Re arithmetic in ZF

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					                                           Re: arithmetic in ZF

Re: arithmetic in ZF

Source: http://sci.tech−archive.net/Archive/sci.logic/2005−04/msg00083.html



      • From: "george" <greeneg@xxxxxxxxxx>
      • Date: 7 Apr 2005 08:02:02 −0700


H. Enderton wrote:
> In a recent post, Bhup pointed out (correctly) that we can interpret
> PA into ZF. But then the post went on to say:
>
> >6. Hence, if ZF is consistent, then every Arithmetical proposition
is
> >syntactically (i.e., independently of the semantic connotations of
the
> >above definitions of "truth" and "satisfiability") decidable from
the
> >axioms of ZF.
>
> As George Greene remarked, this is wrong. The arithmetic sentences
> provable in ZF (i.e., the arithmetic sentences whose interpretations
> are provable in ZF) make up "merely" an r.e. set of sentences.

One thing that misled me in that presentation was multiple ways of
"interpreting" the act of "interpretation". I was thinking purely
in terms of the Tarskian sense of an interpretation as a mapping from
the language (or its parts) to some model−theoretic structure. But
there is another available and relevant sense of "interpretation" that
does NOT involve interpretations−of−a−theory−into−structures/models at
all, but rather is a purely LINGUISTIC sense of "interpretation" in
which
one takes elements of the language of Arithmetic and simply TRANSLATES
them into language−of−ZF. In particular, if one maps the
interpretation
to the *sentence*, it becomes ambiguous which of these two
"interpretations"
(of "interpretation") is meant. If you take some trivial arithmetical
sentence like s(0)+s(s(0))=s(s(s(0))) (which happens to be a theorem),
you could use ZF to construct a model of PA and ask what the
interpretation
of this sentence was in that model (0 will be the empty set, s(.) will
be some unary−function−represented−as−a−ZF−set, +(.,.) will be a binary
function, = will be a binary predicate, and the predicate will assign
T as the truth−value of the overall sentence). BUt you could
instead/also


Re: arithmetic in ZF                                                          1
                                             Re: arithmetic in ZF
just ELIMINATE everything occurring in this arithmetical language,
including =, by translating it into something involving only e (there
are formulations of ZF where e is the only predicate and where no
function−
symbols are needed at all, although the canonical dialect has them).
s(x) would be x U {x}; x=y would be Az[zex<−>zey]; and + and * would
be,
well, complicated, but basically they would be the least ordinal
bijectible
with some set constructed from the two argument sets (in the case of
multiplication, that set could be, conventionally, the
cartesian−product
of the argument sets).

My point is that if one wants to talk about an "axiomatization"
of arithmetic under ZF, then new "axioms" one needs are instead
DEFINITIONS of the linguistic entities from the language of arithmetic
in the language of ZF. I know, definitions often ARE axioms, but
my point is that axioms are usually NOT definitions.
I mean definitions as conservative, as purely abbreviatory.

> Let's give a name to the set:
> Tzf = the set of sentences of arithmetic whose interpretation is
> provable in ZF.

I guess "interpretation" here must be the "linguistic"
interpretation, but when I was responding to Bhup, I was
thinking in terms of the model−theoretic one.

> Then Tzf is a theory (it is closed under provability) and, being
> r.e., it is a recursively axiomatizable theory extending PA.
> (Tzf proves Con PA.)

But there is a problem with thinking about Con PA
in this context, because Neither PA NOR ANY OTHER PARTICULAR
AXIOMATIZATION of first−order arithmetic IS EVEN RELEVANT to
this particular theory! One important thing about Con(PA) in the
context of G1 was that it was IN THE SAME language as PA.
In this context, though, Con(PA) is in the language of ZF
and there is no reason to think of "the axioms of PA" as axioms.
They are just theorems of ZF, and the question of what a provability−
predicate expressing provability FROM PA would even look like gets
COMPLICATED. The axioms of ZF are available for use in proving
things at every step in this framework. HOw are you supposed to
express about a derivation that it needs the PA−theorems but NOT
the other/additional axioms−of−ZF, when in fact ALL those axioms
are NEEDED to prove the theorem−hood of the PA−axioms??

> And someone once told me a "natural" recursive axiomatization of
> Tzf. Unfortunately, I have forgotten what it is. Does someone
> out there in sci.logic land know what it is?

Re: arithmetic in ZF                                                     2
                                           Re: arithmetic in ZF


I don't, but I share your hope that somebody does.
The only point I wanted to make (aside from confessing
my own confusion under ambiguity of "interpretation")
was that beyond the ZF axioms themselves, any axioms
you even MIGHT possibly need to add are just definitions.

.



      • Follow−Ups:
             ♦ Re: arithmetic in ZF
                     ◊ From: Bhupinder Singh Anand

      • References:
             ♦ arithmetic in ZF
                     ◊ From: H. Enderton

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Re: arithmetic in ZF                                              3