Thread: Integer Construction

I once posted this a long time ago. But I starting thinking about this again.

Definition: A set is "finite" if and only if and a bijective map .

Warning: Everything from here might be considered Naive Set Theory. This is why I will use the term "proper class" instead. (And I will not be so careful, because I do not want to spend time being careful).

Defintion: Let be the proper class of all finite sets. So that if then .

Definition: Define an relation on as for iff there exists a bijection .

Theorem: The above relation is an equivalence relation.

Definition: Define to be the proper class of modulo .

Definition: Define the cardinality of each element in the proper class of to be an "natural".

Definition: For and naturals we define to be the cardinality of the union of the two equivalence classes.
(And this is well-defined).

Now with this with continue to define some properties ....

Originally Posted by ThePerfectHacker

I once posted this a long time ago. But I starting thinking about this again.

Definition: A set is "finite" if and only if and a bijective map .

????

Isn't this the definition of a "non-finite" set?

RonL

Originally Posted by CaptainBlank

????
Isn't this the definition of a "non-finite" set?

Yes, sorry. (When I was writing this I had a lot to drink). A infinite set is exactly as described above. So a finite set is a non-infinite set. The negation of that definition.