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Re: Natural numbers embedded in other sets  Followup
Posted:
Mar 14, 2013 10:08 AM


On Wednesday, February 13, 2013 6:13:08 PM UTC5, Dan Christensen wrote: > Here is a formal proof (112 lines in DC Proof format), that proves the existence of natural numberlike structures in every set S on which there is (1) a onetoone (injective) mapping f, and (2) at least one element that has no preimage under f. > > > > http://www.dcproof.com/ProofByInduction.html >
Revised version proves uniqueness of the numberlike structure.
THEOREM: Consider an injective (onetoone) function f defined on a set s. 11 f: s > s For every element s1 of s with no preimage 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 Proof
Thus, the principle of mathematical induction is not just a rule that seems to work. If the other Peano Axioms (axioms 14 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



