Re: Godel cant tell us what makes a mathematical statement true
- From: Nam Nguyen <namducnguyen@xxxxxxx>
- Date: Fri, 05 Sep 2008 06:11:23 GMT
MoeBlee wrote:
On Aug 22, 10:18 pm, Nam Nguyen <namducngu...@xxxxxxx> wrote:herbzet wrote:
MoeBlee already gave a good reply here. Truth in a structure is
a rigorously defined concept, not a matter of intuition.
"Rigorously defined concept" still doesn't help that between any 2 thinkers,
one could choose (1) while the other (2), which of course is a *matter
An intrepretation for a language is a mathematical function. Two
thinkers could diverge as to which mathematical function is more of
interest to them.
I'd use "mapping" instead of "function" since(1) and (2) aren't 1st order
expressions and "function" is a well known 1st order expression in ZF.
Let's talk about the "diverging" point you've just alluded above,
and review what (1) and (2) are (I wished you hadn't cut them out,
while discussing them!):
(1) ((e < m), R={(c,d}}) -> True [This basically says R is a model of T1]
(2) ((e < m), R={(c,d}}) -> False [This basically says R isn't a model of T1]
You could see clearly the subjectivity/relativity/intuitive-nature of
a 1st order mathematical truth by the fact that the left sides of '->'
are one, while the right sides are not! Mathematical used not to be
like that: given a fixed set R = {(c,d)} and a fixed formula 'e < m',
there used to be *only one* interpretation and hence only one truth value.
So what you said here basically doesn't dispel the intuitive nature of
a 1st order mathematical truth. In fact your "diverge" seems to be in
agreement with a mathematical truth's being a relative notion. Right?
They might have subjective, intuitive, or
philosophical reasons for preferring to study one interpretation as
opposed to another; they might even consider a particular
interpretation to be a reasonable, fruitful, or even a correct one.
See my respond above.
But the actual intrepretations themselves are mathematical objects;
they're functions. And the definition of 'the sentence S is true in
the model M' is a rigorous mathematical definition.
What is your definition of a "mathematical object"? and of a
"rigorous mathematical definition"? As I mentioned in the other thread,
a (sport) free agent is not necessarily free. So the 1st order formula
"X is free" could be either true or false - at will! What kind of "rigor"
of truth definition is that when a simple statement could have a negating
truth value, as one wishes? Contrast to that, the provability value of
"X is free" in {"X is free"} is fixed: no one could do anything about it,
right? A statement is being true is not as rigor as it's being provable
in a theory!
That aside, it seems you've missed the key point of the debate here:
it's not the rigor of any truth definition per se that's an issue;
it's that the truth-paradigm (based on intuition) isn't as strong as
the syntactical provability paradigm!
In brief, Moeblee was wrong here. What he seems to fail to realize is that
the very definition of true sentences would depend on the *intuition* of the
natural numbers, hence can not be a "rigorously defined concept"!
No, because I NEVER MENTIONED a definition of 'true sentence'. What I
have talked about is 'true sentence PER a given model'. And the
definition of THAT is rigorous. Of course, the definition may be
motivated by an intuition or sense or even conviction as to what it
means for a sentence to be true per a given state of affairs. But the
definition itself is entirely rigorous mathematics.
"Entirely rigorous"? As alluded to above, is a free agent free? or not free?
or it could be either way, defeating the notion of "rigor"?
--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.
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