Re: Torkel Franzen on truth

On Jan 4, 1:46 am, herbzet <herb...@xxxxxxxxx> wrote:
That's what's "a new thought to me": that an arbitrary structure
is a model of some non-null theory. I don't happen to know whether
that's true.

Of course it's true.
Every structure decides every sentence.
Pick 1 sentence.
Assert the theory with that 1 sentence decided the
way the structure decides it as an axiom.
Obviously, the structure is a model of that theory.
But this depends rather sillily on how you go about
describing a structure BEFORE you know what names
the axioms are going to use, before you know what
signature the language is going to have.
Perhaps you have to attach that, first, too.


Shall we agree as to what sets of sentences constitute "a theory" first?

For simplicity, let's confine to classical first order.

Authors such as Enderton take a theory to be any set of sentences
closed under entailment (which, thanks to the completeness theorem, is
a set of sentences closed under provability).

That is the canonical definition.
"Canon" meaning what it means, that is the ONLY definition
you are going to get to use, without first explicitly attacking
that definition.

I adopt Enderton's definition.

OK.

It's NOT ok.
It does violence to the word (theory).
If you can't tell what's an axiom and you therefore can't
tell even APPROXIMATELY *whether* some sentence is "in"
(declared "provable") by the theory OR NOT, then you don't
ACTUALLY HAVE any coherent *theory* (NON-local-technical sense)
of what makes sentences "in" the theory true!

I usually think of "a theory" as having a recursive (or at least r.e.)
set of axioms.

THAT *IS* the *CORRECT* definition.
As MoeBlee is explaining, it is NOT the standard one.
The point is that the standard is just broken.

.

Relevant Pages

  • Re: Torkel Franzen on truth
    ... a set of sentences closed under provability). ... If you can't tell what's an axiom and you therefore can't ... it is NOT the standard one. ... Yes, well, it's of a piece with tolerating arbitrary subsets of N or ...
    (sci.logic)
  • Re: How to tell if a theory is a good one
    ... >> can never prove that any axiom system is consistent within the system ... That is my exact take as well Bob. ... > provability cannot catch up with truth. ... > formalism such as FOL is purely a syntatical matter. ...
    (sci.physics.particle)
  • Re: How to tell if a theory is a good one
    ... >> can never prove that any axiom system is consistent within the system ... That is my exact take as well Bob. ... > provability cannot catch up with truth. ... > formalism such as FOL is purely a syntatical matter. ...
    (sci.physics)
  • Re: How to tell if a theory is a good one
    ... >> can never prove that any axiom system is consistent within the system ... That is my exact take as well Bob. ... > provability cannot catch up with truth. ... > formalism such as FOL is purely a syntatical matter. ...
    (sci.physics.relativity)
  • Re: arithmetic in ZF
    ... >> that it is a meta-theorem about provability under IPC that ... the axiom that produces this most easily ... In either case, the point is, the fact that a disjunction ... disjunction (being an element of a finite set is equivalent ...
    (sci.logic)