Re: Indefinite Extensibility and Computationalism
- From: MoeBlee <jazzmobe@xxxxxxxxxxx>
- Date: Mon, 02 Jul 2007 12:02:34 -0700
On Jul 2, 10:34 am, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
Daryl McCullough wrote:
Nam D. Nguyen says...
MoeBlee wrote:
On Jun 29, 6:49 am, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
MoeBlee wrote:And here we go again...trying to get you to understand that a model
On Jun 28, 8:42 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:"A model" of what arithmetic formal axiom-system? Until you could answer
How do you know that if ~GC is not true, GC would be true?If ~GC is not true, then GC is true. That follows from the definition
of a truth function per a model.
this question, the discussion is pointless.
for a language does not require first specifying a theory in that
language.
You did: "per the standard model of ... first order PA"!I didn't evoke the consistency of PA.Here, we may regard the the truthHow do you even know PA is consistent?
function per the standard model of, say, first order PA, as being from
the set of sentences of the language of first order PA into {0 1} (or
{false true}, whatever). The function is on the entire set of
sentences of the language, and GC is one of them, so the function
assigns to GC either 0 or1 and not both; and we prove that for any
sentence S of the langauge, the truth function assigns 0 to ~S iff it
assigns 1 to S.
It was clear that he meant a model of the *language*
of PA.
Don't know where you're coming from, here!
Exactly! You don't understand the basics of this subject.
I was addressing *his denial*
"I didn't evoke [invoke] the consistency of PA." of an earlier statement
of his own ("per the standard model of, say, first order PA"). All it needs
for him to say here is something very simple, like "It's my bad I didn't
really mean it"! But he hasn't said it, and based in his last post,
it doesn't seem he has any intention at all!
After Daryl explained to you that I had made a typo of omission (which
you could have known for yourself if you were the least bit paying
attention to what I was saying and the abundant context that makes it
clear I was speaking of the LANGUAGE), I let that stand on its own,
and I DID post subsequently to make clear and EMPHATIC (as Daryl
already had anyway) that whether PA has a model or not, there are
structures for the LANGUAGE of PA.
I mean for the guy who is
very "strict" about deviating from text books (as he *often wrongly* accuses
me of),
NO. That's WRONG. I don't fault anyone for merely deviating from
textbooks. Textbooks deviate among one another, and sometimes
textbooks even deviate inside themselves (as reasonable informality
allows), so it is virtually impossible to talk about these subjects
without deviatiating in some way from some textbook or another.. What
I've stressed with you is that you cited the textbook you refer to,
yet that textbook goes directly against what you are saying - and in
SUBSTANCE, not just as to minor points of terminological difference,
and there is no textbook I've ever seen that agrees with your view.
If you want to set up your own mathematical logic, or cite some
alternative mathematical logic, then fine; but your notion we've been
talking about does contradict the notion as found in ordinary
mathematical logic as in Shoenfield or any number of other textbooks.
he got to admit his own deviation! I'm easy here and would like
to argue the key issues at the foundation of FOL, and not want to engage
pick-on-textbook-minor-issue war here. But I didn't start this war: I just
play along!
Oh for Pete's sake, you're too much. Daryl posted the correction
himself. It's OBVIOUS that I made a typo of omission. You can see it
from the ABUNDANT context in which I made clear that I was talking
about structures for the LANGUAGE. But now you're trying to make some
kind of issue about that. How really silly that is.
So for what it's worth, I remember a Bob Dole's sentiment about
Newt Gingrich:
"A man who plays by the sword will die by the sword"
Boy, talk about inflating the importance of a typo of omission! And
incorrectly as to the tu quoque issue, since I DON'T keep going at
someone after it's clear the mistake was merely a typo.
Even if PA is inconsistent, the language has
models.
MoeBlee's statement
If ~GC is not true, then GC is true. That follows from the definition
of a truth function per a model.
doesn't have *anything* to do with whether PA is consistent. Your
question
"A model" of what arithmetic formal axiom-system?
Is *irrelevant* to MoeBlee's statement. You can choose *any*
model of the *language* of PA, and either GC will be true
in that model, or GC will be false in that model.
Here's a model of the language of PA: The domain consists
of two elements {0,1}. The interpretation of "0" is 0.
The interpretation of "S" is the function s given by
s(0) = 1, s(1) = 0. The interpretation of "+" is
the function p given by p(0,0) = 0, p(0,1) = 1,
p(1,0) = 1, p(1,1) = 0. The interpretation of "*" is
the function t given by t(0,0) = 0, t(0,1) = 0, t(1,0) = 0, t(1,1) = 1.
With this interpretation, there are no prime numbers, so GC
is false.
Notice, you blew right past that generous explanation by Daryl on the
SUBSTANTIVE matter here, as you harp on a typo instead.
And notice that Daryl even provided such an example as you're now
challenging me to provide. And even simpler examples could be
provided. Anyone who has passed a midterm exam in a symbolic logic
class can give us an example of a sentence true in a model without
having to specify any theory or axioms from which the sentence is a
theorem. It's a trivial excercise. If you don't know how to do it,
then you need to figure out how to do it if you want to talk
meaningfully about mathematical logic.
MoeBlee
.
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- From: Nam D. Nguyen
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