Le 29-août-07, à 23:11, Brent Meeker a écrit :
>
> Bruno Marchal wrote:
>>
>> Le 29-août-07, à 02:59, marc.geddes.domain.name.hidden a écrit :
>>
>>> I *don't* think that mathematical
>>> properties are properties of our *descriptions* of the things. I
>>> think they are properties *of the thing itself*.
>>
>>
>> I agree with you. If you identify "mathematical theories" with
>> "descriptions", then the study of the description themselves is
>> metamathematics or mathematical logic, and that is just a tiny part of
>> mathematics.
>
> That seems to be a purely semantic argument. You could as well say
> arithmetic is metacounting.
? I don't understand. Arithmetic is about number. Meta-arithmetic is
about theories on numbers. That is very different. Only, Godel has been
able to show that you can translate a part of meta-arithmetic into
arithmetic, but that is not obvious (especially at Godel's time when
the idea of "programming" did not exist). Obvious or not the
disctinction between metamathematics and mathematics is rather crucial.
It is as different as the difference between an observer and a reality.
>
>> After Godel, even formalists are obliged to take that distinction into
>> account. We know for sure, today, that arithmetical truth cannot be
>> described by a complete theory, only tiny parts of it can, and this
>> despite the fact that we can have a pretty good intuition of what
>> arithmetical truth is.
>
> But one would not expect completeness of descriptions.
Why? After all complete theories exist (like the first order theory of
real numbers for example). Incompleteness of ALL axiomatizable theories
with respect to arithmetical truth has been an unexpected shock.
Hilbert predicted the contrary.
> So the incompleteness of mathematics should count against the
> existence of mathematical Truth - as opposed to individual
> propositions being true.
I don't understand. Incompleteness of a theory is understandable only
with respect to some interpretation or model, that is notion of truth.
I do follow Godel on this question.
>
> Doesn't it strike you as strange that arithmetic is defined by formal
> procedures,
Only a *theory on* arithmetic or number is defined by formal procedure
(and does constitute an abstract machine).
> but when those procedures show it to be incomplete, mathematicians
> resort to intuition justify the existence of some whole? Theology
> indeed!
I don't understand. All mathematicians (except few minorities like
ultrafinitists) accept the notion of arithmetical truth, which can be
represented by the set of all true sentences of arithmetic (or to be
even more specific, it can be represented by the set of godel numbers
of the arithmetical sentences). But no theory at all can define
constructively that set. That set is not recursively enumerable. No
algorithm can generate it.
A rich lobian machine, like a theorem prover for a theory of set like
Zermelo-Fraenkel, can define that set, but still not generate it, and
it can be proved that this remains true for all the effective extension
(where an extension is effective when the extension is still an
axiomatizable theory.
So yes, arithmetical truth is a purely theological matter for a simple
lobian machine like Peano Arithmetic, but is just simple usual math
(despite non effectivity, but this you get once you accept classical
logic) for a super-rich lobian machine like ZF.
Although sometime you say correct thing in logic, I get the feeling
that you miss something about incompleteness ... (to be frank). Are
you aware that the set of true arithmetical sentences is a well defined
set in (formal or informal) set theory, yet that it cannot be generated
by any (axiomatizable) theory.
(note: Axiomatizable theory = theory such that the theorems can be
generated by a machine. You can take this as a definition, but if you
know the usual definition of "axiomatizable theory", then this is a
consequence by a theorem due to Craig).
I have to go. I will say more to David tomorrow.
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Thu Aug 30 2007 - 10:49:36 PDT