> On 17 May 2019, at 23:24, 'Brent Meeker' via Everything List > <[email protected]> wrote: > > > > On 5/17/2019 5:10 AM, Bruno Marchal wrote: >>> On 16 May 2019, at 01:40, 'Brent Meeker' via Everything List >>> <[email protected]> wrote: >>> >>> >>> >>> On 5/15/2019 9:01 AM, Bruno Marchal wrote: >>>>> On 13 May 2019, at 23:46, 'Brent Meeker' via Everything List >>>>> <[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. Which is why it's undefinable within the >>>>> system. And also why it's not the same as the "true" in "It is true that >>>>> snow is white.” >>>> ? >>>> >>>> I don’t see the difference. The standard model of arithmetic is given by >>>> the intersection between all models. >>> Isn't the intersection of all models just the provable part? >> By incompleteness that is not the case. The provable part is much smaller >> than the true part. > > Isn't that what I said?
No, what you can prove is true in all models, but what is true in all models can be proved (by completeness), but that is not equal to what is true in the standard model. Consistent(PA) is true in the standard model, but is not provable, for example. All what you can prive is sigma_1 ([] is a sigma_1-complete predicate), but it is not pi_1-complete, nor sigma_i or pi_i-complete for any big i). The standard arithmetical truth is highly not computable. It is bigger than any sigma_i or pi_i complete sets. > >> The true undecidable sentences are true in the standard sense, just possibly >> false in the non start sense. > > Right. All the models make the provable part "true"; otherwise they wouldn't > be models. What you mean by the "true undecidable sentences are true in the > standard sense" is that they are true in the standard model, which is the > abstraction from empirically counting, adding, subtracting, and multiplying > sets of objects. It is that empirical basis which makes the standard model > standard and is the reason everyone agree on "it”. Maybe. Maybe not. The discovery of the distinction between standard and not standard has waited for the discovery of Löwenheim, Skolem, Gödel, etc. The human conception of numbers is the standard one, almost by definition, and there is few doubt that Nature has an important teaching role in this, but that does not entail that Nature could not be an hallucination by (sheaves of) consciousness flux arising from the universal numbers in arithmetic. At this stage, that could be invalid; and we know with mechanism that this cannot be the case. Bruno > > Brent >> >> Typical example: the consistency of PA. Everyone familiar with natural >> numbers believe that PA is consistent, but PA cannot prove this, and thus >> there is a model of PA where “PA is inconsistent” is true. It means that >> some “omega” (see my preceding posts) is a proof of “0=1”; but as omega is >> not accessible by the successor relation, that they is still consistent. PA >> + (PA is inconsistent) is a consistent theory of natural numbers, but it is >> not a sound theory. It is false in the standard model. >> >> Bruno >> >> >> >>> Brent >>> >>>> See my other recent explanations. >>>> >>>> Bruno >>>> >>>> >>>> >>>>> Brent >>>>> >>>>> -- >>>>> 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]. >>>>> To view this discussion on the web visit >>>>> https://groups.google.com/d/msgid/everything-list/816e0c3c-76cc-82c8-a3b3-18f6af30b042%40verizon.net. >>> >>> -- >>> 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]. >>> To view this discussion on the web visit >>> https://groups.google.com/d/msgid/everything-list/ab8bbe0f-a236-9628-c030-4de2928ca181%40verizon.net. > > > -- > 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]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/d0ad3345-c0f4-878f-9271-07db96cb5190%40verizon.net. -- 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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/3E48ED4B-75CC-4B6A-B69D-F80D29291248%40ulb.ac.be.
