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

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From: Nam Nguyen on 2 Apr 2010 20:32

Nam Nguyen wrote:
> Jesse F. Hughes wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> Daryl McCullough wrote:
>>>> Newberry says...
>>>>
>>>>> What I was getting at is how we know that the system is consistent.
>>>> In the case of PA, it's because we know that everything it says
>>>> about the natural numbers is true.
>>> What is the natural numbers collectively? One in which "There are
>>> infinitely many examples of GC" is true? Or one in which "There are
>>> infinitely many counter examples of GC" is true? Are you sure you
>>> know what you're talking about, in talking about the "natural
>>> numbers"?
>>
>> You do not have to know everything about N in order to know what N is.
>>
>> I don't know whether the moon has hundreds of craters 100 yards across
>> or thousands. Or dozens. Yet, I'm sure I know what the moon is. And
>> I also know certain things about the moon.
>>
>
> Right. We'd have an _incomplete_ knowledge about the moon, in this example.
> That would enable us to know, say, there's at least 1 crater on its
> surface.
> But that wouldn't allow us to know truth of one simple quantification
> predicate statement "All the moon craters are at least 100 yard across".
>
> The issue here is not "what N is" per se. One could easily _define_ N as
> a model of some modulo arithmetic. The issue is how we can know the
> _syntactical consistency_ of PA, using strictly definition of syntactical
> consistency.
>
> Just defining "things" that we don't completely know does not _prove_
> in meta level that PA is syntactically consistent.
>
> We shouldn't invent something that isn't compatible with rules of
> inference, and then "prove" things that the rules themselves
> can *not* prove.

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.
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].

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: Nam Nguyen on 2 Apr 2010 20:34

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].
>
> 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: Transfer Principle on 2 Apr 2010 21:38

On Apr 1, 5:30 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Aatu Koskensilta wrote:
> > How do I know that Peano arithmetic is consistent? I know it the way I
> > know any mathematical theorem I have personally proved.
> So what you're saying is you just _intuit_ PA system be consistent,
> no more no less. Of course anyone else could intuit the other way too!

I agree wholeheartedly. One poster's intuition is another poster's
counterintuition, and just because the standard theorists believe in
all
the theorems of PA that they "have personally proved," it doesn't mean
that all posters must share in that belief.

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.

From: Transfer Principle on 2 Apr 2010 21:53

On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Nam Nguyen wrote:
> > We shouldn't invent something that isn't compatible with rules of
> > inference, and then "prove" things that the rules themselves
> > can *not* prove.
> To know the natural numbers then is to know the value of Un, which we
> can not know.

I can come up with similar "unknown" natural numbers, even easier
than Nguyen just did.

Let Un = n, if n is the smallest counterexample to GC
= 0, if no such counterexample exists

Or even:

Let Un = 1, if a counterexample to GC exists
= 0, if no such counterexample exists

The so-called "crank" WM often came up with similar examples, such
as Un = the googol'th digit of pi. Also, I can consider yet another
example, f2(n), where n = Moser's number.

Since PA, if consistent, must be incomplete, it's possible that we can
never determine the value of some of these unknown numbers.

Although not knowing the value of some of these unknown numbers
doesn't in itself prove that PA is inconsistent, we must still
remember
Ed Nelson and his skepticism that PA is consistent. It's possible that
we may be able to prove both Un = 1 and Un = 0 for one of these
definitions of Un above. The proof of ~Con(PA) on which Ed Nelson is
working involves some of these large numbers.

I've said it before and I'll say it again -- every standard theorist
believes
it's _theoretically_ possible that PA is inconsistent, but no standard
theorist believes it's _actually_ possible that PA is inconsistent.

From: Nam Nguyen on 2 Apr 2010 22:30

Transfer Principle wrote:
> On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Nam Nguyen wrote:
>>> We shouldn't invent something that isn't compatible with rules of
>>> inference, and then "prove" things that the rules themselves
>>> can *not* prove.
>> To know the natural numbers then is to know the value of Un, which we
>> can not know.
>
> I can come up with similar "unknown" natural numbers, even easier
> than Nguyen just did.
>
> Let Un = n, if n is the smallest counterexample to GC
> = 0, if no such counterexample exists
>
> Or even:
>
> Let Un = 1, if a counterexample to GC exists
> = 0, if no such counterexample exists

The problem with your version of "Un" is should there
be a counter example of GC (and that's still a possibility!),
then your Un is known (in principle).

My definition of Un would work - in all cases!