Re: infinity

>> 1. If a set is finite, then its size is equal to some finite natural.
>> 2. For all n, if n is a finite natural, then so is n+1.
>> 3. For all n, if n is a finite natural, then n+1 > n.
>> 4. For all n, the set A_{n+1} = { 1, 2, 3, ..., n+1 } has size n+1.
>> 5. For all n, A_{n+1} is a subset of the set of all finite naturals.
>> 6. For all X and Y, if X is a subset of Y, then size(X) <= size(Y).
>> 7. For all x,y and z: if x > y, and y > z, then x > z.
>> 8. For all x, z: If x > z, then x is not equal to z.

All 8 statements listed above are true. But so also is the statement:

9. There is an infinite number of finite natural numbers.

That infinite number is, of course, outside the range of finite natural
numbers and is in fact the first infinite cardinal number Aleph-0.
Hope that helps!

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