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Path: ...!weretis.net!feeder9.news.weretis.net!news.quux.org!eternal-september.org!feeder2.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott <polcott333@gmail.com> Newsgroups: comp.theory Subject: =?UTF-8?Q?Re=3A_G=C3=B6del=27s_actual_proof_and_deriving_all_of_the?= =?UTF-8?Q?_digits_of_the_actual_G=C3=B6del_numbers?= Date: Thu, 31 Oct 2024 07:15:42 -0500 Organization: A noiseless patient Spider Lines: 274 Message-ID: <vfvsdf$2mcse$2@dont-email.me> References: <ves6p1$2uoln$1@dont-email.me> <vf1stu$8h0v$1@dont-email.me> <592109c757262c48aaca517a829ea1867913316b@i2pn2.org> <vf37qt$fbb3$1@dont-email.me> <vf5430$sjvj$1@dont-email.me> <vf5mat$v6n5$4@dont-email.me> <vf7jbl$1cr7h$1@dont-email.me> <vf8b8p$1gkf5$3@dont-email.me> <vfa8iu$1ulea$1@dont-email.me> <vfassk$21k64$4@dont-email.me> <vfdjc7$2lcba$1@dont-email.me> <vfdlij$2ll17$1@dont-email.me> <vffj9k$33eod$1@dont-email.me> <vfg6j4$36im7$1@dont-email.me> <dcc4d67737371dbac58b18d718b2d3b6613f1b24@i2pn2.org> <vfh3vp$3bkkv$1@dont-email.me> <040cd8511c02a898516db227faa75dbc5f74a097@i2pn2.org> <vfh8ad$3cdsr$1@dont-email.me> <17cad36a46956f00484737183121e8a2c9e742ef@i2pn2.org> <vfish6$3ner2$8@dont-email.me> <vfkvk2$8h64$1@dont-email.me> <vflio2$fj8s$3@dont-email.me> <vfnicm$to2h$1@dont-email.me> <vfo5l8$10s4m$1@dont-email.me> <vfq3dq$1fj4d$1@dont-email.me> <vfqnoe$1iaob$3@dont-email.me> <vfsvv1$23p4t$1@dont-email.me> <vft7tn$25aio$1@dont-email.me> <vfvjjm$2l1fl$1@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 31 Oct 2024 13:15:43 +0100 (CET) Injection-Info: dont-email.me; posting-host="a20cf42bc93637678342112c0763e5cc"; logging-data="2831246"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+5F2GEvggiSIBEcGTRfmeN" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:YdQH61/hLUjlY2YJALcNV0Spnao= X-Antivirus: Norton (VPS 241031-2, 10/31/2024), Outbound message X-Antivirus-Status: Clean In-Reply-To: <vfvjjm$2l1fl$1@dont-email.me> Content-Language: en-US Bytes: 14642 On 10/31/2024 4:45 AM, Mikko wrote: > On 2024-10-30 12:13:43 +0000, olcott said: > >> On 10/30/2024 4:57 AM, Mikko wrote: >>> On 2024-10-29 13:25:34 +0000, olcott said: >>> >>>> On 10/29/2024 2:38 AM, Mikko wrote: >>>>> On 2024-10-28 14:04:24 +0000, olcott said: >>>>> >>>>>> On 10/28/2024 3:35 AM, Mikko wrote: >>>>>>> On 2024-10-27 14:29:22 +0000, olcott said: >>>>>>> >>>>>>>> On 10/27/2024 4:02 AM, Mikko wrote: >>>>>>>>> On 2024-10-26 13:57:58 +0000, olcott said: >>>>>>>>> >>>>>>>>>> On 10/25/2024 11:07 PM, Richard Damon wrote: >>>>>>>>>>> On 10/25/24 7:06 PM, olcott wrote: >>>>>>>>>>>> On 10/25/2024 5:17 PM, Richard Damon wrote: >>>>>>>>>>>>> On 10/25/24 5:52 PM, olcott wrote: >>>>>>>>>>>>>> On 10/25/2024 10:52 AM, Richard Damon wrote: >>>>>>>>>>>>>>> On 10/25/24 9:31 AM, olcott wrote: >>>>>>>>>>>>>>>> On 10/25/2024 3:01 AM, Mikko wrote: >>>>>>>>>>>>>>>>> On 2024-10-24 14:28:35 +0000, olcott said: >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> On 10/24/2024 8:51 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>> On 2024-10-23 13:15:00 +0000, olcott said: >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> On 10/23/2024 2:28 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>> On 2024-10-22 14:02:01 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> On 10/22/2024 2:13 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>> On 2024-10-21 13:52:28 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> On 10/21/2024 3:41 AM, Mikko wrote: >>>>>>>>>>>>>>>>>>>>>>>>> On 2024-10-20 15:32:45 +0000, olcott said: >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>>> The actual barest essence for formal systems >>>>>>>>>>>>>>>>>>>>>>>>>> and computations >>>>>>>>>>>>>>>>>>>>>>>>>> is finite string transformation rules applied >>>>>>>>>>>>>>>>>>>>>>>>>> to finite strings. >>>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>>> Before you can start from that you need a >>>>>>>>>>>>>>>>>>>>>>>>> formal theory that >>>>>>>>>>>>>>>>>>>>>>>>> can be interpreted as a theory of finite strings. >>>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> Not at all. The only theory needed are the >>>>>>>>>>>>>>>>>>>>>>>> operations >>>>>>>>>>>>>>>>>>>>>>>> that can be performed on finite strings: >>>>>>>>>>>>>>>>>>>>>>>> concatenation, substring, relational operator ... >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> You may try with an informal foundation but you >>>>>>>>>>>>>>>>>>>>>>> need to make sure >>>>>>>>>>>>>>>>>>>>>>> that it is sufficicently well defined and that is >>>>>>>>>>>>>>>>>>>>>>> easier with a >>>>>>>>>>>>>>>>>>>>>>> formal theory. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>>> The minimal complete theory that I can think of >>>>>>>>>>>>>>>>>>>>>>>> computes >>>>>>>>>>>>>>>>>>>>>>>> the sum of pairs of ASCII digit strings. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> That is easily extended to Peano arithmetic. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>>> As a bottom layer you need some sort of logic. >>>>>>>>>>>>>>>>>>>>>>> There must be unambifuous >>>>>>>>>>>>>>>>>>>>>>> rules about syntax and inference. >>>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> I already wrote this in C a long time ago. >>>>>>>>>>>>>>>>>>>>>> It simply computes the sum the same way >>>>>>>>>>>>>>>>>>>>>> that a first grader would compute the sum. >>>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>>> I have no idea how the first grade arithmetic >>>>>>>>>>>>>>>>>>>>>> algorithm could be extended to PA. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> Basically you define that the successor of X is X + >>>>>>>>>>>>>>>>>>>>> 1. The only >>>>>>>>>>>>>>>>>>>>> primitive function of Peano arithmetic is the >>>>>>>>>>>>>>>>>>>>> successor. Addition >>>>>>>>>>>>>>>>>>>>> and multiplication are recursively defined from the >>>>>>>>>>>>>>>>>>>>> successor >>>>>>>>>>>>>>>>>>>>> function. Equality is often included in the >>>>>>>>>>>>>>>>>>>>> underlying logic but >>>>>>>>>>>>>>>>>>>>> can be defined recursively from the successor >>>>>>>>>>>>>>>>>>>>> function and the >>>>>>>>>>>>>>>>>>>>> order relation is defined similarly. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>>> Anyway, the details are not important, only that it >>>>>>>>>>>>>>>>>>>>> can be done. >>>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> First grade arithmetic can define a successor function >>>>>>>>>>>>>>>>>>>> by merely applying first grade arithmetic to the pair >>>>>>>>>>>>>>>>>>>> of ASCII digits strings of [0-1]+ and "1". >>>>>>>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Peano_axioms >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> The first incompleteness theorem states that no >>>>>>>>>>>>>>>>>>>> consistent system of axioms whose theorems can be >>>>>>>>>>>>>>>>>>>> listed by an effective procedure (i.e. an algorithm) >>>>>>>>>>>>>>>>>>>> is capable of proving all truths about the >>>>>>>>>>>>>>>>>>>> arithmetic of natural numbers. For any such >>>>>>>>>>>>>>>>>>>> consistent formal system, there will always be >>>>>>>>>>>>>>>>>>>> statements about natural numbers that are true, but >>>>>>>>>>>>>>>>>>>> that are unprovable within the system. >>>>>>>>>>>>>>>>>>>> https://en.wikipedia.org/wiki/ >>>>>>>>>>>>>>>>>>>> G%C3%B6del%27s_incompleteness_theorems >>>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>>> When we boil this down to its first-grade arithmetic >>>>>>>>>>>>>>>>>>>> foundation >>>>>>>>>>>>>>>>>>>> this would seem to mean that there are some cases >>>>>>>>>>>>>>>>>>>> where the >>>>>>>>>>>>>>>>>>>> sum of a pair of ASCII digit strings cannot be >>>>>>>>>>>>>>>>>>>> computed. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> No, it does not. Incompleteness theorem does not >>>>>>>>>>>>>>>>>>> apply to artihmetic >>>>>>>>>>>>>>>>>>> that only has addition but not multiplication. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>>> The incompleteness theorem is about theories that >>>>>>>>>>>>>>>>>>> have quantifiers. >>>>>>>>>>>>>>>>>>> A specific arithmetic expression (i.e, with no >>>>>>>>>>>>>>>>>>> variables of any kind) >>>>>>>>>>>>>>>>>>> always has a well defined value. >>>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> So lets goes the next step and add multiplication to >>>>>>>>>>>>>>>>>> the algorithm: >>>>>>>>>>>>>>>>>> (just like first grade arithmetic we perform >>>>>>>>>>>>>>>>>> multiplication >>>>>>>>>>>>>>>>>> on arbitrary length ASCII digit strings just like >>>>>>>>>>>>>>>>>> someone would >>>>>>>>>>>>>>>>>> do with pencil and paper). >>>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>>> Incompleteness cannot be defined. until we add >>>>>>>>>>>>>>>>>> variables and >>>>>>>>>>>>>>>>>> quantification: There exists an X such that X * 11 = 132. >>>>>>>>>>>>>>>>>> Every detail of every step until we get G is >>>>>>>>>>>>>>>>>> unprovable in F. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>>> Incompleteness is easier to define if you also add the >>>>>>>>>>>>>>>>> power operator >>>>>>>>>>>>>>>>> to the arithmetic. Otherwise the expressions of >>>>>>>>>>>>>>>>> provability and >>>>>>>>>>>>>>>>> incompleteness are more complicated. They become much >>>>>>>>>>>>>>>>> simpler if >>>>>>>>>>>>>>>>> instead of arithmetic the fundamental theory is a >>>>>>>>>>>>>>>>> theory of finite >>>>>>>>>>>>>>>>> strings. As you already observed, arithmetic is easy to >>>>>>>>>>>>>>>>> do with >>>>>>>>>>>>>>>>> finite strings. The opposite is possible but much more >>>>>>>>>>>>>>>>> complicated. >>>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>>> The power operator can be built from repeated operations of >>>>>>>>>>>>>>>> the multiply operator. Will a terabyte be enough to store >>>>>>>>>>>>>>>> the Gödel numbers? >>>>>>>>>>>>>>>> >>>>>>>>>>>>>>> >>>>>>>>>>>>>>> Likely depends on how big of a system you are making F. >>>>>>>>>>>>>>> ========== REMAINDER OF ARTICLE TRUNCATED ==========