Path: csiph.com!xmission!news.alt.net!feeder.usenetexpress.com!tr1.iad1.usenetexpress.com!border1.nntp.dca1.giganews.com!nntp.giganews.com!buffer1.nntp.dca1.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Wed, 15 Jul 2020 20:44:37 -0500 Subject: =?UTF-8?Q?Re=3a_Simply_defining_G=c3=b6del_Incompleteness_and_Tarsk?= =?UTF-8?Q?i_Undefinability_away_V24_=28Are_we_there_yet=3f=29?= Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics References:

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From: olcott Date: Wed, 15 Jul 2020 20:44:36 -0500 User-Agent: Mozilla/5.0 (Windows NT 10.0; WOW64; rv:68.0) Gecko/20100101 Thunderbird/68.10.0 MIME-Version: 1.0 In-Reply-To: Content-Type: text/plain; charset=utf-8; format=flowed Content-Language: en-US Content-Transfer-Encoding: 8bit Message-ID: Lines: 92 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-JNvqvoQumiVlc2OUAn4RtxGwm5jr9LGsYQDp357PDmlkaRrAyNCWJD0dokyDgPlsgHrJ6al5vmxaCU+!lAqK1ZQxU3R6eXp1MQwwMEcU1YHsHXvxdFzKXzdGjXon9nxABlqokQRZW2yFXWXrOIP5jp/Ge+I= X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 X-Original-Bytes: 6539 Xref: csiph.com comp.theory:21695 comp.ai.philosophy:22044 comp.ai.nat-lang:2424 On 7/15/2020 7:44 PM, David Kleinecke wrote: > On Wednesday, July 15, 2020 at 5:28:23 PM UTC-7, olcott wrote: >> On 7/15/2020 7:16 PM, Ben Bacarisse wrote: >>> olcott writes: >>> >>>> On 7/15/2020 6:35 PM, Ben Bacarisse wrote: >>>>> olcott writes: >>>>> >>>>>> On 7/15/2020 10:48 AM, Ben Bacarisse wrote: >>>>>>> olcott writes: >>>>>>> >>>>>>>> On 7/14/2020 8:56 PM, Ben Bacarisse wrote: >>>>> >>>>>>>>> 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. >>>>>>>>> >>>>>>>> Here is what I know of model theory: >>>>>>>> Satisfiability >>>>>>>> A formula is satisfiable if it is possible to find an interpretation >>>>>>>> (model) that makes the formula true. >>>>>>>> https://en.wikipedia.org/wiki/Satisfiability >>>>>>>> >>>>>>>> Interpretation (logic) >>>>>>>> An interpretation is an assignment of meaning to the >>>>>>>> [non-logical] symbols of a formal language. >>>>>>>> https://en.wikipedia.org/wiki/Interpretation_(logic) >>>>>>>> >>>>>>>> Model theory >>>>>>>> A model of a theory is a structure (e.g. an interpretation) >>>>>>>> that satisfies the sentences of that theory. >>>>>>>> https://en.wikipedia.org/wiki/Model_theory >>>>>>> >>>>>>> That tells me you know how to cut and paste. Do you really know what a >>>>>>> model is? For example, can you give two essentially different models of >>>>>>> Q? This a hard question, and I don't expect you to be able to answer it >>>>>>> yet, but you could have a stab at it and I'll try to steer you towards >>>>>>> an answer based on how far you get. >>>>> >>>>> From this refusal to engage, I have to assume that you don't know what a >>>>> model is, at least not well enough to give two distinct models for Q. >>> >>> You can't understand what we mean by true until you understand >>> interpretations and models. I've offered to help, but you need to do >>> some work. You could start by giving just one model of Q. >>> >>>>> The sentence in question is ∀x ∀y x + y = y + x or just x + y = y + x. >>>> >>>> If it is not a WFF of Q (and I suspect that you are correct in this >>>> assessment) then is ceases to be useful for my purposes. >>> >>> It is a well formed formula of Q. It can be used to show that Q is >>> incomplete. It's the version with "∈ ℕ" that was using the notation is >>> a poetic rather than a literal way. >>> >>>> My purpose is to prove that true and uprovable cannot possibly ever >>>> coexist, not in any formal system what-so-ever or even in the whole >>>> body of analytical knowledge. >>> >>> You don't yet know what true means. You've asserted that PO-true and >>> PO-unprovable can't "coexit", but no one cares about that. If you want >>> to talk about theories, truth and incompleteness you need to know what >>> these words mean. Making up your own meanings (or borrowing dictionary >>> definitions from other contexts) is simply pointless. >>> >> >> Unless you can understand things from the 10,000 foot level all the >> trees will get in the way of seeing the forest. >> >> Ultimately for the entire body of all analytical knowledge truth is the >> mathematical mapping from expressions of language to Boolean true. > > If that is what you are aiming at prepare for disappointment. > I am not aiming at that. That is merely the most basic truism beyond the state of existence itself exists. In other words that is what it is !!! Here are some more details so that math guys can begin to tie it into their understanding: There is no way to show that formula φ of theory T is satisfied in T that does not require a mathematical mapping from φ through the axioms and rules-of-inference of T to an element of the set of {true, false}. -- Copyright 2020 Pete Olcott