Re: Torkel Franzen on truth

george wrote:
On Dec 7, 11:27 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
My personal opinion is that mathematical respect flows out of the
completeness theorem: IF it is inconsistent, THEN THAT MUST be
provable.
Why "must", a _subjective_ verb?

That is NOT a subjective verb. That is an OBJECTIVE must.

There is nothing as an "objective must" in mathematical reasoning.
Given a T and an F in L(T), F is or is not a theorem of T. No "must"
is *required* here, of course!

It really must. Just as surely as 2+2 MUST be 4 and not 3 or 5.

One could *easily* come up with a formal system in which 2+2 is 3.
So your "as sure as" is not a mathematical certainty.


What happens if such inconsistency proof
is beyond human reach?

You have no way of knowing that ANYthing is beyond human reach.

On the contrary, We always know *something* (like the *existence of something*)
that is beyond human reach - even human with *an* infinite-knowledge!
I give you 2 hints:

- Think of the phrases such as "cardinality" and "Power Set Axiom"
- Any reasoning framework must necessarily be based on knowledge
of certain cardinality. (In the case of FOL, for example, proofs
are based on *finite* cardinalities.)

You don't know how many humans there are going to
eventually be. You don't know how big human brains are eventually
going to get. It is at least theoretically possible that there is no
upper limit on the number of humans, that the human population will
keep increasing exponentially throughout all currently known galaxies
and that just about the time it appears we are about to fill them all
up,
some sort of cosmic revelation will occur and more space will become
available. Or a similar sort of revelation will cause there to be an
infinite
number of humans.


Or cause some individual finite-brained human parents
to have human children with an infinitary brain-part.

"human ... with an infinitary brain-part" is at best imaginative
(i.e. not real) and at worst is ignorant of what we currently know
about our Universe: in a finite number years, it either would vanish
in a Big Crunch, or would settle down in a state in mode in which
the only thing left is not matter but quanta at lowest possible
states, and in which case no "human" could possibly exist!


"beyond human reach" IS JUST STUPID.

Think twice before capitalizing any word! What word one capitalizes
might de-capitalize one's knowledge on the backfire!


The issue in any case IS NOT what "humanity" can do BUT RATHER what
A TURING MACHINE can do.

In mathematical reasoning context, "what A TURING MACHINE can do" is just
another way of saying "there exists ...". But since mathematical reasoning
is meaningless without human beings [think about the word "reasoning"!
Who would be the reasoner(s)?], brining the catch-phrase "TURING MACHINE"
doesn't make this case less about "human"!


Therefore, the BURDEN of proof rests ALWAYS UPON people expressing
doubts about consistency.
But what happens if the theory is genuinely consistent but it's *impossible*
to know that?

It by definition CANNOT be impossible.
If a recursive set of first-order sentences is inconsistent then THERE
IS A FINITE proof of that. That is a FACT. That is a THEOREM (the completeness
theorem).

Why is it that I talked about _consistency_ and you about _inconsistency_?

Sorry your ignorant ass was still too ignorant to know that after all
these years.

Seems you've not changed in all these years: vulgar in communication, impulsive
in quick reaction to see what the opponents might be *really* talking about!

You just plain should've QUIT a long time ago. You recently promised
to. Please hold yourself to it.

I do quit sharing the plan of discussing with this forum on the "revamp"
of FOL, mainly because people don't seem to care much about any change
to FOL. But this is Holiday Season and I got some spare time so I don't
think it's much of a "violation" of my pledge, to occasionally jump in
when I think the poster is too wrong on something I think important.
(Of course I stand to be corrected where I'm technically wrong).
.

Relevant Pages

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  • Re: Torkel Franzen on truth
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