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From: Nam Nguyen on 5 Nov 2009 02:33
Nam Nguyen wrote:
> Marshall wrote:
>> On Nov 4, 10:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>> Do you really think the notion of natural numbers is a syntactical
>>> notion?
>>
>> Do you think a syntactic treatment of the natural numbers is
>> impossible?
>
> Of course it's impossible: you can't never demonstrate the syntactical

Ah! "can't never" is a double-negative: I of course meant "can never".

> treatment - which is simply just a formal system as strong as Q - be
> consistent, purely by syntactical means (axioms and rules of inference!
>
> (Naturally, an in consistent system is also as strong as Q!).
>
From: Nam Nguyen on 5 Nov 2009 02:58
Nam Nguyen wrote:
> Marshall wrote:
>> On Nov 4, 10:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>> Do you really think the notion of natural numbers is a syntactical
>>> notion?
>>
>> Do you think a syntactic treatment of the natural numbers is
>> impossible?
>
> Of course it's impossible: you can never demonstrate the syntactical
> treatment - which is simply just a formal system as strong as Q - be
> consistent, purely by syntactical means (axioms and rules of inference!
>
> (Naturally, an in consistent system is also as strong as Q!).
>

In other words, from the syntactical point of view, the concept
of the natural numbers is just the concept of a consistent Q.

And there, is our problem: there's no syntactically-based decision
procedure to let us know which side of (in)consistency Q is in.

If we get angry with the strictness of the syntactical paradigm and
gamble with the intuitive-and-loose concept of natural numbers as
arithmetic truths, then we'd quickly get into a foundational problem:

(1) For any concept as strong as the natural numbers, there's an
arithmetic truth that it is impossible to know its truth value.

So there we have it: stay with the strictness of syntactical-ism and
we have to admit some consistency we *can't know* how to prove syntactically;
or venture into the looseness of intuition and face-to-face with
with a truth we *can't know* the truth value!

Either way we've lost and infinity has won. Why don't we be honest and
admit our limitation and move on to a better reasoning framework?
From: Nam Nguyen on 7 Nov 2009 17:38
Nam Nguyen wrote:
> Rupert wrote:
>
>> But the notion of a numeral is a syntactic notion.
>
> You've defended that belief of syours 3 time by now ("so on", "various",
> and "possible"), but so far to no avail!

Correction: I meant you've defended 3 times your belief "Omega-consistency
is a syntactical notion".
From: Nam Nguyen on 8 Nov 2009 23:56
Nam Nguyen wrote:
>
> Assuming we're using the language of arithmetic, a numeral can be defined
> as a _syntactical_ term where only the syntactical symbols are either
> '0' or 'S'. Or if you prefer a simpler definition, it's part of a formula
> where the leftmost symbol is either '0' or 'S' while the rightmost one
> must be '0', with "leftmost", "rightmost", "symbol" are priori one
> would accept as part of FOL's syntacticalism regrading formula.
>
> The point is the definition(s) of a term above is purely syntactical
> _without mentioning_ anything about "truth", "natural numbers",
> "arithmetic",
> "arithmetical recursion", etc.. at all!
>
> The question for you, Rupert, is then can you _similarly_ define omega-
> consistency at all? In other words, can you _purely using symbols_ of the
> language of arithmetic to define omega-consistency, _without mentioning_
> the phrases such as "truth", "natural numbers", "arithmetic", "arithmetical
> recursion", "so on", "various", "possible", "semantic", ...?
>
> So far you have *not* been able to do so!

I have a hint which of course one doesn't have to take it if one chooses so.

The reason why the definition of a numeral can be solely syntactically based
is because it's a _finite concept_, being part of a formula which is finite.
But the concept of omega-consistency is *not* a finite concept. (I'm very
sure we all know what "omega" means!).

Now the *typical* notion of a formal system's being inconsistency can be
solely based on syntactical notion because its definition syntactically
requires a proof, which is syntactical and finite. The *typical* notion of
formal system's consistency is in some sense not syntactical: since no one
can disprove a formula in a given system T. However, since consistency in
meta level is just the _negation_ of inconsistency [which is a syntactical
notion] and since we live in a binary world where we're supposed to know the
meaning of "negation" as a priori, formal-system-consistency could be said
to be syntactically based, because of the syntactical inconsistency if nothing
else.

Omega-consistency, as you mentioned, is the negation of omega-inconsistency.
However, it's unfortunate for your arguments, that in this case the definition
of omega-inconsistency is based on the concepts of *all* the numerals of the
naturals, something which is impossible to characterize _syntactically_!

Just so what I've said above is a bit clearer, let me present an example.
There are formal systems written in the language of arithmetic in which
you'd see the following theorems:

- Ax[S(x)=0] or
- Ex[S(x)=0] or
- ... which are not "natural".

In those cases, the semantic of the numerals 0, S0, SS0, for example
would not necessarily be what one might have in mind. It's therefore
impossible to _syntactically characterize_ something as true or false
or provable _in all the numerals of the same kind_, the "natural kind"
in this case. Therefore, the omega-inconsistency can't be defined
syntactically and with that neither can omega-consistency be.

If we throw in, as Godel did in his Incompleteness Theorem, the intuitive
concepts of arithmetic truths of the natural numbers that would be
a different story, naturally.

But you have to abandon the notions you have defended unsuccessfully if you
want to go that round. But by then you and I wouldn't have anything to argue
here!

From: Nam Nguyen on 9 Nov 2009 09:56
Alan Smaill wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Rupert wrote:
>>
>>> A theory T in the first-order language of arithmetic is said to be
>>> omega-inconsistent if and only if, for some predicate P(x), the theory
>>> T proves ExP(x) but also proves ~P(0), ~P(1), and so on. When I say
>>> that T proves "~P(0), ~P(1), and so on", I mean: it proves every
>>> sentence resulting from the substitution of a numeral into the
>>> predicate ~P(x).
>>> And a theory T in the first-order language of arithmetic is said to
>>> be
>>> omega-consistent if and only if it is not omega-inconsistent.
>>> Are we *done* yet?
>> No. Not yet. You are back to your non-syntactical notion of "so on"
>> again, for the millionth time!
>
> Some exaggeration there.

Not really.

>
>> Keep trying and one day you _might_ get it right.
>
> "and so on" here means "for every numeral".
>
> For some predicate P(x), T proves ExP(x),
> and also, for every numeral n, T proves ~P(n).

What does "every numeral n" differ from "so on", syntactically
speaking?

>
>
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