Re: Simple enough for every reader?

On 2025-07-03 13:08:25 +0000, WM said:

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.

Being involved is not the same as being known.

The domain must be complete by the definition of mapping, and the codomain must be complete by the definition of surjectivity

Cantor's opinion was that everything is complete, and at lest every set is.

"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.

Only becase he could support his beliefs with proofs. Mathematicians
don't care about beliefs but they do care about proofs.
--
Mikko