Re: FOL & completeness


"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:
iff, for a first order theory

The other direction is usually called the 'soundness theorem'. So "iff"
is equivalent to the conjunction of the completeness and soundness
theorems.


Of course.




The incompleteness theorem is that for any consistent, recursive set of
axioms, there is a sentence that is true in the standard model of
number theory (but NOT true in all models) that is not provable from
said axioms.

So, can we say that the same Goedel theorem affirms that G is not
logically
valid ?

I don't think the parenthetical part "(but NOT true in all models)" is
usually mentioned as itself a clause in the incompleteness theorem. I
just added the parenthetical to highlight for you the difference
between the completeness theorem (which addresses formulas that are
satisfied by every model, (or, for sentences, that are true in every
model)) and the incompleteness theorem (which addresses sentences that
are true in the standard model of number theory, but which, I'm
parenthetically now adding, are not necessarily true in all models and
which we can see that they are not true in all models).

That such sentences are not true in all models follows from the
completeness theorem. If a sentence is not provable from certain sets
of axioms, then the sentence is not valid.


Ok, that is just what I meant with my question. Thank you.

P.S. I wonder what could be model of PA in which G does not hold...


.

Relevant Pages

  • Re: Gödels Incompleteness vs. completeness theorem s
    ... > I have a question about Gödel's incompleteness theorem and his Completeness ... My lack of understanding leaves me wondering why/how the ... Completeness says that for any set of sentences, D and some sentence A, ... objects which satisfies the set of axioms. ...
    (sci.logic)
  • Re: Second order inferences
    ... incompleteness theorem shows only that what is formally provable coincides ... *extensionally* with what follow from the axioms". ... Surely it was the COMPLETEness and NOT the INcompleteness ...
    (sci.logic)
  • Re: Gödels Incompleteness vs. completeness theorem s
    ... >I have a question about Gödel's incompleteness theorem and his Completeness ... Completeness says that for any set of sentences, D and some sentence A, ... a semantic consequence of D, then A can be deduced D). ... it is possible to define the full second-order consequence ...
    (sci.logic)
  • Re: Relational lattice completeness
    ... Jan Hidders wrote: ... completeness in the context of the first incompleteness theorem. ... Not necessarily because the syntactical notion of truth is not the ...
    (comp.databases.theory)
  • Re: Gödels Incompleteness vs. completeness theorem s
    ... I have a question about Gödel's incompleteness theorem and his Completeness ... Completeness says that for any set of sentences, D and some sentence A, ... is PA, which is an axiom-set. ... (a class of sentences provable from a given recursive axiom-set) ...
    (sci.logic)