Re: Is peano arithmetic inconsistent under the intended interpretation?

On Mar 12, 6:22 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 11, 7:58 pm, Newberry <newberr...@xxxxxxxxx> wrote:

On Mar 11, 12:57 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
Here is how I would communicate it with the knowledgeable people.

You can tell me whether I've paraphrased you correctly:

We
know the meaning of the symbols ~, &, v, ->, E, A, 0, 1, S, +, x,

We're talking about the language of first order PA, I surmise.

In this case, ~, &, v, ->, E, A are logical symbols that are not
directly assigned an interpretation. However, yes, we do have an
understanding of the role these symbols have.

And, yes, there is a model that we call 'the standard model' such that
we know the interpretation of 0, 1, S, +, * per that standard model.

and
we know what the standard model is.

We define a certain model for the language of PA as 'the standard
model for the language of PA'.

We try to capture the above with
syntactical rules as best as we can.

We define a formal theory such that the standard model is a model of
that theory; also, we might have hoped that only models isomorphic to
that standard model are models of the theory, but we found out
otherwise.

First we produce the rules of
propositional and predicate calculus, and then we produce Peano's
axioms. Then we find to our surprise that the system has not captured
the model we had in mind.

We found that the theory is not categorical - along with the standard
model being a model of the theory, we find that there are models not
isomorphic to the standard model that are also models of the theory.

 The model clearly requires



    (Ex)P_ultimate(x, #F) --> F

I don't know what you mean by "a model requires".

Also, I haven't been heeding discussion about the particular formula
you just mentioned. Please tell me what langauge it is in and the
definitions of any of its defined symbols. I take it that it is
supposed to be in a language of PA extended by definitions. Would you
please sketch how you take it to be a well formed formula of such a
language.

T_ultimate can be defined as a fixed point of the following
operation on theories extending PA:

T --> F(T)


where F(T) = that theory whose axioms consist of all the
axioms of T, plus the additional axiom schema


(Ex Pr_T(x,#Phi)) -> Phi


where Pr_T(x,y) is the formalization in PA of the claim
that x is a Godel code for a proof of the sentence whose
Godel code is y, and #Phi means the Godel code of sentence
Phi.


The operation


T --> F(T)


considered as an operation on r.e. sets has a fixed point,
T_ultimate, which satisfies


T_ultimate |- (Ex Pr_T(x,#Phi)) -> Phi


Unfortunately, applying Godel's theorem, we can show that
T_ultimate is inconsistent.

http://groups.google.com/group/sci.logic/browse_frm/thread/e295044b4d99c456/c68502f8d89c094e?lnk=gst&q=P_ultimate#c68502f8d89c094e

so we should be able to add it to our axioms.

We can add any first order sentence to the axioms. If the sentence has
new symbols to the language, then the result will be another language
and another theory.

It is just as compelling
as the rest of them. But we get a contradiction. What went wrong?

I can better comment on that when I know how you arrived at the
formula as a well formed formula of the language of PA extended by
definitions. (If you've already explicated that, then forgive me, as I
do admit that I have not paid attention to that part of the
conversation here.)

In any case, at least we don't need your phrase "consistent under the
interpretation".

MoeBlee

.

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