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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Thu, 3 Jul 2025 15:08:25 +0200
Organization: A noiseless patient Spider
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On 03.07.2025 11:35, Mikko wrote:
> On 2025-07-02 13:51:01 +0000, WM said:

>>>> The definition of bijection requires completeness.
>>>
>>> No, it doesn't.
>>
>> 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
> 
>> "Cantor's belief in the actual existence of the infinite of Set Theory 
>> still predominates in the mathematical world today." [A. Robinson: 
>> "The metaphysics of the calculus", in I. Lakatos (ed.): "Problems in 
>> the philosophy of mathematics", North Holland, Amsterdam (1967) p. 39]
>>
>> Note belief and predominate.
> 
> Mathematics is about definitions and theorems, not beliefs. Peaple may
> have beliefs about open problems or other things but those beliefs have
> no mathematical significance.

Cantor's beliefs have induced a large filed of mathematics. Anyhow your 
"There is nothing religious in Cantor's arguments." is wrong.
> 
> Mathematical existence of many kinds of infinities has a firm mathematical
> basis. 

Yes.

The rule of subset proves that every proper subset has fewer elements 
than its superset. So there are more natural numbers than prime numbers, 
and more complex numbers than real numbers. Even finitely many 
exceptions from the subset-relation are admitted for infinite subsets. 
Therefore there are more odd numbers than prime numbers.

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

>> "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.
>> "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.

Yes, it is a mathematical way.

> 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.
> 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.
 >
> One of the first things
> Cantor specified in the introduction of the concept of set was that sets
> have no order, i.e., the order is not a part of a set.

Then he introduced well-ordered sets.
Die wohlgeordneten Mengen.
Unter den einfach geordneten Mengen gebührt den wohlgeordneten Mengen 
eine ausgezeichnete Stelle; ihre Ordnungstypen, die wir "Ordnungszahlen" 
nennen, bilden das natürliche Material für eine genaue Definition der 
höheren transfiniten Kardinalzahlen oder Mächtigkeiten, eine Definition, 
die durchaus konform ist derjenigen, welche uns für die kleinste 
transfinite Kardinalzahl Alef-null durch das System aller endlichen 
Zahlen geliefert worden ist (§ 6).

Without order there is no order type and no ordinal number (and no means 
to address a number by digits).

Regards, WM
>