> On 18 Dec 2018, at 01:14, Brent Meeker <[email protected]> wrote: > > > > On 12/17/2018 11:02 AM, Bruno Marchal wrote: >> >>> On 17 Dec 2018, at 07:10, Brent Meeker <[email protected] >>> <mailto:[email protected]>> wrote: >>> >>> >>> >>> On 12/16/2018 9:42 PM, Jason Resch wrote: >>>> >>>> >>>> On Sun, Dec 16, 2018 at 10:27 PM Brent Meeker <[email protected] >>>> <mailto:[email protected]>> wrote: >>>> >>>> >>>> On 12/16/2018 4:43 PM, Jason Resch wrote: >>>>> >>>>> >>>>> On Sun, Dec 16, 2018 at 6:02 PM Brent Meeker <[email protected] >>>>> <mailto:[email protected]>> wrote: >>>>> >>>>> >>>>> On 12/16/2018 2:04 PM, Jason Resch wrote: >>>>>> >>>>>> >>>>>> On Sun, Dec 16, 2018 at 4:01 PM Bruce Kellett <[email protected] >>>>>> <mailto:[email protected]>> wrote: >>>>>> On Mon, Dec 17, 2018 at 8:56 AM Jason Resch <[email protected] >>>>>> <mailto:[email protected]>> wrote: >>>>>> On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker <[email protected] >>>>>> <mailto:[email protected]>> wrote: >>>>>> >>>>>> But a system that is consistent can also prove a statement that is false: >>>>>> >>>>>> axiom 1: Trump is a genius. >>>>>> axiom 2: Trump is stable. >>>>>> >>>>>> theorem: Trump is a stable genius. >>>>>> >>>>>> So how is this different from flawed physical theories? >>>>>> >>>>>> Physical theories do not claim to prove theorems - they are not systems >>>>>> of axioms and theorems. Attempts to recast physics in this form have >>>>>> always failed. >>>>>> >>>>>> >>>>>> Physical theories claim to describe models of reality. You can have a >>>>>> fully consistent physical theory that nevertheless fails to accurately >>>>>> describe the physical world, or is an incomplete description of the >>>>>> physical world. Likewise, you can have an axiomatic system that is >>>>>> consistent, but fails to accurately describe the integers, or is less >>>>>> complete than we would like. >>>>> >>>>> But it still has theorems. And no matter what the theory is, even if it >>>>> describes the integers (another mathematical abstraction), it will fail >>>>> to describe other things. >>>>> >>>>> ISTM that the usefulness of mathematics is that it's identical with its >>>>> theories...it's not intended to describe something else. >>>>> >>>>> A useful set of axioms (a mathematical theory, if you will) will >>>>> accurately describe arithmetical truth. E.g., it will provide us the >>>>> means to determine the behavior of a large number of Turing machines, or >>>>> whether or not a given equation has a solution. The world of >>>>> mathematical truth is what we are trying to describe. We want to know >>>>> whether there is a biggest twin prime or not, for example. There either >>>>> is or isn't a biggest twin prime. Our theories will either succeed or >>>>> fail to include such truths as theorems. >>>> >>>> This is begging the question. You taking one piece of mathematics, >>>> arithmetic, and using it as a theory describing another piece of >>>> mathematics, Turing machines. And then you're calling a successful >>>> description "true". But all you're showing is that one contains the other. >>>> >>>> >>>> I'm not following here. >>>> >>>> Theorems are not "truths" except in the conditional sense that it is true >>>> that they follow from the axioms and the rules of inference. >>>> >>>> I agree a theorem is not the same as a truth. Truth is independent of some >>>> statement being provable in some system. >>> >>> OK. >>> >>>> Truth is objective. If a system of axioms is sound and consistent, then a >>>> theorem in that system is a truth. >>> >>> No, c.f. Donald Trump. >> >> Assuming Donald Trump is sound. >> >> We don’t know what truth is, but we can believe that some formula are true >> about our domain investigation. When I assume x + 0 = x, I ask people if >> they agree with this, about the natural numbers. >> >> Then a theory is sound, if the rule of inference preserves truth. >> >> If a theory appears to be unsound, we put it in the trash, simply. That >> happens sometimes, usually when theories manage too much big objects. >> >> >> >> >>> >>>> But we can never be sure that system is sound and consistent (just like we >>>> can never know if our physical theories reflect the physical reality they >>>> attempt to capture). >>> >>> But sometimes we can be sure that our theory does not reflect reality, even >>> if it is sound and consistent. >> >> >> By definition; soundness means that it reflect reality. > > You're now messing with words. What does "reflect reality" mean? It looks > like an appeal to correspondence theory to truth. But that means to know a > theory is sound you need to know what is true.
There are two definition of soundness. Arithmetical soundness: it means that the theorem are true in (N, 0, +, *) Exemple if the theory proves ExEyEx(x^3 + y^3 + z^3), if the theory is sound, it means that there are (standard) natural numbers n, m, r such that their sum of cube gives 33. Soundness (in general): it means that what is proved in the theory is true in all models/intepretations of the theory. Completeness is the reverse of that one; a theory is complete if what is true in all models is provable. Not to confuse with Arithmetical Completeness: a theory is arithmetically complete if it proves all truth in (N, +, *). PA is a complete theory, like all first order theories (Gödel 1930) PA is arithmetically incomplete (Gödel 1931). You can prove that PA is sound using a bit of set induction, like in Analysis. Humans tend to trust Analysis, which assumes much more than arithmetic. > > Sound just means its theorems are tautologies, i.e. they are valid inferences > from the axioms. You might makes number theorists a bit nervous if you say that Fermat is a tautology. We use just “theorem”, and use only tautology for classical propositional logic. Sound means true in all models (arithmetically sound means true in the standard model (N, +, *). > >> >> Soundness implies consistency. But consistency does not imply soundness. The >> robot describing the Venus of Milo in front of another sculpture is >> consistent, but unsound. >> >> All the machines I am talking about are supposed to be arithmetically sound. > > Meaning that what they "believe" are theorems. No, meaning that the theorem/beliefs are true in the structure (N, +, *). It means that if PA would prove ExEyEx(x^3 + y^3 + z^3), then there are really numbers n, m, r, having their cube added to 33. With PA + []f, you might be able to prove ExEyEx(x^3 + y^3 + z^3), but still not know if such standard n, m, and r exist, because x, y and z could be non standard natural numbers. PA + []f is typically unsound. It asserts that there is proof of the false, but can prove that it not 0, nor 1, nor 2, nor 3, nor 4, etc. It is consistent, but not omega-consistent. > Not that they "believe" everything that is true. In fact you have proven > anything about what a person might or might not believe. Your ideal machines > do not include any concept of acting on "beliefs" which is the real test of > beliefs. I have no clue why this could be true, except contingently, as I use all this to derive physics from arithmetic, not for doing machine acting in our local neighbourhood. Bruno > > Brent > >> >> 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 post to this group, send email to [email protected] >>>> <mailto:[email protected]>. >>>> Visit this group at https://groups.google.com/group/everything-list >>>> <https://groups.google.com/group/everything-list>. >>>> For more options, visit https://groups.google.com/d/optout >>>> <https://groups.google.com/d/optout>. >>> >>> >>> -- >>> 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 post to this group, send email to [email protected] >>> <mailto:[email protected]>. >>> Visit this group at https://groups.google.com/group/everything-list >>> <https://groups.google.com/group/everything-list>. >>> For more options, visit https://groups.google.com/d/optout >>> <https://groups.google.com/d/optout>. >> >> -- >> 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 post to this group, send email to [email protected] >> <mailto:[email protected]>. >> Visit this group at https://groups.google.com/group/everything-list >> <https://groups.google.com/group/everything-list>. >> For more options, visit https://groups.google.com/d/optout >> <https://groups.google.com/d/optout>. > > > -- > 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 post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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