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

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From: Nam Nguyen on 3 Apr 2010 12:51

Nam Nguyen wrote:
> Nam Nguyen 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.
>>
>> Let N be the set of natural numbers and R the set of standard reals.
>
> Please disregard my mentioning about the reals (R) here. Thanks.
>
>> Let a natural number n be expressed as n = d0d1d2...dn, where d's are
>> the decimal digits. Let's also define the following functions:
>>
>> f1: N -> N, f1(n=d0d1d2...dn) = dn...d2d1d0
>> f2: N -> N, pE(n) = p, where p is the greatest prime <= the even n
>> [assuming n >= 0].

Of course it was meant: "[assuming n > 0]". Also since we'd apply
f1() only to primes here, the issue of dn = 0 would be a moot point
for f1().

>>
>> Let S1 = {n | n is an example of GC}
>> Let S2 = {n' | n' is a counter example of GC}
>>
>> Note that at least one of S1, S2 must be infinite. Now if S1 is
>> finite of length l > 0, then there is an infinite sequence:
>>
>> Seq1: n1, n2, ..., nl, 0, 0, 0, ...
>>
>> where all terms are either in S1 or 0. (If S1 were empty, then all terms
>> are defined equal to 0). Similarly, an _infinite_ sequence Seq2 would
>> exist, where all terms are either in S2 or defined to be 0.
>>
>> Let's define the set S as:
>>
>> Let S = { m | max(f1(f2(nth-term-of-Seq2)),f1(f2(nth-term-of-Seq1))) }
>>
>> By Well Ordering Principle, S has a minimal number which would be
>> the called Un: the desired "unknown" natural number.
>>
>> To know the natural numbers then is to know the value of Un, which we
>> can not know.

From: Daryl McCullough on 3 Apr 2010 12:53

Newberry says...
>
>On Apr 2, 3:19=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:

>> Then, as I said, it's a true sentence that Gaifman's procedure does
>> not return true for.
>
>Gaifman's procedure is not applicable.

That's exactly right. That's why it makes no sense to identify truth
with provability (or any other procedure).

--
Daryl McCullough
Ithaca, NY

From: Marshall on 3 Apr 2010 19:16

On Apr 2, 6:38 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
>
> To repeat, what may be intuitive to one poster may be counterintuitive
> to another. And I see no reason to favor one poster's intution over
> another's, no matter what the standard theorists try to say.

I don't see any reason to pay much attention to anyone's
intuition, my own included. "Intuition" is just a fancy word
for "hunch." It might be something that suggests areas
to investigate, but as far as these sorts of discussions
go, it's the beginning of the process, not the end.

Marshall

From: Marshall on 3 Apr 2010 19:20

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?

Out of curiosity (not that you've even answered this question
in the past) what sort of thing *would* you accept as a
proof of consistency?

Marshall

From: Nam Nguyen on 4 Apr 2010 12:33

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.

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

>
> Out of curiosity (not that you've even answered this question
> in the past) what sort of thing *would* you accept as a
> proof of consistency?

It's possible that I missed your _specific_ question of the past.
You just have to cite the specific post where I missed your
"this question", otherwise it's impossible for me to make any
comment on this.

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].