Re: Godel's Theorem and Model Theory
- From: poopdeville@xxxxxxxxx
- Date: Mon, 30 Jul 2007 15:10:18 -0700
On Jul 27, 12:42 am, Keith Ramsay <kram...@xxxxxxx> wrote:
On Jul 26, 10:01 am, george <gree...@xxxxxxxxxx> wrote:
|In this context it really would be smart to just STOP saying
|"true(period)",period, and start ALWAYS talking about truth
|IN A MODEL. Outside a model, what matters is what's provable,
|NOT what's true. Of course, under this paradigm, things that are
|provable are going to be in true in every model, so you might want
|to still call them true. That would be a mistake. It would be like
|calling
|a human being an animal. Yes, people are animals, and yes, theorems
|are true, but the point is, they are so MUCH MORE THAN JUST that.
I wouldn't recommend this advice. I mean, if the
original poster finds it helpful, by all means go
ahead and think this way.
It seems to be a common problem, however, to get
stuck imagining that the concept of truth is
dependent on the concept of "model". If one then
tries to retrace one's steps, to work out a
logical sequence of definitions of terms, one
keeps getting stuck wondering how it's possible
to specify a model while being unable to talk
about truth of any kind (but only provability).
This is a metaphysical and epistemological issue. I am sympathetic to
model relativism, as it is a natural consequence of a solipsistic
epistemology. (Allow me to be flaky: we can think of sense
impressions as being analogous to syntactical forms. The "real world"
is one of many non-isomorphic "models" of the "theory" formed from
these syntactical forms. While everything in quotation marks must be
fleshed out, this is a tidy, unifying description of my
understanding.)
One always knows what integers are long before one
has any of these concepts of mathematical logic in
mind. This isn't just a quirk of human psychology
or of the educational system either; the concepts
of mathematical logic are dependent on the concept
of integer (or natural number at least), or some
equivalent stand-in, like the concept of a finite
string of characters.
I'm not so sure about this. In particular, I think it is not
logically necessary for the standard integers (or naturals) to be the
first structure we become familiar with.
This is tricky to phrase. The standard K-12 curriculum introduces
arithmetic. As a child struggles to understand it, he might try to
come up with a "mental model" of the numbers. But there is no a
priori reason why it must be isomorphic (in principle) to the standard
model. Indeed, it seems that many people have non-standard models of
arithmetic in mind, though without the tools of mathematical logic,
they are unable to understand the consequences. Notice how many
people claim that "infinity" is a number.
My objection comes from two aspects of Wittgensteinian thought: his
thoughts on "rule-following considerations" and the Private Language
Argument. At the very least, at the K12 level, the standard and non-
standard models are linguistically indistinguishable.
Once one knows what natural numbers are, what
addition and multiplication of them are, and so on,
one can then understand what this kind of number-
theoretic sentence being referred to means, and
what it means for it to be true, without having
any notion of what a "model" is.
I agree with this. Everyone who's been through the K-12 system will
know a natural when they see it. But it isn't clear that they can't
throw hyperfinite numbers into their conception of the numbers.
'cid 'ooh
.
- References:
- Godel's Theorem and Model Theory
- From: poopdeville
- Re: Godel's Theorem and Model Theory
- From: george
- Godel's Theorem and Model Theory
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