On 5/14/2019 9:10 AM, Jason Resch wrote:
On Mon, May 13, 2019 at 4:46 PM 'Brent Meeker' via Everything List
<[email protected]
<mailto:[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.
Saying truth depends on the model is like saying facts about something
depend on what you are talking about.
When I said arithmetical truth, it should be clear the model is
arithmetic, and so arithmetical truth are the facts concerning arithmetic.
But which arithmetic? There is more than one model of Peano's axioms
for example. But , you say, I mean the natural numbers model of
arithmetic...but the natural numbers are something hypothesized from
empirical observation.
Which is why it's undefinable within the
system.
Could you clarify this point?
There is more than one model of PA and "true" is relative to the model.
And also why it's not the same as the "true" in "It is true
that snow is white."
How is it different?
Snow is defined ostensively, as are the natural numbers. But what
mathematicians (like Goedel) prove theorems about is the axiomatic
system. That's why Bruno makes the point that provability is well
defined but truth isn't (in mathematics).
Brent
Jason
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