Truth and facticity (Was: Re: Recursivity vs. Provability)


"george" <greeneg@xxxxxxxxxx> wrote in message
news:1114537786.919892.21810@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
>> Charlie-Boo <chvol@xxxxxxx> wrote:
>> >Assume that the set of provable statements is
>> >recursively enumerable.
>> >For any given statement, whether it is provable
>> >or not is the same as
>> >whether it is in this r.e. set.
>> >Now what does that tell us about the
>> >statements that we can prove?
> [ in particular, is there...]
>> >Or a statement that is not provable and which is true?
>
>
> H. Enderton wrote:
>> True where?
>
> I just had to quote that for emphasis.
> This is one of the core terminological points
> we sort of have to pound through the heads of
> normal people: here in THIS weird room, there
> is NO SUCH THING as "true", just PLAIN true.
> You only get to be true (or false) UNDER AN
> INTERPRETATION, or IN A MODEL.

Sadly, this is one of the failures of the logical program.

What I mean by this (before you blow your top) is that
everyday, common sense language usage refers as if this
has meaning.

At one time I tried to talk about

ZF + V=L(D) where D is a normal measure

This axiom admits precisely one two-valued measurable
cardinal. That is your formal representation of "faith" George.
It reflects the usual, everyday, common-sense idea that there
is a reality and that truth and falsity have the expected
correspondence to facticity as it refers to reality.

Read Frege or Russell if you have doubts about the role
of this "reality-as-truth" in the the beginnings of the modern
logical program.

In addition, V=L(D) is relatively constructed. It reflects what
can be linguisticaly expressed relative to this facticity assumption.


>
> Statements that are "just plain" true are being
> talked about in the context of 1 particular STANDARD
> model, and what is actually being alleged is that they
> are true in THAT model.

To some extent, this presupposition is an artifact of your
training. It is a correct presupposition from the perspective
of logical training. But, it completely removes the ability
of a human being to express themselves using the mathematical
elements of natural language.

In one fell swoop you have negated most generally accepted
human knowledge. That is what I infer from most of your
objections along these lines. Perhaps you can clarify this if
I have misunderstood.


> They are likely to be false
> in another one (because if they were true in all models,
> we would be saying that they were provable, AS OPPOSED to
> saying that they were true).

Which is why I began by searching for axioms.

If you had ever shut up long enough to use the brain
God gave you (a reference I make freely given your many
references to Jesus), you might have figured out just
how close our positions really are.

If you had any balanced perspective of the issues here, you
would have directed me long ago to Russell's observation
that mathematicians confused the containment relation
and the membership relation since this is evident in the
axioms I wrote.

On the other hand, the issue with identity reduces to the
non-categoricity of classical logic in the paper by Pavicic
and Megill to which I have tried to direct attention.



>
> Sometimes it is well-known in advance that you have
> one intended model in mind, and that the theory's value
> is only in helping you prove things about that model.
> Other times, the theory itself is important and all its
> various models are more or less created equal.

Is this a FACT?

If you comprehend anything I have written above, then you
will realize that FACTICITY collapses Russell's distinction
between extension and intension.

The utility of an assertion rests with its facticity. Most
people speaking of a truth do so in the context of an
assertion as a reliable statement of fact.




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