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.

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> 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 Proof

Thus, 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

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