Re: Indefinite Extensibility and Computationalism

Nam D. Nguyen says...

MoeBlee wrote:
On Jun 29, 6:49 am, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
MoeBlee wrote:
On Jun 28, 8:42 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
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.
"A model" of what arithmetic formal axiom-system? Until you could answer
this question, the discussion is pointless.

And here we go again...trying to get you to understand that a model
for a language does not require first specifying a theory in that
language.

Here, we may regard the the truth
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.
How do you even know PA is consistent?

I didn't evoke the consistency of PA.

You did: "per the standard model of ... first order PA"!

It was clear that he meant a model of the *language*
of PA. 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.

--
Daryl McCullough
Ithaca, NY

.

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