'When Are Relations Neither True Nor False? [Logic]

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From: Nam Nguyen on 4 Apr 2010 16:20

Nam Nguyen wrote:
> Marshall wrote:

>> Why do you think the existence of a specific unknown number
>> should have anything to do with consistency?
>
> Because they (the syntactical proof of consistency and collectively
> many formulas about this unknown natural) both connote the same thing
> in meta level: impossibility of syntactical proof. If you can't prove
> a certain formula related to this number, you can forget about proving
> a consistency, syntactically speaking.
>
> [Imho, it could be said the the later epitomizes the impossibility of
> the former].

Perhaps I also meant the other way around.

From: Newberry on 4 Apr 2010 20:35

On Apr 3, 9:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Apr 3, 6:54=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:
> >> Newberry says...
>
> >> >If it absolutely certain that PA is consistent why don't we formalize
> >> >the reasoning?
>
> >> It has been. It's easily formalized in ZFC.
>
> >I do not know why we are going through this circle again.
>
> >Look it is very simple. All you have to do is to divorce
>
> >~(Ex)(Ey)(Pxy & Qy) (1)
>
> >from
>
> >~(Ex)Pxm (2)
>
> >[No need to repeat that m is the Goedel number of (1).] Then there is
> >no reason why (2) could not be proven.
>
> It can be proven. Just not in PA.

Cool. So we know that the search for a proof of Goedel's sentence will
never terminate. Can we apply this knowledge to Diophantine equations?

> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -

From: Nam Nguyen on 4 Apr 2010 21:12

Newberry wrote:
> On Apr 3, 9:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:

>> It can be proven. Just not in PA.
>
> Cool. So we know that the search for a proof of Goedel's sentence will
> never terminate. Can we apply this knowledge to Diophantine equations?

I don't know the exact wording but there's a saying in mathematics that
we could not go backward forever in proofs: things have to _stop_ somewhere
such as reasoning framework, axioms, etc...

I don't think the "standard theorists" would insist on not stopping.
Unfortunately by not being humble on what they can possibly know in
reasoning, they inadvertently allow themselves defend-less against
infinite regression of truth and provability.

It wouldn't be a surprise if we learn there were those who would
defend physics against SR to the bitter end. I think there are those
today who'd similarly defend "the natural numbers" foundation in
mathematical logic - to the bitter end - even though to no avail.

Courageous perhaps. But kind of a "sad" story.

From: Marshall on 4 Apr 2010 21:49

On Apr 4, 6:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>
> It wouldn't be a surprise if we learn there were those who would
> defend physics against SR to the bitter end.

This is somewhat off-topic, and I may be misreading you,
but am I to understand from this sentence that you are
of the opinion that special relativity is somehow incorrect,
or not the best currently available theory?

Marshall

From: Marshall on 4 Apr 2010 22:05

On Apr 4, 9:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Marshall wrote:
> > On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> Let me put to rest the idea we know enough about the natural numbers,
> >> to prove important thing such as the consistency of PA. I'll do that
> >> by pointing out the existence of a specific unknown natural number.
>
> > Why do you think the existence of a specific unknown number
> > should have anything to do with consistency?
>
> Because they (the syntactical proof of consistency and collectively
> many formulas about this unknown natural) both connote the same thing
> in meta level: impossibility of syntactical proof. If you can't prove
> a certain formula related to this number, you can forget about proving
> a consistency, syntactically speaking.

Both of these sentences are just a restatement of the position that
I asked you for support of. My question is not, do you consider
a proof of consistency of PA and the existence of an unknown natural
number to be mutually incompatible. You have clearly stated so.
Rather, my question is WHY do you see them as mutually
incompatible. Is this some theorem of which I am unaware?
If so, can you provide a reference? Or is it some theorem
that you yourself have proved? If so, can you provide the
proof for our inspection?

Is it only *syntactic* consistency proof that are so affected?
What about model-theoretic proofs of consistency. I seem
to recall that you do not accept them, but I could be wrong.

Does this mutual incompatibility generalize, or is it something
specific to the natural numbers?

> [Imho, it could be said the the later epitomizes the impossibility of
> the former].

Can you explain how?

> That aside, it's actually my position that it's impossible to
> to syntactically prove a consistency: simply because the rules
> of inference won't let us do that; hence it's a _delusion_ that
> we could have any "sort of thing" that we could "accept as a
> proof of consistency"!

> [That's why I'd would be surprised if in the past I had said something
> that has caused you to think there be a criteria to accept a proof
> of inconsistency].

As far as I am aware, you have not given the conditions under
which you would consider the consistency of a theory to be
proven. As far as proving INconsistency, I was not under the
impression that you disagreed with the usual method of
deriving a contradiction. For the sake of completeness and
clarity, and because I suspect you meant "consistency" above
where you typed "inconsistency" would you clarify if/how you
consider it possible to prove a theory inconsistent.

Marshall