Re: Godel cant tell us what makes a mathematical statement true

MoeBlee wrote:
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
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.

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'.

[It's minor, but let's cut the chase and let me ask you one simple question:
are (1) and (2) mappings?]

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.

Right. And what are the difference between them? Sheer subjective
interpretation/mapping to either True or False, right? What this
means, which you've either failed or resisted to notice, whatever
you could legitimately claim as a model of T1, I could legitimately
claim as *not* a model of T1.

If you don't believe me, specify one that you think as a model M1 of
T1 = {e < m}, and I will demonstrate to you that M1 is not a model of
T1. Are you up to that simple challenge?


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.

Correct. I've never disputed the 1st clause, so you didn't have to cite
it in the first place!

As for the 2nd clause, I've given in details the some detailed reasons
for my disputes using T1, (1) and (2). I have yet to see your reasons why
my disputes are wrong! Why don't you take my proposed simple challenge
above, and I think could settle the issue of subjectivity of model truths.

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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