Path: ...!eternal-september.org!feeder2.eternal-september.org!i2pn.org!i2pn2.org!.POSTED!not-for-mail From: Richard Damon 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: Wed, 30 Oct 2024 19:35:34 -0400 Organization: i2pn2 (i2pn.org) Message-ID: References: <592109c757262c48aaca517a829ea1867913316b@i2pn2.org> <040cd8511c02a898516db227faa75dbc5f74a097@i2pn2.org> <17cad36a46956f00484737183121e8a2c9e742ef@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Wed, 30 Oct 2024 23:35:34 -0000 (UTC) Injection-Info: i2pn2.org; logging-data="215954"; mail-complaints-to="usenet@i2pn2.org"; posting-account="diqKR1lalukngNWEqoq9/uFtbkm5U+w3w6FQ0yesrXg"; User-Agent: Mozilla Thunderbird In-Reply-To: Content-Language: en-US X-Spam-Checker-Version: SpamAssassin 4.0.0 Bytes: 13921 Lines: 258 On 10/30/24 8:13 AM, olcott wrote: > 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. >>>>>>>>>>>>>> >>>>>>>>>>>>> >>>>>>>>>>>>> I am proposing actually doing Gödel's actual proof and >>>>>>>>>>>>> deriving all of the digits of the actual Gödel numbers. >>>>>>>>>>>>> ========== REMAINDER OF ARTICLE TRUNCATED ==========