> On 14 May 2019, at 22:47, 'Brent Meeker' via Everything List > <[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.
The standard model. It is the intersection of all models. We cannot define the “real”, or “standard” natural number in first order logic, but we can do it in second order arithmetic, or in set theory, etc. The non standard natural numbers are very weird, their addition and multiplication are not computable. The only role they could “logically” play is a role of oracle, but to invoke them is again introducing non computable infinities, for what? The whole of science assume the standard number. The non standard numbers are like the limit of analysis, and indeed can be used to reintroduce the infinitesimal in analysis, but I think that the arithmetical epsilon-delat definition is better. > But , you say, I mean the natural numbers model of arithmetic...but the > natural numbers are something hypothesized from empirical observation. In the Aristotelian theology. But eventually this one is inconsistent with Digital Mechanism, which needs to explain the persistence of the hallucination from the logic of the dreams of the universal numbers. > >> >> 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. But all standard natural numbers are standard in all models. What you prove in PA is true in all models. The machines behave the same, and think the same in all model of RA, PA (and ZF, ZFC, ZFC+Ind, etc.). The notion of natural numbers is the simplest and clearest notion in the whole of science. It is bad philosophy to introduce doubt on 2+2=4. That would put Mechanism in jeopardy, but that would put all theories in jeopardy, with some exception in the non Turing universal small realities. >> >> 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. No. The natural numbers are defined by simple axioms and rules. I have given their definition in the combinators languages, and very often directly in predicate calculus. Our familiarity with comes from some use we lake of them with respect to distinguishable object, but to define the natural number by referring to a physical world or to human psychology is typically not how the mathematicians operates. Our intuition of the natural numbers is already as complex to explain as to explain consciousness. But we do have that intuition, like most animals, and like all Löbian Turing universal machine, actually. > 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). Not really. The mathematical logician does define the arithmetical truth. Model theoryis developed in set theory. The truth that Bruno and Brent cannot define is not the arithmetical truth, but the bruno-truth and the Brent-truth, assuming that we have similar complexity. Now with mechanism we cannot really define the arithmetical truth, but that reflect only that we have to use a richer theory than arithmetic to define it, and mathematician do use much richer theories, all the time, with a level of confidence which is common and plausibly deserve on some large portion of set theory. Above, you need God, which is probably an abstract and ultimate form of your definition per-ostension, except that with mechanism, the ostension is not on the physical universe, but on the arithmetical reality we look inward, notably by doing math. Bruno > > 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. 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