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Math Help - Integer Construction

  1. #1
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    Integer Construction

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

    Definition: A set S is "finite" if and only if \exists \  T\subset S and a bijective map \phi: T\mapsto S.

    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 \Omega be the proper class of all finite sets. So that if S_1,S_2 \in \Omega then S_1\cap S_2 = \{ \}.

    Definition: Define an relation R on \Omega as (X,Y)\in R for X,Y\in \Omega iff there exists a bijection \phi: X\mapsto Y.

    Theorem: The above relation is an equivalence relation.

    Definition: Define \Phi to be the proper class of \Omega modulo R.

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

    Definition: For n_1 and n_2 naturals we define n_1+n_2 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 ....
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post
    I once posted this a long time ago. But I starting thinking about this again.

    Definition: A set S is "finite" if and only if \exists \ T\subset S and a bijective map \phi: T\mapsto S.
    ????

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

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlank View Post
    ????
    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.
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