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Math Help - Von Neumann Numbers

  1. #1
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    Von Neumann Numbers

    here's the problem:
    Consider the following inductively defined function:
    s(0):= 0 ; s(S):= s(S) U S
    Prove with an appropriate interpretation of 0 and s that the Peano axioms 1-4 hold for your interpretation.
    thanks a lot for ur help !!

    ps: i think i'm supposed to prove this by induction
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by yaszine View Post
    here's the problem:
    Consider the following inductively defined function:
    s(0):= 0 ; s(S):= s(S) U S
    Prove with an appropriate interpretation of 0 and s that the Peano axioms 1-4 hold for your interpretation.
    thanks a lot for ur help !!

    ps: i think i'm supposed to prove this by induction
    The first four Peano axioms are:

    1. 0 is a natural number
    2. every natural number a has a successor sa
    3. no natural number has 0 as its successor
    4. a=b iff sa=sb.

    Now there seems to be a problem here if I recall Johny's
    model of the naturals we start with φ (the null set) being
    in N, and

    sφ = {φ}

    and

    sS = S U {S}.

    Perhaps you will have better luck showing this construction satisfies
    the first four axioms.

    RonL
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