Re: Simple enough for every reader?

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

On 02.07.2025 09:45, Mikko wrote:

On 2025-06-30 18:21:09 +0000, WM said:
 

On 29.06.2025 12:25, Mikko wrote:

 

That is potential infinity. But Cantor claimed complete enumeration.

 There is no mathematical definiton of "complete enumeration"

 Obviously you don't know much of mathematics.

Mathematics is a so large topic that it is hard to say what could be
called "much" of it.

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. For some
purposes it is sufficient to show that at least one bijection exists
without identifying anu particular bijection.

However, that doesn't really matter as the distinction between complete
and incomplete is not mathematical.

 Obviously you don't know the most important parts of mathematics.

Importance is a matter of opinion.

"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.
Mathematical existence of many kinds of infinities has a firm mathematical
basis. Other kind of actual existence has no mathematical significance.

"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. Which other arguments
don't need but may use infinity is not said in the quote.

"Should we briefly characterize the new view of the infinite introduced by Cantor, we could certainly say: In analysis we have to deal only with the infinitely small and the infinitely large as a limit-notion, as something becoming, emerging, produced, i.e., as we put it, with the potential infinite. But this is not the proper infinite. That we have for instance when we consider the entirety of the numbers 1, 2, 3, 4, ... itself as a completed unit, or the points of a line as an entirety of things which is completely available. That sort of infinity is named actual infinite." [D. Hilbert: "Über das Unendliche", Mathematische Annalen 95 (1925) p. 167]
 

It means that no further element can be found later on.

 Whether an element is "found" has no mathematical meaning and in particular
does not affect its being or not a member of some set.

 "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.

Then it cannot be. If it is that all natural numbers are subtracted in their order, then it is that a last one is subtracted.

 Given two sets there is a set that is their difference. There is no
opeartion of subtraction in order.

 The set ℕ has an intrinsic order which can be used at any time. Bijecting sets presupposes and requires order. Further the difference of sets depends strongly on the order assumed.

That N has an order and can be given other orders is irrelevant. The
difference of sets does not depend on the order. 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. Consequently, the
set operations yield the same result whether the sets have an order or
not.

You are wrong. Here are only few pages of my Book Transfinity:
     4.1 Cantor on theology

 Theology is not mathematics.

 Set theory is theology. You are right, set theory is not mathematics.

Set theory may have some theological applications but is not theology.
Perhaps it is not a part of mathematics that you know but it is a part
of mathematics. Arithmetic and geometry can be regarded as theories
about certain kinds of sets.
--
Mikko