Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: Ben Bacarisse Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics Subject: Re: Simply defining =?iso-8859-1?Q?G=F6del?= Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Followup-To: comp.theory Date: Wed, 15 Jul 2020 02:56:56 +0100 Organization: A noiseless patient Spider Lines: 88 Message-ID: <875zapk0bb.fsf@bsb.me.uk> References: <2tCdnb0urbddzpfCnZ2dnUU7-b_NnZ2d@giganews.com> <87k0z85tt0.fsf@nosuchdomain.example.com> <87d0505kmk.fsf@nosuchdomain.example.com> <5Lmdnehh4P6hLZbCnZ2dnUU7-LdQAAAA@giganews.com> <87365vnik3.fsf@bsb.me.uk> <87a703lz5c.fsf@bsb.me.uk> <87pn8ykrwq.fsf@bsb.me.uk> <7e-dnQpoj9jkoZPCnZ2dnUU7-UHNnZ2d@giganews.com> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="3e1a40821f578e828081ad10d5e7abdf"; logging-data="12796"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19awN2vHno4buNyguWWlXRLYNNRbMwMTFc=" Cancel-Lock: sha1:tDBZnuQF4ZpWlRvuRO2IcXhSL38= sha1:tUwCv8aFamWLqQAi6Yda+MT5rEg= X-BSB-Auth: 1.a3bdfd44c5cd38c83fc7.20200715025656BST.875zapk0bb.fsf@bsb.me.uk Xref: csiph.com comp.theory:21667 comp.ai.philosophy:22012 comp.ai.nat-lang:2402 olcott writes: > On 7/14/2020 11:00 AM, Ben Bacarisse wrote: >> olcott writes: >> >>> On 7/13/2020 7:26 PM, Ben Bacarisse wrote: >>>> olcott writes: >>>> >>>>> On 7/13/2020 5:42 PM, Ben Bacarisse wrote: >>>>>> olcott writes: >>>>>> >>>>>>> φ = ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) >>>>>>> >>>>>>> φ is not true or false in Q because Q lacks a mapping in Q from φ to a >>>>>>> Boolean value. >>>>>> >>>>>> Would you like to learn why that's wrong, or would you rather just keep >>>>>> repeating it? >>>>>> >>>>>> If you'd like to learn, you have to be a student. I'd ask a student to >>>>>> consider an instance of x + y = y + x in Q, for example this one: >>>>>> >>>>>> S0 + SS0 = SS0 + S0. >>>>>> >>>>>> and I'd ask them: what can you say about this formula in Q? >>>>>> >>>>> >>>>> How do you get from point "A" to point "B" when no path from point "A" >>>>> to point "B" exists? YOU DON'T !!! >>>> >>>> This does not appear to answer my question. Are you implying that >>>> nothing can be said about S0 + SS0 = SS0 + S0 in Q? Because that is not >>>> the case. Here's a hint: apply the axioms for + to the LRS and to the >>>> RHS. >>> >>> That you can prove that 1 + 2 = 2 + 1 in Q merely diverts attention >>> away from the fact that this expression ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) >>> cannot be proved in Q. >> >> Your rant about paths made me think you did not know that S0 + SS0 = SS0 >> + S0 is a theorem of Q. I will assume you accept that every formula of >> the form S^n0 + S^m0 = S^m0 + S^n0 is also a theorem of Q. (Do say if >> you dispute this.) >> >> My next questions (I continue to assume you want to learn) would be: do >> you know what a model of Q is? Could you give an example of a model of >> Q? This is crucial. I need you to understand a bit about models of Q >> so I can explain why ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) is, in fact, a rather >> confusing formula. It has two meanings and they are subtly different. > > This is the subject of the last few threads: > > https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq= > > Ultimately True(T, φ) is a mathematical mapping in T from φ to a > Boolean value. I had hoped you wanted to learn what logicians mean by "true". The example of Robinson arithmetic is an enlightening one (that's why it was devised) but if you'd rather just re-state you opinions (as it anyone should care about them), go ahead. My offer to explain still stands. Just say if you know what a model is, and we can take it from there. > What I just said is correct and true and a proper use of all the terms > that I used. That is not an opinion that is widely shared. > It is not, however the usual way that these sort of > things are typically described. Indeed. It makes it impossible for you to talk about these theorems, because you simply insert your meanings without alerting anyone. I've advised before that you put a PO- prefix onto words you've made up, but that would make it clear that you have nothing to say about any of the actual theorems. > If provable(T, φ) is a required element of the only way that φ can be > mathematically mapped in T to true, then True(T, φ) and Uprovable(T, > φ) are impossble. What you mean is that if PO-provable(T, φ) is a PO-required PO-element of the only way that φ can be mathematically PO-mapped in T to PO-true, then PO-True(T, φ) and PO-Unprovable(T, φ) are PO-impossible. But if you actually said that, no one would care. -- Ben.