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

MoeBlee wrote:
On Sep 8, 9:26 pm, Nam Nguyen <namducngu...@xxxxxxx> wrote:

Where did I indicate in my simple and rather straight forward paragraph
above that "{e < m} [is] being a model"? Didn't I state clearly
"specify ... a model M1 of T1 = {e < m}"?

I was unclear. That's why I ASKED.

Since you didn't seem to comprehend such simple English statement
(without me spelling out everything), let me assist you then.

But first let me tell you that you are an insufferably rude and quite
unjustifiablly arrogant little twit you are.

Typical reader and debater in this forum would understand what "T1 = {e < m}"
means but you don't: re require to use single quote (') as in 'e < m'
before you could see T1 as a theory. Why helping you to understanding
a simple statement (that almost everyone else beside you would understand)
be an act of being rude?


Let there be a language L(a,m,<) where a,m are individual constants
and < is is a binary predicate symbol. Let F be a formula as
F df= a < m, and let a theory T1 be defined as T1 df= {F}.

What you mean is that only non-logical axiom of F is 'a<m'.

You really have a reading comprehension problem, MoeBlee! Where did
I state that F is a theory?

That's fine, but I was not clear what you intended. So I ASKED you.

It's one think to ask for what you didn't understand, it's quite another
thing you misread what your opponent clearly said, and blamed your not
comprehending on your opponent. In other words, it's *your own fault*
that F's being a formula wasn't clear to you! (One more time, I stated
"Let F be a formula as..."! You really ought to take your time and
*read technical argument carefully*. And please, that's not an insult,
that's a fact!)

In return I got your stupid insult.

How would that be an insult when it's so simple and you should have
understood that a) "T1 = {e < m}" just means T1 is a theory, and b)
you shouldn't have said "...non-logical axiom of F..." when I clearly
stated F is a formula!

You got to realize how frustration one would have discussing technical
foundational issue with you because instead of making headway on
difficult issues of truth-subjectivity, what we end up with back-and-forth
mediocre-pendatic-misunderstanding such as "T1 = {e < m}" as-opposed-to
"T1 = {'e < m'}", etc...


Now could you construct for me a model M1 of T1? (That's *all* I was
asking for?)

Sure, I can perform this trivial exercise. You can perform it
yourself. Will you be providing some enlightening comment on this?

Nothing "major" at all: given your misunderstanding of some foundational
issues, I just want you to clarify what you think a model of T1 might be.


Here is a model of T1 (I'll use parentheses for tupes rather than
angle brackets):

Let 'A' stand for the universal quantifier.

{ (A {0}) (a 0) (e 0) (< {(0 0)}) }

Given you occasional misunderstanding, you ought to be a bit clearer,
and clarifying in advance some of the notations *you* used here:

- What is 'e' doing here?
- From what you typed above, how would "TRUE" and "FALSE" come to play?
In other words, from what you typed, how could you prevent one in
meta level saying "e < m" is *FALSE*?


By the way, I should have followed up on another matter. I gave an
analogy in a previous post. But my previous analogy still deserves
your response:

(1) Statement S is provable from set of statements G.

(2) Statement S is true in model M.

Why do you think (2) is subjective but not (1)?

I already stated that before. (It would help *if* you're interest in discussing
in the conversation, you should read other recent posts in the very same
thread, *on the very same issue*!). In any rate this here is my answer
to your question above, using T1 as an example of formal system:

>> Now, say, we both come across the a particular ordered-pair (c,d)
>> (which is, say, a ZF set). Now to you, your intuition might map
>> (c,d) to the formula (a < b), but to me my intuition would map
>> the very same set (c,d) to ~(a < b). So for the same set (c,d),
>> we now have 2 different models, simply because we have
>> *2 opposite but valid intuitions*! In brief, model is based on truth,
>> and truth can legitimately vary per (subjective) intuitions.

>> On the other hand, e.g., from T1 = {a < b}, we can prove the theorem
>> Ex(x < b). But each of us either knows or doesn't know the proof, but
>> we could never conclude *2 opposite theorems*. In brief, unlike model
>> truth, provability is based on syntactical rules and formulas.

*and*

>> 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!


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

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