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Science Forum Index » Logic Forum » Newberry's Theses
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| george |
 Posted: Sun Apr 27, 2008 10:00 am |
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On Apr 26, 4:16 pm, Aatu Koskensilta >
Quote: What is special about models of PA?
PA itself, as an axiom-set, is special because
it is prominent; it is the one that generally gets
used. It is likely to remain so since it is, at 2nd-order,
correct. If you count schemata as 1 thing then it is
also only 3 axioms long.
If you have a competing axiomatization that actually
SUCCEEDS in competing, in terms of compactness
and correctness, then it is equally special for the
same reasons. |
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| george |
| Posted: Sun Apr 27, 2008 10:23 am |
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On Apr 26, 4:16 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
wrote:
Quote: What is special about models of PA?
What is "special" about the standard model?
I am using scare-quotes because obviously we wouldn't
even be able to speak of "the" standard model if there were
not in fact a plethora of things that make it distinctive.
Our argument here is about direction of fit.
Your perspective takes N as given and uses theories to
approximate it. Mine takes the theories as given and doesn't
concede the relevance of N at all, except to the extent that
the machinery is founded on it. |
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| Daryl McCullough |
| Posted: Sun Apr 27, 2008 3:16 pm |
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Newberry says...
Quote: Let me clarify then. Goedel's theorem establishes that there is no
proof that establishes with absolute certainty the truth of Goedel's
sentence.
No, it does not. Godel made no effort to formalize the notion
of "establishing something with absolute certainty". His proof
said nothing about that.
If you want, *you* can try to formalize the notion of "absolute
certainty" and attempt to prove that it follows that the Godel
sentence for PA cannot be established with absolute certainty.
But that's not what Godel's theorem showed.
--
Daryl McCullough
Ithaca, NY |
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| Daryl McCullough |
| Posted: Sun Apr 27, 2008 3:19 pm |
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Newberry says...
Quote: How certain are we about the consistency of propositional calculus and
predicate calculus?
Different people are certain of different things. There is no
mathematical consensus about what things are certain and what
things are not.
If you are asking about me, personally, then my attitude is
that there are plenty of things in the world to be uncertain
about. I don't have the time or energy to doubt propositional
logic.
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Daryl McCullough
Ithaca, NY |
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| Newberry |
| Posted: Sun Apr 27, 2008 4:31 pm |
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On Apr 27, 1:16 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
wrote:
Quote: Newberry says...
Let me clarify then. Goedel's theorem establishes that there is no
proof that establishes with absolute certainty the truth of Goedel's
sentence.
No, it does not. Godel made no effort to formalize the notion
of "establishing something with absolute certainty". His proof
said nothing about that.
Let me clarify again. When I said "Goedel's theorem establishes that
there is no proof that establishes with absolute certainty the truth
of Goedel's sentence" I did not mean that this is what Goedel's
theorem says. I meant that that is a consequence of Goedel's theorem.
Quote: If you want, *you* can try to formalize the notion of "absolute
certainty" and attempt to prove that it follows that the Godel
sentence for PA cannot be established with absolute certainty.
But that's not what Godel's theorem showed.
--
Daryl McCullough
Ithaca, NY |
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| Jesse F. Hughes |
| Posted: Sun Apr 27, 2008 9:07 pm |
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"Nam D. Nguyen" <namducnguyen@shaw.ca> writes:
Quote: "Whoever undertakes to set himself up as a judge of Truth and Knowledge
is shipwrecked by the laughter of the gods".
Albert Einstein
Kinda flowery words for Einstein, no? I searched Wikiquote and they
attribute it to Edmund Burke.
--
Jesse F. Hughes
"I'm not sure whether I'm not thinking clearly or clearly not
thinking." -- Ling Cheung, said on some day other than our wedding
day. Honest. |
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| Nam D. Nguyen |
| Posted: Sun Apr 27, 2008 11:14 pm |
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Jesse F. Hughes wrote:
Quote: "Nam D. Nguyen" <namducnguyen@shaw.ca> writes:
"Whoever undertakes to set himself up as a judge of Truth and Knowledge
is shipwrecked by the laughter of the gods".
Albert Einstein
Kinda flowery words for Einstein, no? I searched Wikiquote and they
attribute it to Edmund Burke.
It's in the following link, among other places:
http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html |
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| Nam D. Nguyen |
| Posted: Sun Apr 27, 2008 11:18 pm |
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Nam D. Nguyen wrote:
Quote: Jesse F. Hughes wrote:
"Nam D. Nguyen" <namducnguyen@shaw.ca> writes:
"Whoever undertakes to set himself up as a judge of Truth and Knowledge
is shipwrecked by the laughter of the gods".
Albert Einstein
Kinda flowery words for Einstein, no? I searched Wikiquote and they
attribute it to Edmund Burke.
It's in the following link, among other places:
http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html
There's also a caveat that the lister couldn't vouch for authenticity of any quotes.
So it seems possible this particular quote might be from someone else! |
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| Aatu Koskensilta |
| Posted: Mon Apr 28, 2008 6:02 am |
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On 2008-04-26, in sci.logic, Nam D. Nguyen wrote:
Quote: 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)?
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.
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. 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.
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. 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. 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)!
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. 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.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Aatu Koskensilta |
| Posted: Mon Apr 28, 2008 6:04 am |
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On 2008-04-27, in sci.logic, Nam D. Nguyen wrote:
Quote: "Whoever undertakes to set himself up as a judge of Truth and Knowledge
is shipwrecked by the laughter of the gods".
Do you think I've set myself up as a judge of Truth and Knowledge?
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| Daryl McCullough |
| Posted: Mon Apr 28, 2008 8:58 am |
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Newberry says...
Quote: Let me clarify again. When I said "Goedel's theorem establishes that
there is no proof that establishes with absolute certainty the truth
of Goedel's sentence" I did not mean that this is what Goedel's
theorem says. I meant that that is a consequence of Goedel's theorem.
No, it is *not* a consequence of Godel's theorem. Godel's theorem
does not imply that. Perhaps Godel's theorem, together with some
additional assumptions about the nature of "absolute certainty"
implies that, but Godel's theorem alone doesn't. So please quit
saying that your philosophical conclusions are consequences of
Godel's theorem.
--
Daryl McCullough
Ithaca, NY |
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| Aatu Koskensilta |
| Posted: Tue Apr 29, 2008 9:43 am |
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On 2008-04-26, in sci.logic, Newberry wrote:
Quote: Are you saying that we have two sets of proofs? One set of formal
proofs for shuffling around meaningless marks on paper and another set
of informal mathematically compelling arguments?
Formal proofs are mathematical structures by means of which we can
analyse some aspects of our mathematical practice. In particular, by
the completeness theorem and certain conceptual considerations, we can
establish that a mathematical statement is provable, in the ordinary
sense, from some bunch of statements, if and only if the formalisation
of that statement is formally derivable from the formalisations of the
bunch of statements. From this observation, we may sometimes conclude
something about provability in the ordinary sense from purely
mathematical results about formal derivability.
What we find if we look at any substantial piece of mathematics is
"informal" proofs, not in the sense that there were any vagueness,
unclarity, etc. to them, but in the sense that they are not presented
in the form of any formal derivations in any formal theory. In judging
the correctness of such proofs we must decide whether the principles
invoked in them are acceptable or not, something that corresponds to
no mathematical property of formal derivations. In saying that we can
prove the Godel sentence of this or that theory, we have of course
this in mind, that we can produce a mathematically compelling argument
from principles we accept as correct.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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| george |
| Posted: Tue Apr 29, 2008 9:48 am |
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It really does NOT take a whole lot of art or skill
TO KNOW HOW TO HAVE A RATIONAL DISCUSSION.
For some reason AK is wilfully hell-bent on never
having one.
I said:
Quote: PA itself, as an axiom-set, is special because
it is prominent; it is the one that generally gets
used.
AK replied:
This is A STUPID question.
There is basically only ONE way in which ANY
axiom-set can be used: you prove theorems from it.
Quote: PA is the most famous of the various theories of arithmetic
that abound in the logical literature,
Exactly, and this CONSTITUTES YOUR ANSWER to your
question "used how?", which IS WHAT MAKES your question
STUPID.
Quote: and unlike many other such theories
Oh, please. Your whole QUESTION was, what makes PA
*special*! In the first place, THERE ARE *NO OTHER* "such"
theories! There is ONLY ONE PA! PA is the ONLY thing
that plays the roles PA plays, in the literature OR ANYWHERE
ELSE! Precisely because, as you just said, it is ALONE in being
"the most famous", it has NO peers!
Quote: we can expect that most mathematicians have heard of it. (We
might also expect that most of them would be unable to
spell in detail
what is or is not a formal proof in PA).
That is patently ABSURD!
EVERY mathematician KNOWS WHAT A PROOF is!
EVERY mathematician knows in principle what does or
doesn't constitute a proof from some axioms!
That applies TO EVERY finitary axiom-set (as well as
every mathematician) and PA is NOT special in THAT regard!
Quote: If you have a competing axiomatization that actually
SUCCEEDS in competing, in terms of compactness
and correctness, then it is equally special for the
same reasons.
Take for example second-order arithmetic
This is NOT a competing axiomatization, dumbass.
In the first place, it's SECOND ORDER, and the second place,
when I said that PA only had 3 axioms, I was TALKING about
2nd-order *PA* and NOT "second-order arithmetic"!
Or maybe I was talking about that too, since you may mean
by 2nd-order arithmetic what *I* meant by 2nd-order PA (and
since PA is categorical at 2nd-order anyway, there is a sense
in which they canNOT possibly be different). But my point
is that there IS a choice of AXIOMS to be had.
Your removal of the definitions in the 2nd-order case is
simply irrelevant; definitions are abbreviations.
To actually USE that formulation, you would have to prove
the existence of operators like multiplication and addition,
and then invoke some sort of new or temporary name for
the operator after proving that it existed. |
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| george |
| Posted: Tue Apr 29, 2008 9:49 am |
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On Apr 29, 12:02 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
wrote:
Quote: The question was, however,
why should the fact that the Godel sentence
of PA has different truth-values in different models of PA tell us
anything about whether it's true or false.
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. |
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| Aatu Koskensilta |
| Posted: Tue Apr 29, 2008 11:02 am |
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On 2008-04-27, in sci.logic, george wrote:
Quote: PA itself, as an axiom-set, is special because
it is prominent; it is the one that generally gets
used.
Used how? PA is the most famous of the various theories of arithmetic
that abound in the logical literature, and unlike many other such
theories we can expect that most mathematicians have heard of it. (We
might also expect that most of them would be unable to spell in detail
what is or is not a formal proof in PA).
Quote: If you have a competing axiomatization that actually
SUCCEEDS in competing, in terms of compactness
and correctness, then it is equally special for the
same reasons.
Take for example second-order arithmetic -- it's even more compact
since we don't need any special axioms for addition and multiplication!
The question was, however, why should the fact that the Godel sentence
of PA has different truth-values in different models of PA tell us
anything about whether it's true or false. Stipulating that models of
PA are in some fashion special with regards the truth or falsity of
arithmetical statements is entirely arbitrary, unless this speciality
is explained somehow, in terms not involving mere popularity.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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