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

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From: Tim Golden BandTech.com on 31 Mar 2010 10:23

On Mar 30, 11:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Tim Golden BandTech.com wrote:
> > On Mar 27, 11:25 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> Tim Golden BandTech.com wrote:
> >>> On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >>>> Nam Nguyen wrote:
> >>>>> Alan Smaill wrote:
> >>>>>> Nam Nguyen <namducngu...(a)shaw.ca> writes:
> >>>>>>>> Seriously, if you could demonstrate a truly absolute abstract truth
> >>>>>>>> in mathematical reasoning, I'd leave the forum never coming back.
> >>>>>>> If you can't (general "you") then I'm sorry: my duty to the Zen council,
> >>>>>>> so to speak, is to see to it that "absolute" truths such as G(PA) is a
> >>>>>>> thing of the past, if not of oblivion.
> >>>>>> one day you will realise that your duty to the Zen council
> >>>>>> is to overcome your feeling of duty to what is purely subjective ...
> >>>>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which
> >>>>> you'd need to overcome - someday. Each of us (including Godel) coming to
> >>>>> mathematics and reasoning has our own subjective "baggage".
> >>>>> Is it FOL, or FOL=, that you've alluded to? For example.
> >>>> Note how much this physical reality has influenced and shaped our
> >>>> mathematics and mathematical reasonings. Euclidean postulates had their
> >>>> root in our once perception of space. From P(a) we infer Ex[P(x)]
> >>>> wouldn't be an inference if the our physical reality didn't support
> >>>> such at least in some way. And uncertainty in physics is a form
> >>>> relativity.
> >>>> The point is relativity runs deep in reality and you're not fighting
> >>>> with a lone person: you're fighting against your own limitation!
> >>>> Any rate, enough talk. Do you have even a single absolute truth you
> >>>> could show me so that I'd realize I've been wrong all along? Let's
> >>>> begin with the natural numbers: which formula in the language of
> >>>> arithmetic could _you_ demonstrate as absolutely true?
> >>> There is a fairly straightforward construction that can yield both
> >>> boolean logic and continuous higher forms, and even a lower form that
> >>> I will call universal.
> >>> Constrain the real numbers to those values whose magnitude is unity.
> >>> We see two options
> >>> +1, -1 .
> >> It's relative as to how many real numbers one could "constrain". So
> >> "constraint" is a relative notion, not an absolute one.
>
> >> In any rate, in all the below (including the URL) I still couldn't
> >> see an absolute truth. Could you state such truth here?
>
> > By accepting the generalization of sign the existence of dimension
> > follows directly.
> > That is the most absolute truth that I've come up with.
>
> So what would happen if one doesn't accept the "the generalization
> of sign"? Would we get a relative truth, or an absolute falsehood?

An argument can be made that by denying the generalization of sign
that you then have denied sign altogether, for the two-signed real
number is but an instance of the generalization. One would need a
conflict of the three-signed number which does not exist for the two-
signed number, but no such conflict exists. Instead what we witness is
that the three-signed number is the complex number, the next after the
real number, even historically speaking.

From a physical perspective denial of sign is actually fairly
sensible. We can see the physical world does not really yield negative
distance using measuring rods, nor negative temperatures when absolute
zero is accepted, nor negative masses. Some of the few things that do
seem to have positive and negative happen to be discrete (e.g. charge)
and so are actually consistent with the spherical one dimensional
construction.

Nam, your annoying responses do not expose any processing on your
part. That sign and dimension are closely linked is the most profound
portion and now we are approaching it, or at least I am approaching it
here as paragraph three where these two prior paragraphs are exposing
a divergence. The fact that we exist in a three dimensional space is
well developed, and that time is a feature of existence as well. Here
is the deepest marriage of sign, for the polysign numbers allow
natural arithmetic support for spacetime, including unidirectional
time. Polysign numbers dictate that spacetime is structured. This then
becomes a paradigm consistent with Maxwell's equations and is the edge
that I am working at. We have the freedom to manually fix the deck of
cards for the card trick. We are even free to build our own cards;
draw whatever you want on them. Still I am missing at least one card.

- Tim

From: Aatu Koskensilta on 31 Mar 2010 10:30

Newberry <newberryxy(a)gmail.com> writes:

> No.

I see. So you agree that for any formal theory T which is an extension
of Robinson arithmetic, either directly or through an interpretation,
and in which we can express statements of the form "the Diophantine
equation D(x1, ..., xn) = 0 has no solutions", there are infinitely many
true statements (of the form "the Diophantine equation D(x1, ..., xn) =
0 has no solutions") that are unprovable in T if T is consistent?

> You never answered my question what you ment by "Goedel."

You have asked me what I mean by "Goedel"?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Daryl McCullough on 31 Mar 2010 10:35

In article <87eij05ztb.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta says...
>
>Newberry <newberryxy(a)gmail.com> writes:
>
>> No.
>
>I see. So you agree that for any formal theory T which is an extension
>of Robinson arithmetic, either directly or through an interpretation,
>and in which we can express statements of the form "the Diophantine
>equation D(x1, ..., xn) = 0 has no solutions", there are infinitely many
>true statements (of the form "the Diophantine equation D(x1, ..., xn) =
>0 has no solutions") that are unprovable in T if T is consistent?
>
>> You never answered my question what you ment by "Goedel."
>
>You have asked me what I mean by "Goedel"?

When people write "Goedel", they usually mean "G�del".

--
Daryl McCullough
Ithaca, NY

From: Aatu Koskensilta on 31 Mar 2010 10:45

stevendaryl3016(a)yahoo.com (Daryl McCullough) writes:

> In article <87eij05ztb.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta
> says...
>
>>Newberry <newberryxy(a)gmail.com> writes:
>>
>>> You never answered my question what you ment by "Goedel."
>>
>>You have asked me what I mean by "Goedel"?
>
> When people write "Goedel", they usually mean "G�del".

This is true, but I don't think I've ever used the spelling "Goedel".

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Frederick Williams on 31 Mar 2010 15:45

Aatu Koskensilta wrote:

> This is true, but I don't think I've ever used the spelling "Goedel".

Thank heavens for the use-mention distinction!

--
I can't go on, I'll go on.