Re: Which theories hold a true but unprovable statement?
- From: "Rupert" <rupertmccallum@xxxxxxxxx>
- Date: 9 Feb 2007 02:30:24 -0800
On Feb 9, 6:05 am, "george" <gree...@xxxxxxxxxx> wrote:
On Feb 7, 6:34 pm, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:
Before you can make sense of the idea of
a sentence being true, you
have to specify a model for the theory.
Self-refuting as usual. I continue to be amazed by who
graduated when I didn't.
What I said is quite correct. In the study of first-order languages,
to define the notion of truth for a sentence in a given first-order
language, you have to specify in which model. This is not self-
refuting at all.
Consider the theory of real-closed fields, for example.
This makes the EXACT OPPOSITE of the point you were
TRYING to make.
No, it does not. Provability in that theory happens to equivalent to
truth in any one of the models. This does not affect the point I am
trying to make.
This theory is recursively axiomatizable,
It's also DECIDABLE, DUMBASS.
Yes, I know that. I really don't see the call to insult me.
and a sentence is true in the model consisting
of the real numbers if
and only if it is provable in the theory.
Therefore you NEVER NEED TO CONSIDER ANY models
of the theory in order to discern whether something is true:
In order to discern whether something is provable in the theory. The
notion of "truth" doesn't make sense except relative to some model. It
happens to be the case that provability, and truth in any one of the
models, are equivalent.
YOU JUST PROVE IT. The model of the real numbers
DECIDES EVERYthing. EVERYthing is either true or false
in that model (models are complete BY DEFINITION).
Therefore your "only if" direction implies that the model's
decision WILL ALWAYS BE ACCOMPANIED by a corresponding
proof.
The OP really did NEED somebody to explain to him that
truth comes from models and not from theories.
He seemed to me to have the point that provability is not equivalent
to truth down pretty well, given his understanding of Goedel's
theorem. He seemed to understand the case where the set of true
sentences is not recursively enumerable. I wanted to show him a case
where the set of true sentences is recursive, so that the Goedelian
phenomenon does not hold in this case. This directly addressed his
query.
If you want to emphasize other things, then you give your own
explanation. I don't see why you always have to come along and
obnoxiously try to pick holes in what other people say. Why don't you
try not being a tiresome prat?
You have
totally botched this, by picking a theory where truth really
DOES come from the theory. The point there is that THOSE
theories are trivial. They are not hard enough to be worth
bothering with, and are not powerful enough to do anything
as basic as Robinson Arithmetic.
Yes, but I wanted to explain that Goedel's theorem only applies to a
very special kind of theory. This seemed to me to be the point that he
did not understand.
.
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