Re: This makes all Analytic(Olcott) truth computable

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.
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).
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.
--
Mikko