Re: A Question About Consistent Theories

On 28 Dec 2006 05:52:41 -0800, "Bob Stewart"
<bobstewart_III@xxxxxxxxxxx> wrote:


David C. Ullrich wrote:
On 27 Dec 2006 11:41:59 -0800, "Bob Stewart"
<bobstewart_III@xxxxxxxxxxx> wrote:

Thank you both Aatu and David. Actually Mendelson's book defines the
set of symbols of a first-order theory to be countable (David's
assumption).

Ok. Does he really make countability part of the _definition_,
or does he just say that he will only be considering countable
languages?

Yes, all languages of all formal theories are assumed to have a
countable set of symbols, at least in my model of the book, on page 28:
Elliott Mendelson, Introduction to Mathematical Logic, Third Edition,
Wadsworth & Brooks/Cole mathematics series, 1987, ISBN 0-534-06624-0

Of course a countable language is enough for most purposes,
and as we've seen here it can sometimes simplify proofs, but
you might note that there _are_ situations where a larger
language is needed.

Perhaps, if you have the time and patience, you could explain to me
what these situations are. But don't bother, if its too much trouble, I
believe you!

The example I had in mind is using the compactness theorem to set up
non-standard analysis (see Enderton's book, for example). You start
with a _large_ language, containing a constant c_x for each real
number x, a function symbol phi_f for each function f : R -> R, etc.
You say T is the theory of everything that's true in R (so T contains
sentences like c_2 + c_2 = c_4, phi_sin(c_0) = c_0, etc).

Now say T' is T together with the sentences c > c_1, c > c_2, etc.
The compactness theorem says that T' has a model...


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

David C. Ullrich
.

Relevant Pages

  • Re: A Question About Consistent Theories
    ... David C. Ullrich wrote: ... set of symbols of a first-order theory to be countable (David's ... Does he really make countability part of the _definition_, ... all languages of all formal theories are assumed to have a ...
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  • Re: A Question About Consistent Theories
    ... Does he really make countability part of the _definition_, ... Of course a countable language is enough for most purposes, ... expressions in any first-order theory is denumerable. ... or equal to alpha. ...
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