| Deutsch English Français Italiano |
|
<va79s8$e6ht$2@dont-email.me> View for Bookmarking (what is this?) Look up another Usenet article |
Path: ...!news.mixmin.net!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: Mikko <mikko.levanto@iki.fi> Newsgroups: sci.logic Subject: Re: This makes all Analytic(Olcott) truth computable --- ZFC Date: Thu, 22 Aug 2024 15:10:48 +0300 Organization: - Lines: 149 Message-ID: <va79s8$e6ht$2@dont-email.me> References: <v86olp$5km4$1@dont-email.me> <v92q4f$37e9$1@dont-email.me> <v94l1p$ldq7$1@dont-email.me> <v95c2j$p5rb$4@dont-email.me> <v95cke$p5rb$5@dont-email.me> <v977fo$gsru$1@dont-email.me> <v97goj$ielu$1@dont-email.me> <v9c93e$35sg6$1@dont-email.me> <v9d3k1$3ajip$1@dont-email.me> <v9ffpr$3s45o$1@dont-email.me> <v9fkd4$3se8c$1@dont-email.me> <v9kg66$tdvb$1@dont-email.me> <v9nbjf$1dj8q$1@dont-email.me> <20b1dea98eda49e74e822c96b37565bb3eb36013@i2pn2.org> <v9o4p2$1h5u4$1@dont-email.me> <cd12fb81fcd05d2e112fc8aca2f5b791c521cfc9@i2pn2.org> <v9oddf$1i745$2@dont-email.me> <7f2a1f77084810d4cee18ac3b44251601380b93a@i2pn2.org> <v9ogmp$1i745$6@dont-email.me> <662de0ccc3dc5a5f0be0918d340aa3314d51a348@i2pn2.org> <v9oj4r$1i745$8@dont-email.me> <v9sibq$2bq1o$1@dont-email.me> <v9sn85$2c67u$6@dont-email.me> <v9v0u0$2qajg$1@dont-email.me> <v9vgbu$2rjt1$13@dont-email.me> <va1qmi$3biht$1@dont-email.me> <va27ge$3cvgv$7@dont-email.me> <va4a0u$3q89u$1@dont-email.me> <va4n2u$3s0hu$3@dont-email.me> MIME-Version: 1.0 Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Thu, 22 Aug 2024 14:10:48 +0200 (CEST) Injection-Info: dont-email.me; posting-host="b61e73617b836bb657aca38e501dcccb"; logging-data="465469"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+0LCyAfl80Ph4LsOZrmfui" User-Agent: Unison/2.2 Cancel-Lock: sha1:tk8zVZYpRU6S1ndKhy15hg6PiIg= Bytes: 8079 On 2024-08-21 12:37:50 +0000, olcott said: > On 8/21/2024 3:54 AM, Mikko wrote: >> On 2024-08-20 13:59:42 +0000, olcott said: >> >>> On 8/20/2024 5:21 AM, Mikko wrote: >>>> On 2024-08-19 13:12:30 +0000, olcott said: >>>> >>>>> On 8/19/2024 3:49 AM, Mikko wrote: >>>>>> On 2024-08-18 11:51:33 +0000, olcott said: >>>>>> >>>>>>> On 8/18/2024 5:28 AM, Mikko wrote: >>>>>>>> On 2024-08-16 22:16:59 +0000, olcott said: >>>>>>>> >>>>>>>>> On 8/16/2024 5:03 PM, Richard Damon wrote: >>>>>>>>>> On 8/16/24 5:35 PM, olcott wrote: >>>>>>>>>>> On 8/16/2024 4:05 PM, Richard Damon wrote: >>>>>>>>>>>> On 8/16/24 4:39 PM, olcott wrote: >>>>>>>>>>>>> >>>>>>>>>>>>> ZFC didn't need to do that. All they had to do is >>>>>>>>>>>>> redefine the notion of a set so that it was no longer >>>>>>>>>>>>> incoherent. >>>>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> I guess you haven't read the papers of Zermelo and Fraenkel. They >>>>>>>>>>>> created a new definition of what a set was, and then showed what that >>>>>>>>>>>> implies, since by changing the definitions, all the old work of set >>>>>>>>>>>> theory has to be thrown out, and then we see what can be established. >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> None of this is changing any more rules. All >>>>>>>>>>> of these are the effects of the change of the >>>>>>>>>>> definition of a set. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> No, they defined not only what WAS a set, but what you could do as >>>>>>>>>> basic operations ON a set. >>>>>>>>>> >>>>>>>>>> Axiom of extensibility: the definition of sets being equal, that ZFC is >>>>>>>>>> built on first-order logic. >>>>>>>>> >>>>>>>>> >>>>>>>>>> >>>>>>>>>> Axion of regularity/Foundation: This is the rule that a set can not be >>>>>>>>>> a member of itself, and that we can count the members of a set. >>>>>>>>>> >>>>>>>>> This one is the key that conquered Russell's Paradox. >>>>>>>>> If anything else changed it changed on the basis of this change >>>>>>>>> or was not required to defeat RP. >>>>>>>> >>>>>>>> That is not sufficient. They also had to Comprehension. >>>>>>>> >>>>>>>>>> Axiom Schema of Specification: We can build a sub-set from another set >>>>>>>>>> and a set of conditions. (Which implies the existance of the empty set) >>>>>>>> >>>>>>>> This is added to keep most of Comprenesion but not Russell's set. >>>>>>>> >>>>>>> >>>>>>> All they did was (as I already said) was redefine the notion of a set. >>>>>>> That this can still be called set theory seems redundant. >>>>>> >>>>>> They did, as both Richard Damon and I already said, much more. They >>>>>> also explained their rationale, worked out various consequnces of >>>>>> their axioms and compared them to expectations, and developed better >>>>>> sets of axioms. >>>>>> >>>>> >>>>> They made no other changes to the notion of set theory >>>>> than redefining what a set is. Even then it seems they >>>>> did less than this. >>>> >>>> That is so obvious that needs not be mentined. There is nothing >>>> in the set theory expept what a set is so obviously nothing else >>>> can be changed. >>>> >>> >>> There are at least two tings in set theory: >>> (a) What a set is >>> (b) How a set works >> >> They are the same thing. There is nothing in a set other than how >> a set works. And it does not work in any way other than having >> certain relations to other sets. >> >>> When how a set is constructed is changed this single >>> change has great impact yet is still only one change. >> >> That is true. Therefore one must be careful with the construction >> rules and ensure that non-existent or undesiderable sets cannot >> be constructed but all sets that are regarded necessary can be >> constructed. >> >>>>> From what I recall it seems that they only changed how >>>>> sets can be constructed. The operations that can be >>>>> performed on sets remained the same. >>>> >>>> There are axioms about exstence and non-existence of certain kind of >>>> sets. For example, the axiom of regularity (aka foudation) specifies >>>> that ill-founded sets (e.g., Quine's atom) do not exist. >>>> >>>>>> One consequence of ZF axioms is that there is no set that contains all >>>>>> other sets as members. Some regard this as a defect and have developed >>>>>> set thories that have a universal set that contains all other sets as >>>>>> members (and usually itself, too). >>>>> >>>>> Then maybe they did this incorrectly. They only needed to >>>>> specify that a set cannot be a member of itself when a >>>>> set is constructed. This would not preclude a universal >>>>> set of all other sets. >>>> >>>> The power set axiom prevents the existence of a set that contains >>>> all other sets. >>> >>> In mathematics, the axiom of power set[1] is one of the >>> Zermelo–Fraenkel axioms of axiomatic set theory. It >>> guarantees for every set x the existence of a set P(x) >>> the power set of x consisting precisely of the subsets of x. >>> https://en.wikipedia.org/wiki/Axiom_of_power_set >>> >>> *It simply corrected the error of this* >>> In mathematics, the power set (or powerset) of a set S >>> is the set of all subsets of S, including the empty set >>> and S itself. >>> https://en.wikipedia.org/wiki/Power_set >> >> What was the error and what was the correction? >> Anyway, the pawer set axiom of ZF ensures that for every set S >> that is neither its own member nor a member of its member there >> is another set cointaing a member that is not S and not a member of S. >> >>>> Set theories with an unversal set need to restrict >>>> the construction operations more than what is usually considered >>>> reasonable. >>> >>> I don't see how. The set of all sets that do not contain >>> themselves simply becomes the set of all sets. >> >> The set of all sets that do not contain themselves is the Russell set >> that revealied the inconsistency of the naive set theory. The main >> improvment in ZF was the non-existence of this set. >> > > So basically you agreed with me on everything. No, in particular not with message that says that one thing is two. -- Mikko