Re: Simple enough for every reader?

Am Thu, 03 Jul 2025 15:08:25 +0200 schrieb WM:

On 03.07.2025 11:35, Mikko wrote:

On 2025-07-02 13:51:01 +0000, WM said:

The function is injective, or one-to-one, if each element of the
codomain is mapped to by at most one element of the domain,
The function is surjective, or onto, if each element of the codomain
is mapped to by at least one element of the domain; Wikipedia
Bijection = injection and surjection.
Note that no element must be missing. That means completeness.

 
It does not mean that the bijection is completely known.

 
It means that every element of the domain and of the codomain is
involved.
The domain must be complete by the definition of mapping, and the
codomain must be complete by the definition of surjectivity

Which are the case for Cantor's function.

The rule of subset proves that every proper subset has fewer elements

No such rule for infinite sets.

than its superset. So there are more natural numbers than prime numbers,

No, you can number the primes.

The rule of construction yields the number of integers |Z| = 2|N| + 1
and the number of fractions |Q| = 2|N|^2 + 1.

Those numbers are equal.

"The arguments using infinity, including the Differential Calculus of
Newton and Leibniz, do not require the use of infinite sets." [T.
Jech: "Set theory", Stanford Encyclopedia of Philosophy (2002)]

Differential calculus does not require sets at all.

But it needs potential infinity. Therefore your "the distinction between
complete and incomplete is not mathematical." is wrong.

It doesn't need "actual infinities".

"Numerals constitute a potential infinity. Given any numeral, we can
construct a new numeral by prefixing it with S." [E. Nelson:
"Hilbert's mistake" (2007) p. 3]

 
That is a possible way to view them.
But a different view does not lead to different mathematical conclusion
as they are irrelevant to inferences from axioms and postulates.

 
Potential infinity is based upon other axioms than actual infinity and
has other results.

Uh, no?

That N has an order and can be given other orders is irrelevant.

Not for bijections. The enumeration of the rational numbers is
impossible in the natural order by size for instance.

That's a different function.

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Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.