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April 10th, 2016, 11:50 AM   #1
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Contradiction about the Set of Natural Numbers?



It is said that there is an infinite number (aleph 0) of elements in the set of natural numbers. If we are counting the natural numbers in such a way as to have a matching of a one-to-one correlation where n = x; for example, if the 5th element in the set of natural numbers is n = 5, then x = 5.

But, if we have an infinite number of elements, where x = aleph 0, then n = aleph 0, but n must be a natural number.

Last edited by Mathbound; April 10th, 2016 at 12:09 PM.
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April 10th, 2016, 12:03 PM   #2
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You are confusing ordinals and cardinals. You are also ignoring the definition of the natural numbers. In particular, there is no axiom or definition that says that either cardinal numbers or ordinal numbers must be natural numbers.
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April 10th, 2016, 12:20 PM   #3
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But n must be a natural number: N = {1x=1, 2x=2, 3x=3, ...}

I should have been clearer.
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April 10th, 2016, 12:31 PM   #4
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Why must it? I can match the natural numbers to the natural numbers easily enough, but there are no infinite natural numbers, so no natural number describes the size of the set of natural numbers because the set is infinite.
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April 10th, 2016, 02:42 PM   #5
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Quote:
Originally Posted by v8archie View Post
Why must it? I can match the natural numbers to the natural numbers easily enough, but there are no infinite natural numbers, so no natural number describes the size of the set of natural numbers because the set is infinite.
That's the problem; there can't be a match of aleph 0 number of elements x to anything in the set. But the set is said to have an aleph 0 number of elements.
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April 10th, 2016, 02:58 PM   #6
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Yes. What's the problem? $\{0,1,2,3\}$ has four members, but 4 is not a member of the set. And some people define the natural numbers to include zero (indeed, it's the most natural set-theoretic definition).

To have a natural number represent the size of the set of natural numbers, you'd need a natural number that is bigger than all the rest. And that isn't possible by the definition of natural numbers.
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April 10th, 2016, 06:28 PM   #7
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Quote:
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Yes. What's the problem? $\{0,1,2,3\}$ has four members, but 4 is not a member of the set. And some people define the natural numbers to include zero (indeed, it's the most natural set-theoretic definition).
I specifically want to talk about the natural set from 1 and greater.

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To have a natural number represent the size of the set of natural numbers, you'd need a natural number that is bigger than all the rest. And that isn't possible by the definition of natural numbers.
I don't want x to be restricted to only the natural numbers. In this case, let's add aleph 0 to the possible elements that x can be.
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April 10th, 2016, 06:38 PM   #8
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All you are doing is trying to change the definitions so that they fit your assertion. In particular, if $x$ doesn't have to be a natural number, why should $n$ have to be a natural number?

Not that is makes any difference. The set $\{\aleph_0, 1, 2, 3, ...\}$ is also infinite and has cardinality $\aleph_0$.

I think you might be trying to play with ordinals, in which case you should use $\omega$ rather than $\aleph_0$.
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April 10th, 2016, 07:29 PM   #9
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All you are doing is trying to change the definitions so that they fit your assertion. In particular, if $x$ doesn't have to be a natural number, why should $n$ have to be a natural number?
Because there is a claim that {1,2,3...} has an infinite number of elements. So as far as I can tell, my argument is only going for this exact set.

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Not that is makes any difference. The set $\{\aleph_0, 1, 2, 3, ...\}$ is also infinite and has cardinality $\aleph_0$.
But just think about what this would mean on, say, an x, y graph that extends to include infinity, where x is the x axis and n is the y axis. It seems to me that you are essentially saying that we can have an infinite domain, but we can't have an infinite range for f(x) = n, or just x = n.

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I think you might be trying to play with ordinals, in which case you should use $\omega$ rather than $\aleph_0$.
I don't know if aleph 0 is an ordinal; is it?
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April 10th, 2016, 08:13 PM   #10
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Quote:
Originally Posted by Mathbound View Post
Because there is a claim that {1,2,3...} has an infinite number of elements. So as far as I can tell, my argument is only going for this exact set.
It does have an infinite number of elements. But that just means that there isn't a last one. There's no need to add another number to make that happen.

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It seems to me that you are essentially saying that we can have an infinite domain, but we can't have an infinite range for f(x) = n, or just x = n.
No, I'm just saying that for the function you are trying to describe, the range is not the natural numbers, it's the ordinal numbers.

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I don't know if aleph 0 is an ordinal; is it?
No, it's a cardinal number which means it describes the size of a set. An ordinal describes the ordering of elements of a set.

Perhaps this video will help. (With thanks to whoever posted it this morning).
Thanks from Mathbound

Last edited by skipjack; April 11th, 2016 at 11:24 PM.
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