Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott Newsgroups: comp.theory,sci.logic Subject: =?UTF-8?Q?Re=3A_ZFC_solution_to_incorrect_questions=3A_reject_them_?= =?UTF-8?Q?--G=C3=B6del--?= Date: Tue, 12 Mar 2024 22:14:15 -0500 Organization: A noiseless patient Spider Lines: 199 Message-ID: References: MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 13 Mar 2024 03:14:16 -0000 (UTC) Injection-Info: dont-email.me; posting-host="aa13334f329e2006d1dfb90f9960e443"; logging-data="805596"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+ZpmfD6AXjgKsf/wEyeXJ1" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:kVBku1TOxXSaEOCIjw52HX+5rkE= Content-Language: en-US In-Reply-To: Bytes: 9742 On 3/12/2024 9:51 PM, Richard Damon wrote: > On 3/12/24 4:14 PM, olcott wrote: >> On 3/12/2024 6:00 PM, Richard Damon wrote: >>> On 3/12/24 2:44 PM, olcott wrote: >>>> On 3/12/2024 4:31 PM, Richard Damon wrote: >>>>> On 3/12/24 1:38 PM, olcott wrote: >>>>>> On 3/12/2024 3:31 PM, immibis wrote: >>>>>>> On 12/03/24 20:02, olcott wrote: >>>>>>>> On 3/12/2024 1:31 PM, immibis wrote: >>>>>>>>> On 12/03/24 19:12, olcott wrote: >>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders >>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  | >>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>>> >>>>>>>>>> There is some input TMD to every H such that >>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD) >>>>>>>>> >>>>>>>>> And it can be a different TMD to each H. >>>>>>>>> >>>>>>>>>> When we disallow decider/input pairs that are incorrect >>>>>>>>>> questions where both YES and NO are the wrong answer >>>>>>>>> >>>>>>>>> Once we understand that either YES or NO is the right answer, >>>>>>>>> the whole rebuttal is tossed out as invalid and incorrect. >>>>>>>>> >>>>>>>> >>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts >>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does >>>>>>>> not halt >>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ >>>>>>>> >>>>>>> >>>>>>> Once we understand that either YES or NO is the right answer, the >>>>>>> whole rebuttal is tossed out as invalid and incorrect. >>>>>>> >>>>>>>>>> Does the barber that shaves everyone that does not shave >>>>>>>>>> themselves shave himself? is rejected as an incorrect question. >>>>>>>>> >>>>>>>>> The barber does not exist. >>>>>>>> >>>>>>>> Russell's paradox did not allow this answer within Naive set >>>>>>>> theory. >>>>>>> >>>>>>> Naive set theory says that for every predicate P, the set {x | >>>>>>> P(x)} exists. This axiom was a mistake. This axiom is not in ZFC. >>>>>>> >>>>>>> In Turing machines, for every non-empty finite set of alphabet >>>>>>> symbols Γ, every b∈Γ, every Σ⊆Γ, every non-empty finite set of >>>>>>> states Q, every q0∈Q, every F⊆Q, and every δ:(Q∖F)×Γ↛Q×Γ×{L,R}, >>>>>>> ⟨Q,Γ,b,Σ,δ,q0,F⟩ is a Turing machine. Do you think this is a >>>>>>> mistake? Would you remove this axiom from your version of Turing >>>>>>> machines? >>>>>>> >>>>>>> (Following the definition used on Wikipedia: >>>>>>> https://en.wikipedia.org/wiki/Turing_machine#Formal_definition) >>>>>>> >>>>>>>>> The following is true statement: >>>>>>>>> >>>>>>>>> ∀ Barber ∈ People. ¬(∀ Person ∈ People. Shaves(Barber, Person) >>>>>>>>> ⇔ ¬Shaves(Person, Person)) >>>>>>>>> >>>>>>>>> The following is a true statement: >>>>>>>>> >>>>>>>>> ¬∃ Barber ∈ People. (∀ Person ∈ People. Shaves(Barber, Person) >>>>>>>>> ⇔ ¬Shaves(Person, Person)) >>>>>>>>> >>>>>>>> >>>>>>>> That might be correct I did not check it over and over >>>>>>>> again and again to make sure. >>>>>>>> >>>>>>>> The same reasoning seems to rebut Gödel Incompleteness: >>>>>>>> ...We are therefore confronted with a proposition which >>>>>>>> asserts its own unprovability. 15 ... (Gödel 1931:43-44) >>>>>>>> ¬∃G ∈ F | G := ~(F ⊢ G) >>>>>>>> >>>>>>>> Any G in F that asserts its own unprovability in F is >>>>>>>> asserting that there is no sequence of inference steps >>>>>>>> in F that prove that they themselves do not exist in F. >>>>>>> >>>>>>> The barber does not exist and the proposition does not exist. >>>>>>> >>>>>> >>>>>> When we do this exact same thing that ZFC did for self-referential >>>>>> sets then Gödel's self-referential expressions that assert their >>>>>> own unprovability in F also cease to exist. >>>>>> >>>>> >>>>> And you end up with a very weak logic system that can't even have >>>>> the full properties of the Natuarl Numbers. >>>> >>>> Natural numbers never really did have the property of provability. >>>> This was something artificially contrived that never really belonged >>>> to them. >>>> >>> >>> No, Godel showed (or maybe used a previous proof) that you can use >>> the Mathematics of Natural Numbers to test if a proof is valid. >>> >>> You just don't understand it. It really is very related to how Turing >>> Machines work, which can be converted to a mathematical model. >>> >>> There is a field that looks at the comparability of Computations to >>> Logic, so they are all really quite related. >> >> This is refuted. >> ...We are therefore confronted with a proposition which >> asserts its own unprovability. 15 ...(Gödel 1931:43-44) > > Right, seen in the meta-Theory from F. > >> >> based on immblis Russell's Paradox reply >> ¬∃G ∈ F | G ↔ ~(F ⊢ G)  // is simply false > > Then a proof must exist in F that G is True, So G can't be false. > > Note, The statement G is NOT a statement about itself being provable, > that is only a semantic revealed in the RIGHT meta-theory. > *Not in my actual example where F has its own provability operator* >> >> Any G in F that asserts its own unprovability in F is >> asserting that there is no sequence of inference steps >> in F that prove that they themselves do not exist in F. >> > > No FINITE sequence of inference steps. > No one can prove that they themselves do not exist. Thus G cannot possibly derive a sequence of inference steps that prove that they themselves do not exist. > Note, the key is that G doesn't assert that, G is a statement about > math, that only when interpreted in a meta-theory sees that. > I am not referring to that one. > G is actually a statment about the existance of a number that matches a > complected and carefully constructed relationship, that is fully > computable. The Existance or non-existance of such a number is a pure > binary thing, either it WILL or it WON'T. > > The complicated relationship deals with a way to encode as a number ANY > analyitic statement in F (since all statements are just strings, and > strings can be encoded into a number), and a calculation to see if that > statement actually is a proof starting with the enumerated truth makers > of F, through the valid and allowed logical operatons to the statement > of G. A number that satisfies this relation, encodes a proof of G, and > any such proof, can always be encoded in to a number. > No formal system can possibly have any sequence of inference step that prove that they themselves do not exist because this is self-contradictory, not because the system is incomplete. In my G this is obvious. Whether or not Gödel's G is isomorphic to mine I have proved this this is false by providing the counter-example of my G. ....14 Every epistemological antinomy can likewise be used for a similar undecidability proof...(Gödel 1931:43-44) > G is the statement that no such number exist. So, if G is false, then > such a number exists, and then in the Meta-Theory, we can decode that > number into a proof in F that G must be true. > > If this is the case, then F must be inconsistant, and the proof starts > with the presumption that we are dealing with a consistant logic system. > Or it could be the epistemological antinomy of trying to find a sequence of inference steps in F that prove that they themselves do not exist. ========== REMAINDER OF ARTICLE TRUNCATED ==========