>two sets have the same size if and only if there exists a bijection between them
>two sets have the same size if and only if all injective mappings are surjective
Both of these definitions are valid for finite sets. In order to use the second definition, you simply need to show the existence of a bijection, and it can be proven that a bijection implies that all injective mappings are surjective. So is it the first “definition” that is implied from the second, or the second that is implied from the first? Why is one more valid than the other?
Infinite sets rely on using the first definition while ignoring the second. For this reason, infinite sets can appear to have the same size while also having a different size. Example:
Set A = {1, 2, 3, …}
Set B = {0, 1, 2, …}
The mapping x~x-1 is injective and surjective, but the mapping x~x is injective and not surjective. Intuitively, if every element in A maps to an element in B, but 0 still isn’t mapped to, set B is bigger than set A. This contradicts the mapping x~x, for which it seems that the sets have the same size.
According to definition 1, we simply ignore the contradiction and declare them equal. That is the cause of paradoxes related to infinity. Why do we not admit there is a contradiction and say that it makes no sense to talk about the cardinality of an infinite set? Or that infinite sets themselves are an unfounded axiom?
>two sets have the same size if and only if all injective mappings are surjective
Both of these definitions are valid for finite sets. In order to use the second definition, you simply need to show the existence of a bijection, and it can be proven that a bijection implies that all injective mappings are surjective. So is it the first “definition” that is implied from the second, or the second that is implied from the first? Why is one more valid than the other?
Infinite sets rely on using the first definition while ignoring the second. For this reason, infinite sets can appear to have the same size while also having a different size. Example:
Set A = {1, 2, 3, …}
Set B = {0, 1, 2, …}
The mapping x~x-1 is injective and surjective, but the mapping x~x is injective and not surjective. Intuitively, if every element in A maps to an element in B, but 0 still isn’t mapped to, set B is bigger than set A. This contradicts the mapping x~x, for which it seems that the sets have the same size.
According to definition 1, we simply ignore the contradiction and declare them equal. That is the cause of paradoxes related to infinity. Why do we not admit there is a contradiction and say that it makes no sense to talk about the cardinality of an infinite set? Or that infinite sets themselves are an unfounded axiom?
