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From: joes <noreply@example.org>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Thu, 3 Jul 2025 14:12:29 -0000 (UTC)
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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.

-- 
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.