Re: Interesting (IMO) question about the reals...

On 7 Feb 2007 12:46:22 -0800, "MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:


(Technicality: This assumes that we're talking about the
theory of complete ordered fields as a first-order
theory using set theory, so the Completeness Theorem
from logic applies...)

Well first, I should say that I have a little uneasiness
about this myself. But:

This is something I dont' understand. What is the theory of complete
ordered fields? All I know of, as you mentioned, is set theory. If we
take some proper subtheory of set theory to be the theory of complete
ordered fields, then it's not a recursively axiomatizable theory,
right?

So what? We can still talk about the class of theorems of set
theory of the form "if F is a complete ordered field then...".

Meanwhile, just putting it as a first order theory onto itself
(without a primitive and axioms for the membership relation) by using
an axiom schema for the least upper bound principle doesn't, as I've
been told, result in a theory whose models are all and only complete
ordered fields.

Anything you might say to shed some light here is appreciated.

I was actually hoping someone else would shed some light...

MoeBlee


************************

David C. Ullrich
.

Relevant Pages

  • Re: What is the 1st order formal system known as PA?
    ... His hypothesis is that R1 and R2 are complete ordered fields, ... My premise is that they are sets in set theory. ... And the first order formal meta theory is set theory, ... >> at the object level and Z1 is first order set theory at the meta-level. ...
    (sci.logic)
  • Re: What is the 1st order formal system known as PA?
    ... > about structures within set theory. ... > complete ordered fields, there exists an isomorphism of ordered fields ... > at the object level and Z1 is first order set theory at the meta-level. ...
    (sci.logic)
  • Re: What is the 1st order formal system known as PA?
    ... My premise is that they are sets in set theory. ... >> about and your metatheory would be something like set theory. ... > Because my object level set theory is in a formal first order language. ... >> models are precisely the complete ordered fields. ...
    (sci.logic)
  • Re: Interesting (IMO) question about the reals...
    ... theory of complete ordered fields as a first-order ... This is something I dont' understand. ... All I know of, as you mentioned, is set theory. ... an axiom schema for the least upper bound principle doesn't, ...
    (sci.math)