Re: The collection of all sets and proper classes.

On Mar 1, 9:10 pm, "Calvin" <cri...@xxxxxxxxxxxxxx> wrote:
On Mar 1, 9:04 pm, "zuhair" <zaljo...@xxxxxxxxx> wrote:

On Mar 1, 8:48 pm, "Calvin" <cri...@xxxxxxxxxxxxxx> wrote:
There is no collection of all sets, and therefore
no collection of all sets and anything else.
hmmm...., such an absolute statement. That is not a precise statement.
...
IF you have thought that whatever the set theory is there cannot be a
collection of all sets then you are mistaken, for instance in NF there
is a set of all sets.

Are you saying that in NF the assumption that there is a
collection of all sets does not lead to the contradiction
known as Russell's Paradox?

Actually Quine has construced New Foundation 'NF', with the aim to
avoid Russell paradox in his mind.

The answer to your question is: sure , no doubt in NF there is a set
of all sets that do not lead to the contradiction known as Russell's
paradox 'R.P'.

In NBG and Kelley-Morse there is a Proper class of all sets
and NBG is consevative over ZFC, while Kelley-Morse is stronger than
ZFC, and in both of these there is a proper class of all sets, without
raising R.P.

But neither NBG nor K.M have the collection of all proper classes and
sets is not defined.

In this theory which is essentially Kelley-Morse, but with
the collection of all sets and proper classes well defined
and axiomatized.

So actually this proves that the idea that there cannot be a
collection which contains them all is erronous.

Here in this theory which is stronger than ZFC ( since a subset of
this theory is equi-interpretable with M.K, and M.K
is stronger than ZFC) there is T which is the collection of all sets
and proper classes.

Zuhair

.

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