Re: This makes all Analytic(Olcott) truth computable --- ZFC

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.
--
Mikko