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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: This makes all Analytic(Olcott) truth computable
Date: Tue, 20 Aug 2024 13:21:06 +0300
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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.

> 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. Set theories with an unversal set need to restrict
the construction operations more than what is usually considered
reasonable.

Cantor's original presentation did not specify the permitted operations
but all presented constructions were from already constructed sets so
no presented set contained itself. Cantor did not use unlimited
construction by properties but did not prohibit such constructions.

-- 
Mikko