On 5/14/2019 2:33 PM, Jason Resch wrote:
On Tue, May 14, 2019 at 3:47 PM 'Brent Meeker' via Everything List <[email protected] <mailto:[email protected]>> wrote: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. I see a 100% analagous situation to the natural sciences:"There's more than one model of gravitation for example. But, you say I mean the gravitation of our universe...but gravitation is something hypothesized from empirical observation."
"Model" means different, almost complementary, things in physics and mathematical logic. Physicist would call Newton's theory a model of gravity, the physical phenomenon. Mathematicians would axiomatize Newton's theory and then look around for something that satisfied the axioms, which they would call "the model". The physical phenomenon, gravity, would not be a model of Newton's theory, because it doesn't correctly model the advance of the perihelion of Mercury.
Axiomatic systems are just like theories in the sciences. They attempted to systematize what is out there. But we can never be sure our models correctly reflect the reality. We can only hope to improve our models over time to become more powerful in what they can explain.
Explanation is cheap. Prediction is dear.
Which is why it's undefinable within thesystem.Could you clarify this point?There is more than one model of PA and "true" is relative to the model. I think you mean "provable" is relative to the model.
No, provable depends only on the axioms and the rules of inference. "True" depends on the model. Everything provable is true in every model. But the truth value of what isn't provable can vary depending on the model.
In Newton's gravity you could "prove" something about the expected orbital velocity of Mercury in that model. It just wouldn't be true when compared to reality.
Right, and we only have one reality.
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.Do we agree that the true properties of the natural numbers are objective? If so no need to debate this any further.
It's a matter of equivocating on "the natural numbers". If you regard them as a theory of things, the way you learn them at your mother's knee, then there are objective truths "Two garbanzo beans plus two chick peas make four beans." the way "Snow is white." But if you want to evaluate the truth of "2+2=4" that's a proposition in arithmetic. If it's PA then it's true in every model because it's a theorem. But when you say there are true statements of arithmetic that aren't provable in PA, what they are depends on the model.
I'm sure Bruno can explain this better than I can. I only took a couple of semesters of symbolic logic.
Brent
Jason 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). BrentJason-- You received this message because you are subscribed to theGoogle Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected] <mailto:[email protected]>. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUgQZz3nO%2BOKaiWZrtmbVivC8E_0BtwfjhW7hm9PjRoZ_Q%40mail.gmail.com <https://groups.google.com/d/msgid/everything-list/CA%2BBCJUgQZz3nO%2BOKaiWZrtmbVivC8E_0BtwfjhW7hm9PjRoZ_Q%40mail.gmail.com?utm_medium=email&utm_source=footer>.-- You received this message because you are subscribed to the GoogleGroups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected] <mailto:[email protected]>. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/c8ccb7d1-0417-4718-aae2-484716798892%40verizon.net <https://groups.google.com/d/msgid/everything-list/c8ccb7d1-0417-4718-aae2-484716798892%40verizon.net?utm_medium=email&utm_source=footer>. --You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected] <mailto:[email protected]>. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CA%2BBCJUhqBMhAbbO9z4225FsqOUpSqTvK2JvQ7rbYenHAKU5aQA%40mail.gmail.com <https://groups.google.com/d/msgid/everything-list/CA%2BBCJUhqBMhAbbO9z4225FsqOUpSqTvK2JvQ7rbYenHAKU5aQA%40mail.gmail.com?utm_medium=email&utm_source=footer>.
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