On 5/20/2019 2:59 AM, Bruno Marchal wrote:
On 17 May 2019, at 23:24, 'Brent Meeker' via Everything List
<[email protected]> wrote:
On 5/17/2019 5:10 AM, Bruno Marchal wrote:
On 16 May 2019, at 01:40, 'Brent Meeker' via Everything List
<[email protected]> wrote:
On 5/15/2019 9:01 AM, Bruno Marchal wrote:
On 13 May 2019, at 23:46, 'Brent Meeker' via Everything List
<[email protected]> wrote:
On 5/13/2019 8:50 AM, Jason Resch wrote:
But then what is arithmetical truth? We have no label for it. It cannot be
derived from or defined by labels.
And it depends on the model. Which is why it's undefinable within the system. And also why
it's not the same as the "true" in "It is true that snow is white.”
?
I don’t see the difference. The standard model of arithmetic is given by the
intersection between all models.
Isn't the intersection of all models just the provable part?
By incompleteness that is not the case. The provable part is much smaller than
the true part.
Isn't that what I said?
No, what you can prove is true in all models, but what is true in all models
can be proved (by completeness),
So what is true in the all models is what can be proved...which is what
I wrote as a question four lines above.
but that is not equal to what is true in the standard model.
Consistent(PA) is true in the standard model, but is not provable, for example.
All what you can prive is sigma_1 ([] is a sigma_1-complete predicate), but it
is not pi_1-complete, nor sigma_i or pi_i-complete for any big i).
The standard arithmetical truth is highly not computable. It is bigger than any
sigma_i or pi_i complete sets.
The true undecidable sentences are true in the standard sense, just possibly
false in the non start sense.
Right. All the models make the provable part "true"; otherwise they wouldn't be models. What
you mean by the "true undecidable sentences are true in the standard sense" is that they are
true in the standard model, which is the abstraction from empirically counting, adding, subtracting, and
multiplying sets of objects. It is that empirical basis which makes the standard model standard and is
the reason everyone agree on "it”.
Maybe. Maybe not. The discovery of the distinction between standard and not
standard has waited for the discovery of Löwenheim, Skolem, Gödel, etc.
The human conception of numbers is the standard one, almost by definition, and
there is few doubt that Nature has an important teaching role in this, but that
does not entail that Nature could not be an hallucination by (sheaves of)
consciousness flux arising from the universal numbers in arithmetic.
I don't know how to understand things like "hallucination arising from
universal numbers" and "sheaves of consciousness flux". I don't know
whether you're waxing poetic or just talking gibberish.
Brent
At this stage, that could be invalid; and we know with mechanism that this
cannot be the case.
Bruno
Brent
Typical example: the consistency of PA. Everyone familiar with natural numbers
believe that PA is consistent, but PA cannot prove this, and thus there is a
model of PA where “PA is inconsistent” is true. It means that some “omega” (see
my preceding posts) is a proof of “0=1”; but as omega is not accessible by the
successor relation, that they is still consistent. PA + (PA is inconsistent)
is a consistent theory of natural numbers, but it is not a sound theory. It is
false in the standard model.
Bruno
Brent
See my other recent explanations.
Bruno
Brent
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