Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!.POSTED!not-for-mail From: olcott Newsgroups: sci.logic Subject: Re: This makes all Analytic(Olcott) truth computable Date: Mon, 19 Aug 2024 08:12:30 -0500 Organization: A noiseless patient Spider Lines: 89 Message-ID: References: <20b1dea98eda49e74e822c96b37565bb3eb36013@i2pn2.org> <7f2a1f77084810d4cee18ac3b44251601380b93a@i2pn2.org> <662de0ccc3dc5a5f0be0918d340aa3314d51a348@i2pn2.org> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 7bit Injection-Date: Mon, 19 Aug 2024 15:12:31 +0200 (CEST) Injection-Info: dont-email.me; posting-host="a406f5e7706e340b197dee4287cd04b3"; logging-data="3002273"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18FDPUSdYlgTAP2W8chC0Rt" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:YTWQyO47Ga8tq/5RGfteo8eIU+Y= In-Reply-To: Content-Language: en-US Bytes: 5369 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. 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. > 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. > Some common forms of second order logic use sets. Those sets are different > from the sets of ZFC. In ZFC all members of sets are sets but in such > second order logic a set cannot be a memeber of set. > -- Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius hits a target no one else can see." Arthur Schopenhauer