Tomas,
Let me add this. Gödel’s theorem does not add any doubt on arithmetic. On the contrary it free arithmetic from reductionist theories of Arithmetic. Rich theories like PA or ZF proves their own second incompleteness theorem: they prove that if they are consistent then they cannot rove that their are consistent, making “consistency” already a bit like consciousness (indeed conscious reasoner comes up often with the idea that a conscious entity cannot prove its consciousness to another entity). IF PA was inconsistent, it would of course proves its consistency, for no avail, bacuse it would not have any model and be senseless. So Gödel’s theorem is more like giving more trust in the mathematical theories, because it shows that somehow, the mathematical theories and the ‘rich) machine are rather well aware of their limitation, and that explain also why personal semantic (rich model encompassing ourselves) requires some act of faith. We are not conscious of this, because the brain, for survival propose, makes such act of faith instinctive. If the lamb doubt its experience of the wolf, its chance of surviving will decline. That explains also why it is hard for most people to doubt on some matter, because we are all brain wired, so to speak, to not doubt too much. With Mechanism, Gödel’s theorem apply to us, as far as we are consistent/sound. That is beauty of it; it illustrates that reason can discover its own limitation, and why we need some faith to believe in any reality (equivalent with self-consistency). By machine theology, I always mean something very specific: the G* logic discovered by Solovay. It is the true part of the logic of machine (and other entities) provability/consistency. Solovay knew about G, which concerns the provable part of G*. So, science has discovered that for all (enough rich) machine, there is a surrational part in between the rational and the irrational (false and/or refutable). Reason explain the reason why reality is larger than what reason can justify, and this happens even when we limited ourself to arithmetic. Gödel’s theorem can be see as the Hubble of mathematics: it shows that it is *very* big, and that there is an infinity of surprises awaiting us. Bruno > On 1 Jun 2019, at 11:27, Tomas Pales <[email protected]> wrote: > > > > On Saturday, June 1, 2019 at 10:06:31 AM UTC+2, Bruno Marchal wrote: > >> On 31 May 2019, at 22:27, Tomas Pales <[email protected] <javascript:>> >> wrote: >> >> Existence is just the principle of logical consistency or identity. > > Almost. Peano arithmetic is consistent with the proposition that Peano > arithmetic is inconsistent. Consistency is shown rather cheap, and far away > from Truth, which is the key notion, but of course not a very obvious one. > > > If PA is consistent, it exists. If PA is not consistent, it doesn't exist. > But since PA is subject to Godel's second incompleteness theorem, we may > never know, right? > > > Here I disagree. 17 is very concrete. It the successor of 16, which is very > concrete, etc. With mechanism, 0, 1, 2, 3, … are taken as the most concrete > “really existing” object. The moon, and yourself are extremely abstract type, > having only phenomenological existence. > > This is what I mean by "abstract" and "concrete": > > abstract: has instances/examples (is a property) > > concrete: has no instances/examples (is not a property) > > Number 17 has instances/examples in any collection of 17 objects (is a > property of any collection of 17 objects). Therefore number 17 is an abstract > object. > > Our moon has no instances/examples (is not a property of any object, just as > Bruno Marchal is not a property of any object). Therefore our moon is a > concrete object. > > > > > >> >> >> -- >> 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] <javascript:>. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/everything-list/2470590b-373e-4595-970c-40409ee19907%40googlegroups.com >> >> <https://groups.google.com/d/msgid/everything-list/2470590b-373e-4595-970c-40409ee19907%40googlegroups.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/30ae74e7-66d0-47f9-baa2-230001d10843%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/30ae74e7-66d0-47f9-baa2-230001d10843%40googlegroups.com?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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