Re: Simple enough for every reader?

On 04.07.2025 09:51, Mikko wrote:

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:

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The definition of bijection requires completeness.

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No, it doesn't.

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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
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Bijection = injection and surjection.
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Note that no element must be missing. That means completeness.

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It does not mean that the bijection is completely known.

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It means that every element of the domain and of the codomain is involved.

 Being involved is not the same as being known.

I only said: The definition of bijection requires completeness.
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You: No, it doesn't.

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.

And every bijection.

Cantor's beliefs have induced a large filed of mathematics.

 Only becase he could support his beliefs with proofs.

There is no proof of an actual infinity. How should that work?

Mathematicians
don't care about beliefs but they do care about proofs.

His "proofs" contradict mathematics, according to which in every interval (0, n] and i the limit there are twice as many natural nu8mbers as eve numbers. Every correct counting method must confirm this result.
Regards, WM