Path: csiph.com!xmission!news.snarked.org!border2.nntp.dca1.giganews.com!nntp.giganews.com!buffer2.nntp.dca1.giganews.com!buffer1.nntp.dca1.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Tue, 14 Jul 2020 10:11:06 -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: <2tCdnb0urbddzpfCnZ2dnUU7-b_NnZ2d@giganews.com> <87k0z85tt0.fsf@nosuchdomain.example.com> <87d0505kmk.fsf@nosuchdomain.example.com> <5Lmdnehh4P6hLZbCnZ2dnUU7-LdQAAAA@giganews.com> <878sfo5elp.fsf@nosuchdomain.example.com> From: olcott Date: Tue, 14 Jul 2020 10:11:03 -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: 132 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-uICXIrZ+2QPSNI4SVV9DEP5vuTHn6EWyGZdzSvmmZY4e+BdLQFXwKCNbl7TIt1s7PQ/plSmFFpnz4Mf!eNXaS2IDSGwEkcY3HHpChnTRHAvCeJ30v6u5MnFVrn+aLAgsbLh7vJ28gqCUdkTHFUAxdHIa/tk= 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: 7856 Xref: csiph.com comp.theory:21654 comp.ai.philosophy:21994 comp.ai.nat-lang:2389 On 7/14/2020 12:48 AM, André G. Isaak wrote: > On 2020-07-13 18:49, olcott wrote: >> On 7/13/2020 9:05 AM, André G. Isaak wrote: >>> On 2020-07-12 21:47, olcott wrote: >>>> On 7/12/2020 9:32 PM, Keith Thompson wrote: >>>>> olcott writes: >>>>>> On 7/12/2020 7:22 PM, Keith Thompson wrote: >>>>>>> olcott writes: >>>>>>>> On 7/12/2020 4:04 PM, Keith Thompson wrote: >>>>> [...] >>>>>>>>> Robinson Arithmetic cannot prove or disprove commutativity >>>>>>>>> of addition.  We can construct a consistent system based on >>>>>>>>> Robinson Arithmetic in which addition is provably commutative. >>>>>>>> >>>>>>>> Sure just add an axiom: ∀x ∈ ℕ ∀y ∈ ℕ (x + y = y + x) >>>>>>>> >>>>>>>>> Can we construct a consistent system based on Robinson Arithmetic >>>>>>>>> in which addition is provably *not* commutative? >>>> >>>> No. That would be like proving that existence never existed or >>>> finding some integer Crazy_Number such that Crazy_Number > 5 and >>>> Crazy_Number < 3. >>>> >>>> No thing of all things can be proved false that has been defined to >>>> be true. Defined to be true it the ultimate foundation of all truth. >>> >>> What exactly is it that you are claiming is 'defined to be true'? When so ever a Boolean function evaluates a WFF to be true. Predicates are defined to be Boolean functions. Boolean True("2+3=5") Boolean Theorem("2+3=5") Boolean Provable("2+3=5") >>> Certainly not the commutativity of addition in Q. In PA, one can >>> prove that addition is commutative, but it certainly isn't defined to >>> be true. >> >> A Boolean function having a WFF as its argument. > > I don't understand how that is an answer to the question. > > You claimed it was possible to create a system based on Q in which > addition is provably commutative by adding an axiom. > > You also claimed it was not possible to create a system based on Q in > which addition is *not* commutative because nothing can be proven to be > false which is defined to be true. But commutativity is certainly not > defined to be true in Q, so what is it that is "defined to be true" > which would preclude creating a system based on Q in which addition was > not commutative? > In the set of human knowledge there is a set of interrelationships between finite strings that define an algorithm that specifies the meaning of the commutativity of addition, thus commutativity is defined to be true on the basis of its meaning. The commutativity of addition is merely the name of this defined algorithm. >>> >>> If it is not possible to add some axiom to Q which makes addition >>> non-commutative then that would certainly support what everyone other >>> than you is claiming: that addition IS commutative in Q despite the >>> fact that this cannot be proven in Q. >>> >>> André >>> >> >> φ = ∀x ∈ ℕ ∀y ∈ ℕ (y > x) >> Boolean Is_Commutativity_of_Addition(φ) >> >> This function does not exist in Q because Q doesn't know about the >> commutativity of addition. > > That function also does not exist in PA, but commutativity is provable > in PA (whether that means PA 'knows' about commutativity is another > matter -- I have no idea what it means for a system to 'know' about > something). Since everyone here is indoctrinated into believing that Gödel is correct I have to use different terms for provability so that people will carefully analyze my reasoning and not simply dismiss it out-of-hand on the basis of their indoctrination. Ultimately provability in a formal system is a mathematical mapping from an expression of language to a Boolean value. Without this mapping the expression is not a truth bearer and thus neither true nor false. Is the question: "What time is it?" true or false? (or not a truth bearer). > >> I am taking the fact that that the commutativity of addition is not >> provable in Q to mean that φ is undecidable in Q because φ the >> commutativity of addition. >> >> If a sentence is undecidable in Q then this is merely another way of >> saying that it is neither true nor false in Q. > > Sigh. No it is not. It is a way of saying that Q can neither prove nor > disprove that sentence. True and false have nothing to do with it. And > Q, being based on standard logic, does not allow propositions which are > neither true nor false. > > André > In mathematical logic, a sentence of a predicate logic is a boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. https://en.wikipedia.org/wiki/Sentence_(mathematical_logic) The expression of the language of a formal system THAT DOES HAVE FREE VARIABLES apparently does exist in the language of formal system and DOES NOT meet the criteria of a proposition. One can infer from the above quote that the expressions of the language of formal systems with free variables are neither sentences nor propositions thus are not able to be true or false. 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) "If I say that X went to the the store to buy some Y" it remains unevaluatable until X and Y have been defined with values providing an interpretation of the above expression. -- Copyright 2020 Pete Olcott