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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: This makes all Analytic(Olcott) truth computable --- ZFC
Date: Thu, 22 Aug 2024 15:21:51 +0300
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On 2024-08-22 09:55:51 +0000, Wasell said:

> On Wed, 21 Aug 2024 07:37:50 -0500, in article <va4n2u$3s0hu$3@dont-email.me>,
> olcott wrote:
>> 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:
> 
> [...]
> 
>>>>> 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.
> 
> Oh, you blithering imbecile! The universal set V is, by definition,

Often U is used instead of V for the universal set. V is often the
smallest set that conains the empty set and every set that can be
constructed from other members of V.

> an element of itself. It is a set, and therefore an element of the
> set of all sets.
> 
> By Specification, we can split V into the set of all sets that have
> themselves as an element, and its complement, the set of all sets
> that are not elements of themselves. Neither of these two sets are
> empty.
> 
> Do you see where this is going? Or do you need more hand holding?
> 
> There are set theories with a universal set, but they also
> have restricted Specification. (Or, more commonly, no Axiom of
> Specification, but a restricted Comprehension instead.)

If every set has a power set then the univesal has a power set, too,
and the power set of the universal set is a member of the univerasl set.
One must be careful with axioms or one can prove someting false from
the exstence of a set that has its poverset (and the power set of its
powerset) as a member.

-- 
Mikko