that each of these Natural numbers are finite. I can't reconcile these in my head. Surely if there are an infinite number of them and they start from 1, they must go up to infinity.
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4 years ago
#2
(Original post by andythepiano007)
I have it on good authority that there are an infinite number of Natural numbers, and also that each of these Natural numbers are finite. I can't reconcile these in my head. Surely if there are an infinite number of them and they start from 1, they must go up to infinity.
Firstly there are different types of infinite. Namely countably infinite sets, and non countable sets.

It is true that the natural numbers are countably infinite, the standard way we define a countably infinite set is if we can define a bijection mapping from the set to the natural numbers.

Well clearly then the natural numbers are countably infinite because we can just take the identity map id:N ---> N defined by f(n)=n for all n in N.

Something that is not countable would be the set of real numbers (it is not possible to create a bijection from R ---> N).

So yes there are infinitely many natural numbers 1,2,3,4,... and each of these numbers is finite and they do "go up to infinity" of course this is not really a mathematically rigorous statement though.
1
4 years ago
#3
If you want to go for something where it's a bit more obvious: Take the set of all numbers of the form 1/n where n is a non-zero natural number. There's infinitely many of these (one for each natural number: alternatively, if there were an finite number of them, then there would be a smallest one: which one are you claiming is smallest?), but all of them are between 0 and 1.
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4 years ago
#4
Wiki- Cantors Diagonal Proof- this might help.
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#5
(Original post by Anon07079191)
Wiki- Cantors Diagonal Proof- this might help.
Yes, I know about Cantor's Diagonal Proof.

Here is my problem, there are the set of natural numbers (1, 2, 3, ...) and the set of ordinals (1st, 2nd, 3rd, ...), so there is a one-to-one correspondence between them. If there were only 100 ordinals, the natural numbers go up to 100, if a million ordinals, the natural numbers go up to one million. Why then, is it that if there an infinite number of ordinals, that the naturals DON'T go up to infinity?
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#6
(Original post by poorform)
Firstly there are different types of infinite. Namely countably infinite sets, and non countable sets.

It is true that the natural numbers are countably infinite, the standard way we define a countably infinite set is if we can define a bijection mapping from the set to the natural numbers.

Well clearly then the natural numbers are countably infinite because we can just take the identity map id:N ---> N defined by f(n)=n for all n in N.

Something that is not countable would be the set of real numbers (it is not possible to create a bijection from R ---> N).

So yes there are infinitely many natural numbers 1,2,3,4,... and each of these numbers is finite and they do "go up to infinity" of course this is not really a mathematically rigorous statement though.
I'm not arguing with you, but it just sounds like playing with definitions to me.

Here is my problem, there are the set of natural numbers (1, 2, 3, ...) and the set of ordinals (1st, 2nd, 3rd, ...), so there is a one-to-one correspondence between them. If there were only 100 ordinals, the natural numbers go up to 100, if a million ordinals, the natural numbers go up to one million. Why then, is it that if there an infinite number of ordinals, that the naturals DON'T go up to infinity?
0
4 years ago
#7
(Original post by andythepiano007)
I'm not arguing with you, but it just sounds like playing with definitions to me.

Here is my problem, there are the set of natural numbers (1, 2, 3, ...) and the set of ordinals (1st, 2nd, 3rd, ...), so there is a one-to-one correspondence between them. If there were only 100 ordinals, the natural numbers go up to 100, if a million ordinals, the natural numbers go up to one million. Why then, is it that if there an infinite number of ordinals, that the naturals DON'T go up to infinity?
The notion of infinite sets is entrenched in the definition; it is perhaps a matter of "playing with definitions" but this is because definitions are how we know what things are. Rather than defining formally "infinity", we define things like "countably infinite" and "tends to infinity" as self-enclosed things. Therefore the phrase "go up to infinity" doesn't really mean anything. You have to be precise with what you are saying to avoid these kinds of complications.
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#8
(Original post by 13 1 20 8 42)
The notion of infinite sets is entrenched in the definition; it is perhaps a matter of "playing with definitions" but this is because definitions are how we know what things are. Rather than defining formally "infinity", we define things like "countably infinite" and "tends to infinity" as self-enclosed things. Therefore the phrase "go up to infinity" doesn't really mean anything. You have to be precise with what you are saying to avoid these kinds of complications.
OK, imagine you have n natural numbers. That means they go up from 1 to n. Yes? Now let n be infinity. If you have infinite natural numbers, surely they rise without limit. I don't see what's so wrong with that idea that you have to use a set of obscure definitions to refute it.

Also, if they didn't rise without limit, there would be a largest natural number (which is also wrong)

Am I wrong in suggesting that rising without limit is the same as tending to infinity?
0
4 years ago
#9
(Original post by andythepiano007)
I have it on good authority that there are an infinite number of Natural numbers, and also that each of these Natural numbers are finite. I can't reconcile these in my head. Surely if there are an infinite number of them and they start from 1, they must go up to infinity.
I love them naturals. Ain't they lovely.
0
4 years ago
#10
(Original post by andythepiano007)
OK, imagine you have n natural numbers. That means they go up from 1 to n. Yes? Now let n be infinity. If you have infinite natural numbers, surely they rise without limit. I don't see what's so wrong with that idea that you have to use a set of obscure definitions to refute it.

Also, if they didn't rise without limit, there would be a largest natural number (which is also wrong)

Am I wrong in suggesting that rising without limit is the same as tending to infinity?
But you can't let n be infinity, that's the problem, you can only let n tend to infinity. Yes, they rise without limit. But if you count long enough, you can reach any conceivable number, therefore the numbers themselves cannot be infinite.
1
4 years ago
#11
To me it seems like the notion of a limit is the real issue here. The jump between natural numbers is different to the jump between the naturals and . Hence the names successor and limit ordinals.

What you are saying reminds me of one of the famous Zeno "paradoxes": you're shooting an arrow at a board. For the arrow to reach the board, it must first arrive at a point between the bow and the board. It must then arrive at a further point between its position and the board etc. ... so we never reach the board. Another analogy might be an increasing sequence of rationals that tend to an irrational.

In both cases we need to forget the smaller, counting steps, and picture a single jump directly from those steps to the board, or the irrational number, to reach them. Counting in the naturals you will obviously never arrive at : the only way to reach it is to make a leap from the naturals directly onto . The nature of this leap, which can be characterised formally, is precisely why is not part of the naturals, and why this leap does not imply that the set of smaller steps we started with are not finite.
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