Re: The set of All sets

Jonathan Hoyle wrote:
> In ZF Set Theory with no urelements, sets contain only other sets and
> nothing else. The set of natural numbers, or example, is just a set of
> sets.
>
> If you mean *all sets* then ZF and ZFC do not allow for the set of all
> sets. This must be the case, since if the collection of all sets were
> a set, it must therefore contain itself as a member, which violates the
> Axiom of Foundation (AF), one of the axioms of ZF.
>
> In a conservative extension of ZFC called NBG
> (vonNeumann-Bernays-Godel), you can have a collection of all sets, but
> this collection is not a set, but rather a proper class, for the
> reasons stated above. (All sets are classes, and those classes which
> are not sets are called proper classes). NBG is equi-consistent to
> ZFC; that is to say, if NBG is inconsistent, then it is only because
> ZFC is as well.
>
> To have "the set of all sets", you must work within a set theory which
> does not contain AF. These are called "non-Well Founded" Set Theories,
> and sets which violates AF are called "non-well founded sets".
> Obviously non-well founded sets in these theories do not exist in ZF,
> and thus you cannot assume traditional ZF properties to them.
>
> Hope that helps,
>
> Jonathan Hoyle

What's the class of all classes, in lieu of a set of all sets?

Answer: there is none. Proper classes aren't allowed to contain
proper classes.

The order type of ordinals would still be an ordinal, in NBG it's not a
set, but it still contains the structure of an ordinal. That the order
type of ordinals would be an ordinal is called the Burali-Forti paradox
for Cesare Burali-Forti.

For some, having classes, not classes at school but these contents of
classified collections, does not seem to be resolution of the problems
of unrestricted comprehension, which in a sense is logical induction.
The set surcease, class conundrum, group game, shell shuffle, doesn't
have a universe.

To get to talking about a universe, there are some difficulties.

Ross

.

Relevant Pages

  • Re: Cardinality and Proper classes.
    ... NBG does, which can be thought of as the equivalent of ZFC ... without the proper classes. ... The statement "Every nonempty class of ordinals has a least element" becomes the scheme, for any formula fwith one free variable, ...
    (sci.math)
  • Re: still funny
    ... Then there are no proper classes. ... and yet _NBG_ doesn't prove that every ... So there must be some difference between the Axioms ... want to have the class of all ordinals, ...
    (sci.math)
  • Re: Cardinality and Proper classes.
    ... mistake. ... NBG does, which can be thought of as the equivalent of ZFC ... without the proper classes. ...
    (sci.math)
  • Re: Cardinality and Proper classes.
    ... mistake. ... NBG does, which can be thought of as the equivalent of ZFC ... without the proper classes. ...
    (sci.math)
  • Re: still funny
    ... Then there are no proper classes. ... NBG has Axioms of Extensionality ... want to have the class of all ordinals, ...
    (sci.math)
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