Re: arithmetic in ZF
- From: "george" <greeneg@xxxxxxxxxx>
- Date: 18 Apr 2005 07:31:32 -0700
Bhupinder Singh Anand wrote:
> Can ZF define a consistent model for PA?
> ========================================
Terminology matters.
"Consistent model" is redundant.
Every model of a theory, just by existing at all,
confirms the consistency of the theory. The issue
is not whether the MODEL is consistent (it is almost
a grammatical error to apply the adjective "consistent"
to the noun "model"), but rather, whether the THEORY
that the model is a model OF, is consistent. If any old
model of a theory exists, then the theory is consistent.
> Classically, standard, first order PA is interpretable
> in standard, first order ZF, in the sense that the axioms
> of PA translate as theorems of ZF,
It bears stressing just what is being translated into what,
there. PA and ZF are in completely different languages.
ZF only has 1 predicate (e == set-membership).
PA is a much more complicated language.
It has 0, +, *, either 1 or successor, and <.
THESE must FIRST be "interpreted" or "translated"
into the language of ZF. THOSE TRANSLATIONS of PA's
axioms -- NOT PA's axioms themselves -- will
then be provable as theorems from the axioms of ZF.
There is more than one possible translation and it
is non-trivial to get it right; that's why HE had to
ask for the best/right version in renewing the thread.
> and the rules of inference of PA
Again, application error.
You cannot apply "rules of inference" to PA.
PA is an axiom-set. It DOESN'T HAVE rules of
inference. Rules of inference are had by LOGICs,
NOT axiom-sets. The logic that we are using in
the context of this discussion is [classical]first-
order[predicate] logic. IT has rules of inference.
It also has directives defining something called a first-
order language (the sentences that this logic's rules
of inference are defined-as-inferring-among should be
phrased in first-order languages). There are *2nd*-order
versions of both of these axiom-sets, but we are here thinking
about and dealing with first-order PA and first-order ZF.
Since the whole framework of our discussion is first-order
logic, that logic's inference rules are being applied to
BOTH axiom-sets and BOTH of their corresponding first-
order languages.
> are preserved under the translation.
In the sense that we are translating from one first-order
language into another, with intent to apply the same rules
of inference to sentences from both languages, yes.
> However, the significant question arises out of your
> comments:
>
> Is such translation also a formal, Tarskian,
> interpretation?
No. That's why I had to pause to mention two possible
different meanings of "interpretation".
> More precisely, can ZF define a consistent model,
> say Int_ZF, for PA over an
> appropriately defined domain?
ZF can define myriad models for PA. Model-definition
is WHAT ZF IS *for*. It's WHY WE CARE about it. ZF
is the *default* "model-description-language", or meta-
theory, for first-order logic IN GENERAL.
> If so, then, as a consequence of Tarskian satisfaction
> and truth in a model under a formal interpretation, the
> following argument suggests that every Arithmetical
> proposition should be decidable in the model
> from the axioms of a consistent ZF.
It is, but that's not just a property of THAT model:
that's a property of ALL models. If you have a theory
Th and a candidate model Mc, then Mc is not ACTUALLY
a model of Th, the candidate does not WIN election, UNLESS
it in fact decides every sentence of the theory. ANYthing
that leaves some sentence of the theory undecided is, FOR
THAT VERY REASON, NOT a model of the theory. It is at best
a partial specification of a model. Of course, sometimes
in ZF you can fudge this by saying that, sure, your model
decides everything, you just don't YET KNOW HOW it decides
SOME sentences.
> ZF decides all Arithmetical propositions
> ========================================
Well, no, actually, it doesn't.
One problem is that the language of arithmetic
and the language of ZF are different. The other
is that different models of ZF may have different
opinions about some of these questions.
> Let M be a model of ZF.
That's the whole problem: *a* model.
Different models are going to give different answers.
> We denote the domain of M by D_M.
>
> DEF1: We define the union of all D_M
That is hopelessly ambiguous.
D_M is the domain of M, and M was fixed;
there was only 1 M, and therefore there is only
1 D_M. If you meant over all M, as over all models
of ZF, then you can't do that *in* ZF.
> as the domain, namely D_ZF, of the interpretation,
you mean the domain of THAT interpretation, of your
personal chosen particular interpretation (a great
many others are possible)
> namely Int_ZF, of PA in ZF.
You can't do this in ZF.
The problem is that ANY INFINITE SET WHATSOEVER
can be a model for PA, and for ZF. It's very dangerous
to talk about a model of ZF in ZF. The problem
here is that ZF doesn't have a universal set. In every
MODEL of ZF, though, the domain IS the universal set
FOR THAT MODEL.
>
> (In other words, Int_ZF is the model of PA defined by {D_ZF, ZF}.
D_ZF doesn't exist in ZF; it would have to be the universal
set, and ZF doesn't have a universal set.
.
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