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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: This makes all Analytic(Olcott) truth computable --- ZFC
Date: Wed, 21 Aug 2024 07:37:50 -0500
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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.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer