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| Aatu Koskensilta |
 Posted: Tue Apr 29, 2008 2:51 pm |
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On 2008-04-29, in sci.logic, george wrote:
Quote: No, DUMBASS, *That* was NOT the question. THAT is the ANSWER. The
fact that the sentence has different truth-values in different
models simply PROVES THAT IT DOES NOT have "a" truth-value.
Then no sentence in the language of arithmetic that is not a logical
truth or a logical falsity has a truth-value. Why then claim that 2 +
2 = 4 is true, as you did? If the answer is that 2 + 2 = 4 is true in
all models of PA we must once again ask why grant this special status
to PA and not ACA or any other theory?
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Nam D. Nguyen |
| Posted: Wed Apr 30, 2008 1:15 am |
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Aatu Koskensilta wrote:
Quote: On 2008-04-26, in sci.logic, Nam D. Nguyen wrote:
Choose one particular schema axiom, say:
(1) (P0(0) & (x)(P0(x) --> P0(S(x)))) --> (x)P0(x)
Now define P1(x) as: P1(x) df= ~G(Q) & P0(x).
Now we have:
(2) (P1(0) & (x)(P1(x) --> P1(S(x)))) --> (x)P1(x)
It seems true ~G(Q) is provable using (2), and (2) isn't one of
the non-induction axioms listed above. Perhaps you could explain
why (2) shouldn't be an induction axiom?
(2) is an induction axiom. But why do you think ~G(Q) -- which, I take
it, is the negation of the Godel sentence of Robinson arithmetic --
follows from (2)?
Yes. ~G(Q) is the Godel sentence of Robinson arithmetic. (We could
have used CON(Q) as well). All right. I was incorrect about ~G(Q) being
provable using (2). [I overlooked the leftmost parenthesis "(" immediately
before P1!]. But I'll standby my protest about something you stated:
Quote: To conclude that PA is consistent, we show that T(A)
for every axiom of PA, and hence PA does not prove 0=1.
How would you know the given definition of T wouldn't have
T(A) and T(B) where A and B are theorems and A = ~B, for some
A and B?
Quote:
Then why for years you seem to have fought with the "cranks", whose
reasonings seem to thrive from inconstant reasonings?
That consistency is a piddling correctness condition does not entail
that it is not a correctness condition at all.
Then "piddling" is almost meaningless in this context. The precision of
mathematical reasoning requires correctness conditions, which is agnostic
whether or not anything is/isn't piddling.
Quote: In any rate, in "all statements we consider meaningful", what did you
mean by "all statements"? of the language? of the T in question?
All statements in the language of T we take to meaningfully assert
something.
I guess I could live with that, though I'd point out that "a=b /\ a=/=b"
does assert something (all wff's do have meaning and do assert something;
whether or not we'd consider them as having some truth values is another
matter).
Quote: Depending on T, and our personal view on the extent
mathematical talk is meaningful, that might be the whole of the
language of T, or just a part.
I'd agree, as long as "Depending on T" means depending on T's being consistent!
Quote: To the extend that "arithmetical statements" include finite number
of axioms of Q, what's provable is not necessarily true!
What is provable in an arbitrary formal theory need not be true, but
if we are to accept that anything at all is provable in mathematics,
it makes no sense to claim that what is provable in Robinson
arithmetic is not necessarily true.
I think we've gone through this route already. Should Robinson arithmetic
be inconsistent "anything at all is [*still*] provable in mathematics":
the "good" as well as the "bad" theorems! And we have no choice but accept
that; for if we don't, in the meta level we'll have a bigger inconsistency
and incorrectness: our reasoning!
Q is after all is a formal system, like any other ones. If you couldn't
demonstrate it syntactical consistency, what's provable in it is not
necessarily true.
Quote: Please recall that Robinson
arithmetic is extremely weak, and in particular even the most
innocuous principles we use in reasoning about e.g. formal proofs,
formulas, and so on, go beyond what is included in Robinson
arithmetic.
"Weak" and "strong" are relative term. On the account that
Q is finitely axiomatizable and yet the purported natural numbers is
one of its model, then it could be considered as "stronger" (more compact)
than say PA: it describes the same concept with fewer axioms. Secondly,
should Q be inconsistent, all extensions of Q would be of the same
uselessness, not stronger!
Quote: Even proving that the parentheses in a well-formed formula
are always evenly paired is beyond the deductive powers of Robinson
arithmetic (or an equivalent theory of syntax)!
Everything comes with a price! If finite knowledge is the only "money"
we have, then we could define, say, "infinite" as the negation of "finite"!
If on the other hand we still have finite knowledge capability, and yet
we desire to build our reasoning on the natural numbers, then something
"bad" are bound to happen. How much money we have usually determines
the quality of the things we bought! For instance, if we think we understand
the natural numbers enough, I could define a wff formula whose length
is less than one line long and yet nobody could possibly have a clue what
the formula is!
In brief, don't think that just because we have a desire to make our reasoning
more powerful with infinity, we could assume anything and then claim anything
we want.
Quote:
My comment, however, was not on what is or is not correct or
acceptable in mathematics, but was simply an answer to your query as
to what I mean by calling a theory correct in this context; the
answer, once again, is that a theory is correct is what is provable in
it is true.
But to the extend that a formula's being true can be defined with
it being provable in a consistent formal system, the dichotomy
between a mathematical truth and a mathematical (syntactical) provability
is not genuine (i.e. is fake). In other word, a correct theory is
a syntactically consistent theory! (Can we see one otherwise?)
Quote: Meaningfulness enters into this in that if (part of) the
language of the theory is not meaningful, it makes no sense to say of
sentences (in that part) that they are or are not true.
Unfortunately what is meaningful to one is equally not meaningful to other!
For what it's worth, if "Ax" means "for many x's" and "Ex" means "for a few x's",
you'd still have the same FOL framework, syntactically speaking! |
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| Frederick Williams |
| Posted: Wed Apr 30, 2008 5:01 am |
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Newberry wrote:
Quote:
On Apr 21, 3:59 am, David C. Ullrich <dullr...@sprynet.com> wrote:
On Sun, 20 Apr 2008 21:22:33 -0700 (PDT), Newberry
newberr...@gmail.com> wrote:
1. Mathematics (excluding geometry) is analytic.
The fact that you exclude geometry shows you're confusing
geometry and physics. The Pythagorean Theorem does not
prove anything about what's going to happen when you measure
the sides of triangles in the actual physical world.
I do not claim anything about geometry. In fact I have explicitly
excluded it from any claims.
You are implicitly claiming that mathematics consists of (at least two)
disconnected pieces which can be broken off from one another. This is
false.
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