```Date: Mar 14, 2013 10:08 AM
Author: Dan Christensen
Subject: Re: Natural numbers embedded in other sets -- Follow-up

On Wednesday, February 13, 2013 6:13:08 PM UTC-5, Dan Christensen wrote:> Here is a formal proof (112 lines in DC Proof format), that proves the existence of natural number-like structures in every set S on which there is (1) a one-to-one (injective) mapping f, and (2) at least one element that has no pre-image under f.> > > > http://www.dcproof.com/ProofByInduction.html> Revised version proves uniqueness of the number-like structure. THEOREM:   Consider an injective (one-to-one) function f defined on a set s.             1-1      f: s ---> s   For every element s1 of s with no pre-image under f, there exists a UNIQUE subset n that is identical in structure to that of the set of natural numbers as defined by Peano's axioms.     Informally,        n = {s1, f(s1), f(f(s1)), ... }      f = usual successor function   The usual axioms for the natural numbers are shown to apply on n:   1. s1 e n   2. ALL(a):[a e n => f(a) e n]   3. ALL(a):ALL(b):[a e n & b e n => [f(a)=f(b) => a=b]]   4. ALL(a):[a e n => ~f(a)=s1]   5. ALL(b):[Set(b)    & ALL(c):[c e b => c e n]    & s1 e b    & ALL(c):[c e b => f(c) e b]    => ALL(c):[c e n => c e b]] where: e is set membership (epsilon)Set is the "is a set" predicate in DC ProofThus, the principle of mathematical induction is not just a rule that seems to work. If the other Peano Axioms (axioms 1-4 above) hold on some set s, then each axiom, including induction (axiom 5), will hold on a subset of s. Dan   Download my DC Proof 2.0 freeware at: http://www.dcproof.com Visit my new math blog: http://www.dcproof.wordpress.com
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