Re: Peano Algebra and it's base theorem

[Mark Hahn <mhahn@tcbtech.com>]

At 07:42 AM 9/22/98 -0500, Jim Choate wrote:
>Peano Algebra's are based on the following:
>
>If a unary predicate P holds for 0, and if P holds, together with an element
>x, also for it's succesor x', then P  holds for all natural numbers.

Didn't you over simplify this theorem somewhat. The proof that P holds
for the successor x' must be derived from the truth of predicate P for x.
The successor, x', must be shown to satisfy P because x satisfies P.
Restated: x' satisfies P if and only if x satisfies P. 

In this way you can start with P and the natural number 1, and 
show P holds for any natural number by applying your proof recursively. 
(i.e. P is true for 1 and my proof show it is true for 2, then my proof shows
it is true for 3, then it shows its true for 4, the it shows its true for
....)

If the proof of P for x' is not related to the proof of P for x then
you can prove lots of irrational statements. I.e. say the Predicate P is
"is prime". Then 2 holds for P (2 "is prime"). 2's successor is 3. P also
holds for 3 (3 "is prime"). So all P holds for all natural numbers. Ergo,
all natural number are prime. I don't think so. 3's primeness must be
derived from 2's primeness. 

-MpH
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