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Math Help - On an Attempt to Define Natural Numbers

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    On an Attempt to Define Natural Numbers

    I was trying to define "natural numbers" consider my reasoning and maybe you can complete my construction.
    Steps:
    1)Let S be the set of all FINITE sets (note that S itself is infinite)
    2)Consider the following Axiom (Axiom of Disjointness): For any finite set K there exists another set K' with the same cardinality as K having the property that the interestion of K and K' is {}. IMPORTANT- this axiom probably cannot be proven because we do not have a definition for a set.
    3)Define a natural number as the cardinality of any element of set S.
    4)Define addition of two natural numbers as the union of two disjoint sets.
    5)Demonstrate that addition is well-defined by using the Axiom of Disjointness.
    6)Define "zero" as the cardinality of the empty set.
    This is were I will stop. Perhaps you can complete this construction. Maybe use Zorn's Lemma to show there is a minimal element, prove the Well-Ordering Theorem for this set and define partial ordering.
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    Like a stone-audioslave ADARSH's Avatar
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    Smile

    Quote Originally Posted by ThePerfectHacker View Post
    I was trying to define """"natural numbers""""
    ....
    ...
    6)Define "zero" as the cardinality of the empty set.
    Is zero a natural number , I think when you add zero you get set of whole numbers
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    Quote Originally Posted by ADARSH View Post
    Is zero a natural number , I think when you add zero you get set of whole numbers
    Yes zero is a natural number.

    Well that thread is from three years ago.
    Now I know how natural numbers are defined.
    ---
    For any set x define x+1 = x \cup \{ x \}.
    Let  0 = \emptyset, 1 = 0+1 = \{ \emptyset \}, 2=1+1 = \{ \emptyset, \{ \emptyset\} \}, 3=2+1, ...

    That is basically how we define natural numbers. We just need to explain what " ... " means. Once we do that we have the natural numbers.
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    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Yes zero is a natural number.
    Thanks for the definition but, than what are Whole numbers ?
    I have been taught from childhood that whole numbers are defined as the set of natural numbers including zero
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