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

Prev: geometry precisely defining ellipsis and how infinity is in the midsection #427 Correcting Math
Next: Accounting for Governmental and Nonprofit Entities, 15th Edition Earl Wilson McGraw Hill Test bank is available at affordable prices. Email me at allsolutionmanuals11[at]gmail.com if you need to buy this. All emails will be answered ASAP.

From: Marshall on 6 Mar 2010 11:21

On Mar 6, 2:43 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Marshall <marshall.spi...(a)gmail.com> writes:
> > For me, Nam has mostly moved into the same category as AP.
>
> Come now, even if you don't find Nam's posts worth reading comparing him
> to Archimedes Plutonium is surely excessively harsh.

There is indeed a clear difference in quality between NN and AP,
however this quality does not translate into treating their posts
much differently.

Marshall

From: Jesse F. Hughes on 6 Mar 2010 11:16

"Jesse F. Hughes" <jesse(a)phiwumbda.org> writes:

> Newberry <newberryxy(a)gmail.com> writes:
>
>>> > But I would be curious if any real mathematical proof goes
>>> > according to your example. Prime numbers are a subset of the natural
>>> > numbers, but how do you make the implication hold the other way?
>>>
>>> I don't recall the argument, but I don't see why you should be
>>> surprised that a statement like (a) may be provable.
>>
>> Here he devil is in the details. We need to see the actual proof to
>> find where the problem is.
>
> Claims of the form (Ex)Px -> (Ex)(Px & Qx) are not that hard to come
> by. For example, let
>
> Px <-> x is an ordered pair of integers (a,b) and b*y = a
>
> Qy <-> x is an ordered pair of integers that are coprime.
^^ I meant Qx, obviously.
>
> Then, for any real number y, we can prove
>
> (Ex)Px -> (Ex)(Px & Qx)
>
> I leave you to find the proof and see where the "problem" is.
--
"So yeah, do the wrong math, and use the ring of algebraic integers
wrong, without understanding its quirks and real mathematical
properties, and you can think you proved Fermat's Last Theorem when
you didn't." -- James S. Harris on hobbies

From: Marshall on 6 Mar 2010 11:29

From: Jesse F. Hughes on 6 Mar 2010 11:29

"Jesse F. Hughes" <jesse(a)phiwumbda.org> writes:

> Newberry <newberryxy(a)gmail.com> writes:
>
>>> > But I would be curious if any real mathematical proof goes
>>> > according to your example. Prime numbers are a subset of the natural
>>> > numbers, but how do you make the implication hold the other way?
>>>
>>> I don't recall the argument, but I don't see why you should be
>>> surprised that a statement like (a) may be provable.
>>
>> Here he devil is in the details. We need to see the actual proof to
>> find where the problem is.
>
> Claims of the form (Ex)Px -> (Ex)(Px & Qx) are not that hard to come
> by. For example, let
>
> Px <-> x is an ordered pair of integers (a,b) and b*y = a
>
> Qy <-> x is an ordered pair of integers that are coprime.
>
> Then, for any real number y, we can prove
>
> (Ex)Px -> (Ex)(Px & Qx)
>
> I leave you to find the proof and see where the "problem" is.

Note that this sort of reasoning is used in a very familiar proof.

Let

Px <-> (Ea,b) (x = (a,b) in N x N and a^2/b^2 = 2)

Qx <-> (Ea,b) (x = (a,b) in N x N and a and b are coprime)

Then the traditional proof that sqrt(2) is irrational goes something
like this:

(Ex)Px -> (Ex)(Px & Qx)
~(Ex)(Px & Qx)
---------------------------
~(Ex)Px

So, let us forget FLT and focus on this well-known and simple (but
elegant) bit of reasoning.

--
"So now you see a math person coming out to talk about *his* program
which is fast as he says it can count over 89 billions primes in less
than a second. How is that objective? It's childish."
-- James S. Harris, on objective facts.

From: Nam Nguyen on 6 Mar 2010 11:44

Alan Smaill wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Jesse F. Hughes wrote:
>>
>>> So, you want to deny that Goedel's theorem is true.
>> The "crank" tends to assert Goedel's theorem is false.
>> The "standard theorist" would insist GIT is true.
>>
>> That leaves the "rebel" the only side who observes the
>> method in Godel's work is invalid.
>>
>> Except for the relativists, why should we care about invalid
>> truth or falsehood?
>
>
> Because the notions of "truth" and "validity" are not beyond dispute, and
> maybe we/you/I have got those wrong.

In a high level that makes sense, imho. But I'm not after the "absolute"
truth, validity, or correctness. It's the _process and methods_ of
reasoning that I'm really ... really after.

My posting in threads after threads is a means to re-examine, evaluate,
and revise the current methods of reasoning in FOL in general and in
Incompleteness in particular. And my motivation of the re-examination
is basically a following of what Shoenfield said about mathematical logic:

"Logic is the study of reasoning; and mathematical logic is the study
of reasoning done by the mathematicians.

To _discover proper approach to mathematical logic_, we must therefore
_examine the methods of the mathematicians_."

[The highlights are mine].

For what it's worth, the long and short of it is what I've been doing for
years is simply trying to evaluate, for possible shortcomings, of the current
FOL regime and of anyone's reasonings I've come across in some way: myself,
posters in this and other fora, Torkel Franzen, Godel, Hilbert, etc...

I'm less interested for example whether or not, say, PA is consistent
but I'm interested in based from what existing and historical reasoning
backgrounds and by what methods one would logically conclude - with no
emotion or "belief" - PA is consistent - or not.

For instance, here's what MoeBlee said earlier:

>> at least we do know how to formalize PRA. And if we cannot be
>> confident that PRA is consistent then I don't know what substantive
>> mathematical theory we could be confident as to consistency.

This passage, imho, is a typical example of where one would forget that
reasoning is a _process_ (as Shoenfield alluded to above), _not_ a mixture
of intuitive and religious-like _beliefs_. If we agree what "formalize"
means, what methods of reasoning are, what consistency and inconsistency
means, then as a fact it's either PRA is or isn't inconsistent strictly
based on the definitions and the methods, and it's either we do know *or*
don't know about the fact. Period. "Confidence" isn't an issue nor does
it have a role here, in the process of reasoning.

The long and short of it is from what I've gathered, Godel's "proof" is
invalid as a meta proof, because the methods used in reasoning failed
in one of the few intuitive principles about the methods of logical
reasonings. Let me cite the 2 obvious principles:

(1) Principle of Consistency:

No methods shall lead to contradictory conclusions.

(2) Principle of (Method) Compatibility:

If each of any 2 equivalent conclusions is expressed in an independent
method then it shall not be the case that one method would lead to
its perspective conclusion, while the other method wouldn not lead to
the other counterpart conclusion.

In Godel's work, the knowledge of the natural numbers as a sheer intuition
or as a model of a FOL formal system is incompatible with rules of inference
in so far as both of them are methods of reasonings (and they are).

The reason being is "undecidability" of formal systems is purportedly
"equivalently" defined by both methods, but the natural number would lead
to proof of consistency while the rules of inference method would not!