Re: Peano's Axioms from the Field Axioms alone?

Dan Christensen <dchris@xxxxxxxxx> wrote:
Hi,

I have been able to show that the equivalent all but one of Peano's
Axioms can be derived from the fields axioms -- namely that no natural
number has a successor of 1. See:

http://www.dcproof.com/PeanoFieldAxioms.html

I can derive the remaining axiom if I assume the field is ordered. Is
it possible to derive it without using the order axioms?

Perhaps I misunderstood something, but in view of the existence
of finite fields I would be very surprised, if the field axioms
alone were sufficient.


Here, I use the standard construction of N with it being the "smallest"
inductive subset of the field, starting at 1. Intuitively, it seems to
me that it should be possible to show that 0 is not in that "smallest"
set, but the proof eludes me.

See above. In the case of a finite field your construction will
produce the prime field; also in the case of an infinite field
as long as the characteristic is nonzero.

Marc
.

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