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?
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".
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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