Re: Godel cant tell us what makes a mathematical statement true
- From: MoeBlee <jazzmobe@xxxxxxxxxxx>
- Date: Fri, 5 Sep 2008 10:58:39 -0700 (PDT)
On Sep 4, 11:11 pm, Nam Nguyen <namducngu...@xxxxxxx> wrote:
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"Rigorously defined concept" still doesn't help that between any 2 thinkers,
a rigorously defined concept, not a matter of intuition.
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.
The word 'function' where I wrote it is precisely correct. Whether in
informal mathematics, informal Z set theory, or formal Z set theory, I
mean exactly 'function'.
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]
I don't recall the specifics of your notation above, but I surmise
that you mean to give two examples - one of model that is a model of
some theory T1 and the other that is not 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!
Again, I don't propose a notion of 'a sentence being true', but rather
of 'a sentence being true PER a model' (putting aside, for the time
being, also the informal notion of a finitistic sentence being true).
There's no subjectivity in the fact that one model is a model of a
given theory T1 and that another model is not a model of a given
theory T1.
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..
Then we got a more precise formulation.
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?
No, wrong. I said people may diverge as to what models they wish to
study, and I could add that people may diverge as to what models they
think capture certain concepts or are best analogies for certain
empirical facts and all kinds of things like that. But that doesn't
refute that in FORMAL mathematics there is anything subjective about
'true per model'.
I'll say it again just so that you are crystal clear as to what I am
saying: In a formal mathematical theory (such as formal Z set theory),
we define 'S is true per M'. Then whether a given sentence S is or is
not true in a given model M is not subjective.
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.
See mine now.
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"?
I don't have one. Perhaps if forced to give one I would say "a member
of the universe of an interpretation of a formal language." Though,
I'd probably want to think about that more before committing to it.
(Notice that circularity is not at issue since by saying that an
interpretation is a mathematical object, that is not part of any
DEFINITION of 'interpretation'.) Anyway, I can reformulate my remarks
so that the word 'object' is not used. An interpretation for a
language is a function. In a given formal theory, such as Z set
theory, we define 'function', and we do so without need of the word
'object'. I use the word 'object' only to impress upon you the CONTEXT
in which 'interpretation' is given a formal definition.
and of a
"rigorous mathematical definition"?
A certain kind of sentence in a formal language. The specifics include
such things as satisfying the criteria of eliminability and non-
creativity. Those are discussed in many a textbook on mathematical
logic. I've even stated them and explained them in many posts I've
made in various threads.
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!
AGAIN, I'm not talking about "true" without regard to a given model.
What kind of "rigor"
of truth definition is that when a simple statement could have a negating
truth value, as one wishes?
I don't know, since that is not the way the ordinary recursvie
definition works.
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!
AGAIN, I don't refer to a "statement is true" but rather to a
"statement is true per a given model".
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!
"The key point of the debate" as you now define it. And now its the
notion of "strength". When did you decreee that that is the "key
point"? And I don't even know what you mean by "as strong as" in this
context. Both concepts ("S is provable in system T" and "S is true per
model M") are given rigorous formal definitions and the question of
whether a sentence S is provable in a system T is not subjective, and
whether a sentence S is true in a model M is not subjective. You've
not refuted that, and that WAS the key point.
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"?
Your reply there goes RIGHT PAST my remark you just quoted above.
AGAIN, for about the sixth time now: I refer to "sentence S is true
per model M" (or more often said as "sentence S is true in model M")
and not to "S is true". (Putting aside the special case of finitistic
statements, which is another matter of discussion, and and INFORMAL
one, and a matter that is optional in this present context.)
ONCE MORE:
It is not subjective whether a sentence S is provable in a system T;
and it is not subjective whether a sentence S is true in a model M.
And THAT was the proposition I have always stated, and it is the
second clause of that proposition that you have disputed.
MoeBlee
.
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