Discussion on Partial field and Full field Bijections in Set Theory

[西部牛仔]

Dear Professor,

While studying set theory, I have encountered numerous apparent contradictions within the existing framework. After conducting independent research, I propose that there exist two distinct types of bijective mappings: partial field bijections and full field bijections. I believe the failure to differentiate between these two mappings in current theory is the root cause of those contradictions. I have written a paper elaborating on my ideas, and I would greatly appreciate your feedback and corrections.

Article link:

https://doi.org/10.5281/zenodo.21262344

Sincerely,

Jin Qi

[Steven Nguyen]

To be honest, I would prefer a clear warning box in the intro "there is probably a mistake", to prevent uninformed misuse of the paper. Such is the usual approach to any 'proof' that some fundamental math is wrong.

First mistake:

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Section 2:

"No matter how long ... this won't happen"

= Mathematically this is: Choose a finite number, and it won't happen.

So intuitively, this logic cannot be used to logic with an infinite size. That's a hint to look deeper and the issue becomes found:

In "dynamic proper subset" "A proper subset B" "C proper subset D" etc, there is nothing stopping

the natural numbers to grow like {1}, {1, 2}, {1, 2, 3},

and the even naturals to grow like {2}, {2, 4}, {2, 4, 6}.

Logically, a "partial bijection" could have both sides seemingly grow at independent arbitrary rates. There is a disconnect between the definition of "partial bijection" and the strict structure of the thought exercise in section 2.

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The broader classic issue is to understand that things are Exactly their definitions, no more, no less. Infinity is strange. If intuition understandably thinks something is wrong with the following proof I write here, the logic has been machine-verified, so the only potential issue is in the definitions or vocabulary not matching the intuition:

Two sets A and B are "equinumerous" iff there is a bijection between them.

A "bijection" is a two-way function from all of A to all of B; "two-way" meaning you can take the inverse function to get from all of B back to all of A.

Obviously the function f(x) = x/2 maps all even integers to all integers, and can be reversed using g(x) = 2x.

Therefore f(x) is a bijection and the set of even integers is equinumerous to the set of all integers.

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Eventually, working with the topic enough, intuition will update.

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[Steven Nguyen]

to clarify:

"No matter how long ... this won't happen"

= Mathematically this is: Choose a finite number, and it won't happen.

The intuitive reason why this is true is that, the only way to do an infinite amount of steps is to do it all at once. There is no "building to infinity", although note that the subtly different "build for as long a time and as high a number as you could ever want" is possible... because we eventually choose to stop.