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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: This makes all Analytic(Olcott) truth computable
Date: Mon, 19 Aug 2024 08:12:30 -0500
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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