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Path: ...!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: Sun, 27 Oct 2024 09:29:22 -0500 Organization: A noiseless patient Spider Lines: 198 Message-ID: <vflio2$fj8s$3@dont-email.me> References: <ves6p1$2uoln$1@dont-email.me> <vev8gg$3me0u$1@dont-email.me> <eb38c4aff9c8bc250c49892461ac25bfccfe303f@i2pn2.org> <vf051u$3rr97$1@dont-email.me> <e3f28689429722f86224d0d736115e4d1895299b@i2pn2.org> <vf1hun$39e3$1@dont-email.me> <dedb2801cc230a4cf689802934c4b841ae1a29eb@i2pn2.org> <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> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Sun, 27 Oct 2024 15:29:23 +0100 (CET) Injection-Info: dont-email.me; posting-host="aecb0ee9851572be14adcf8faad49f58"; logging-data="511260"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1++i3A/sRyWjALLpOLT+KmF" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:DEF8yz5gUvdnP2OWCoHupNXbUI0= X-Antivirus: Norton (VPS 241027-2, 10/27/2024), Outbound message Content-Language: en-US X-Antivirus-Status: Clean In-Reply-To: <vfkvk2$8h64$1@dont-email.me> Bytes: 10831 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. >>>>>>> >>>>>> >>>>>> I am proposing actually doing Gödel's actual proof and >>>>>> deriving all of the digits of the actual Gödel numbers. >>>>>> >>>>> >>>>> Then try it and see. >>>>> >>>>> You do understand that the first step is to fully enumerate all the >>>>> axioms of the system, and any proofs used to generate the needed >>>>> properties of the mathematics that he uses. >>>>> >>>> >>>> Gödel seems to propose that his numbers are >>>> actual integers, are you saying otherwise? >>>> >>> >>> Not at all, just that they may be very large numbers. >> >> Are they less than one GB each? I want to see the c >> code that computes them. I want to know how many bytes >> of ASCII digits strings they are. > > The memory needs are easier to estimate if you use a different > numbering system: > > 1. Encode all formulas with the 94 visible ASCII characters. ========== REMAINDER OF ARTICLE TRUNCATED ==========