Path: csiph.com!xmission!news.alt.net!feeder.usenetexpress.com!tr3.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: Thu, 09 Jul 2020 23:23:24 -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: From: olcott Date: Thu, 9 Jul 2020 23:23:23 -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: 191 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-OJPDK90L4KFdIHrCqOI5xmjASBeOFRM/5HZ9mLIhLLhQTxka/jJMHtoCAE5fFAM8X2g5ekadokGy2Y5!i80wmRgtoMsWSIjGG49ilHJ1aXpleLzdn8L9yiXn1sJyQ1zC1BKiTn2s3GzVopBzpA/zi3kD/tUT 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: 10541 Xref: csiph.com comp.theory:21535 comp.ai.philosophy:21850 comp.ai.nat-lang:2289 On 7/9/2020 12:33 PM, André G. Isaak wrote: > On 2020-07-09 10:02, olcott wrote: >> On 7/9/2020 7:56 AM, André G. Isaak wrote: >>> On 2020-07-08 10:29, olcott wrote: >>>> On 7/8/2020 10:50 AM, André G. Isaak wrote: >>>>> On 2020-07-08 09:11, olcott wrote: >>>>>> On 7/8/2020 1:14 AM, André G. Isaak wrote: >>>>>>> On 2020-07-07 23:54, olcott wrote: >>>>>>>> On 7/8/2020 12:39 AM, André G. Isaak wrote: >>>>>>>>> On 2020-07-07 23:16, olcott wrote: >>>>>>>>>> On 7/7/2020 11:43 PM, André G. Isaak wrote: >>>>>>>>>>> On 2020-07-07 22:00, olcott wrote: >>>>>>>>>>>> On 7/7/2020 10:52 PM, André G. Isaak wrote: >>>>>>>>>>>>> On 2020-07-07 18:43, olcott wrote: >>>>>>>>>>>>>> On 7/7/2020 7:27 PM, André G. Isaak wrote: >>>>>>>>>>>>>>> On 2020-07-07 16:12, olcott wrote: >>>>>>>>>>>>>>>> On 7/7/2020 3:50 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>> On 2020-07-07 14:25, olcott wrote: >>>>>>>>>>>>>>>>>> On 7/7/2020 3:17 PM, André G. Isaak wrote: >>>>>>>>>>>>>>>>>>> On 2020-07-07 14:00, olcott wrote: >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> For the LHS to have a value of true there must be a >>>>>>>>>>>>>>>>>>>> proof that there is no proof of φ. If there is no >>>>>>>>>>>>>>>>>>>> proof of φ then it does not have a truth value of >>>>>>>>>>>>>>>>>>>> true. CONTRADICTION !!! >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> A contradiction arises when something is both true >>>>>>>>>>>>>>>>>>> and false. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> There is no contradiction in being both true and >>>>>>>>>>>>>>>>>>> unprovable. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> André >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> -- ∃φ (φ ↔ T ⊬ φ) >>>>>>>>>>>>>>>>>> A truth value of true on the RHS of ↔ makes the truth >>>>>>>>>>>>>>>>>> value of the LHS of ↔ false, >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> No, it doesn't. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> André >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>      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) >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Why do you always cite random definitions in your post >>>>>>>>>>>>>>> which have no specific relevance to what you are asking? >>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> By what means could it be verified that φ is true in T >>>>>>>>>>>>>>>> besides T ⊢ φ? >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> This has been answered ad nauseam already. I will give >>>>>>>>>>>>>>> you several different answers and hopefully one will sink >>>>>>>>>>>>>>> in. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Answer 1: Since ∃φ (φ ↔ T ⊬ φ) doesn't appear anywhere in >>>>>>>>>>>>>>> Gödel's proof, your question is irrelevant. Gödel's proof >>>>>>>>>>>>>>> makes no reference to truth at all. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Answer 2: The sentence F ⊢ G_F ↔ ¬Prov_F(G_F), which also >>>>>>>>>>>>>>> doesn't appear in Gödel's proof, but is used by >>>>>>>>>>>>>>> Raatikainen, doesn't require that you know the truth of G_F. >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Answer 3: Your sentence, ∃φ (φ ↔ T ⊬ φ), also doesn't >>>>>>>>>>>>>>> require that you know the truth of φ. There are two >>>>>>>>>>>>>>> possible truth values which can be assigned to φ. If φ is >>>>>>>>>>>>>>> true and (φ ↔ T ⊬ φ) is true then T ⊬ φ is true, meaning >>>>>>>>>>>>>>> φ cannot be proven, meaning T is incomplete. >>>>>>>>>>>>> >>>>>>>>>>>>> That wasn't my Answer 3. That is the first half of my >>>>>>>>>>>>> answer 3. Did you forget to read the second half? >>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>>> You dodged the question. >>>>>>>>>>>>>> In order to know that: φ ↔ T ⊬ φ true we have to have some >>>>>>>>>>>>>> way of deciding whether or not φ is true and then compare >>>>>>>>>>>>>> that Boolean value to see if it has the same value as T ⊬ φ. >>>>>>>>>>>>>> >>>>>>>>>>>>>> The way that you did it you simply assumed that Gödel was >>>>>>>>>>>>>> right and used that as the whole basis of your proof. >>>>>>>>>>>>> Maybe you should read the rest of that answer. >>>>>>>>>>>>> >>>>>>>>>>>>> André >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> Every sentence of any language that asserts its own >>>>>>>>>>>> unprovability cannot possibly be proved to be satisfied >>>>>>>>>>>> within the same formal system. >>>>>>>>>>> >>>>>>>>>>> Which is utterly irrelevant. >>>>>>>>>>> >>>>>>>>>>> We are dealing with φ ↔ (T ⊬ φ), not with some sentence which >>>>>>>>>>> expresses its own unprovability. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> Every sentence of any language that asserts that it is >>>>>>>>>> logically equivalent to its own unprovability is logically >>>>>>>>>> equivalent to a sentence that asserts it own unprovability >>>>>>>>>> within the same formal system. >>>>>>>>> >>>>>>>>> Which, as I pointed out, is irrelevant since the above does not >>>>>>>>> do that. φ ↔ (T ⊬ φ) makes a claims about φ's unprovability, >>>>>>>>> not a claim about its own unprovability. >>>>>>>> >>>>>>>> Yes and a pie in the face it not a pie in the face too. >>>>>>>> >>>>>>>> ∃φ (φ ↔ T ⊬ φ) >>>>>>>> Here it is in English: >>>>>>>> >>>>>>>> There exists an expression of language φ such that φ is >>>>>>>> logically equivalent to its own unprovability in T; >>>>>>> >>>>>>> The above asserts the existence of some sentence φ which is true >>>>>>> if and only if it is not provable in T. It tells you nothing >>>>>>> whatsoever about what the actual *content* of φ is. It certainly >>>>>>> does not entail that φ expresses its own unprovability. >>>>>> >>>>>> One of the reasons that I created the Simple_Arithmetic formal >>>>>> system is so that we could give φ some content and then examine >>>>>> concrete examples. >>>>> >>>>> Your "Simple Arithmetic" is completely worthless here. >>>> >>>> We don't have to use Simple_Arithmetic ordinary FOPL will do >>>> >>>> // true sentences are provable >>>> When φ = "¬(p ∧ q) ↔ (¬p ∨ ¬q)" // De Morgans >>>> ∃φ (φ ↔ T ⊬ φ) >>>> LHS of ↔ is true. RHS of ↔ is false >>>> >>>> // false sentences are unprovable >>>> When φ = "¬(p ∧ q) ↔ (p ∧ q)" // Law of noncontradiction >>>> ∃φ (φ ↔ T ⊬ φ) >>>> LHS of ↔ is false. RHS of ↔ is true >>>> >>>> By concrete examples we have shown that ∃φ (φ ↔ T ⊬ φ) is >>>> unsatisfiable in any model. >>> >>> You've given *two* examples. This doesn't even show that this is >>> unsatisfiable in FOPL, let alone that it is unsatisfiable in *any* >>> model. >> >> Yes you are correct. Even my Simple_Arithmetic only shows that >> Theorem(T,φ) ↔ True(T,φ) for the infinite set of finite strings >> applied to one theory. >> >>> >>> You "argument" is en par with the following, equally >>> not-an-argument-at-all argument. >>> >>> Prime(P) ↔ Odd(P). >>> >>> To "prove" this claim, I will consider two cases: >>> >>> When P = 19 >>> Prime(19) ↔ Odd(19) >>> The LHS is true. The RHS is also TRUE. >>> >>> When P = 18 >>> >>> Prime(18) ↔ Odd(18) >>> The LHS is false. The RHS is also False; >>> >>> Therefore Prime(P) ↔ Odd(P) is TRUE. >>> >>> This obviously proves absolutely nothing. Why do you think the same >>> approach would succeed above? > > Are you not going to address this? McAfee site advisor hijacked my chrome settings and it took me all day (8 hours) to finally get them restored. I want to carefully study Mendelson to make sure that I am using the technical terminology exactly correctly before I reply to this post. I was planning on spending several hours on doing that today. -- Copyright 2020 Pete Olcott