> On 15 May 2019, at 00:17, 'Brent Meeker' via Everything List > <[email protected]> wrote: > > > > 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.
Yes. That vocabulary différence is a source of a lot of confusion. > >> >> 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. Prediction is dear, OK. But explanation is even more dear, if the goal is “understanding”, instead of “surviving” or “making money”. > >> >> >> >>> >>> 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. >> >> >> 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. I guess Jason used the word model in the sense of theory. (Cf above). Sometimes I use model also in that sense, because of the context. The notion of Model models reality, the notion of theory models machines and numbers ... > >> 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. Monism. OK. But when doing metaphysics or theology, we must not decide what that reality is at the start. > >>> >>> 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. Hmm… No. > > I'm sure Bruno can explain this better than I can. I only took a couple of > semesters of symbolic logic. The non provable truth are all well defined in the arithmetical truth, which by definition is the standard model. They have all the shape “the machine x will stop”, when in truth, they will not stop, but PA or ZF cannot prove that it will non stop. Typically, all the arithmetical interpretation of the G* \ G proposition are true in the standard model, and non provable by the corresponding Löbian entity (even the non mechanical one). Bruno > > 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). >> >> Brent >>> >>> Jason >>> -- >>> 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%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 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/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>. > > > -- > 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/dbac195b-06e9-2962-6b04-4a8889e694d6%40verizon.net > > <https://groups.google.com/d/msgid/everything-list/dbac195b-06e9-2962-6b04-4a8889e694d6%40verizon.net?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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