Liste des Groupes | Revenir à s logic |
On 8/19/2024 3:49 AM, Mikko wrote:That is so obvious that needs not be mentined. There is nothingOn 2024-08-18 11:51:33 +0000, olcott said:They made no other changes to the notion of set theory
On 8/18/2024 5:28 AM, Mikko wrote:They did, as both Richard Damon and I already said, much more. TheyOn 2024-08-16 22:16:59 +0000, olcott said:All they did was (as I already said) was redefine the notion of a set.
On 8/16/2024 5:03 PM, Richard Damon wrote:That is not sufficient. They also had to Comprehension.On 8/16/24 5:35 PM, olcott wrote:On 8/16/2024 4:05 PM, Richard Damon wrote:No, they defined not only what WAS a set, but what you could do as basic operations ON a set.On 8/16/24 4:39 PM, olcott wrote:None of this is changing any more rules. AllZFC didn't need to do that. All they had to do isI 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.
redefine the notion of a set so that it was no longer
incoherent.
of these are the effects of the change of the
definition of 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.
This is added to keep most of Comprenesion but not Russell's set.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)
That this can still be called set theory seems redundant.
also explained their rationale, worked out various consequnces of
their axioms and compared them to expectations, and developed better
sets of axioms.
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 howThere are axioms about exstence and non-existence of certain kind of
sets can be constructed. The operations that can be
performed on sets remained the same.
The power set axiom prevents the existence of a set that containsOne consequence of ZF axioms is that there is no set that contains allThen maybe they did this incorrectly. They only needed to
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).
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.
Les messages affichés proviennent d'usenet.