Re: Equal Sets and Identical Sets

On Mar 11, 1:12 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 10, 10:40 am, apoorv <sudhir...@xxxxxxxxxxx> wrote:

So, we have
p=q , but again p and q are not identical.

Yes, the axiom of extensionality works fine for 'e' interpreted as set
membership and doesn't work well at all for 'e' interpreted as certain
other relations.

What allows us to assert that the 'e' we
have in set theory ensures that equal sets are
identical sets ?

By interpreting it as the membership relation on a domain of sets.

"Interpreting 'it' as an UNDEFINED membership relation on
a NON EXISTENT domain of sets." What does that mean?

How do we assert that there is
only one null set ( although all null sets may be equal )?

We prove the theorem

E!xAy ~yex.

You only prove that all null sets are equal. You do not prove that
there is only one
null set.

If 'e' is interpreted as membership on a domain of sets, then the
above theorem is interpreted as there exists a unique set that has no
members.

Only if you assume that equal sets (as defined in set theory ) are
identical.

How do we infer from A = B that {A} ={B}, unless we
assume that A and B are identical ?

Now, that is not a matter of set theory alone, but rather of identity
theory more basically. First, identity theory has axioms (including
one that reads roughly as "equals can be substituted for equals) from
which we prove

x=y -> {x}={y}.

Then, usually, '=' is taken to have a fixed interpretation so that '='
is always interpreted as the indicating the identity relation on the
universe of discourse.

We just saw that in the context of 'e' undefined and '=' as defined,
'=' is
not necessarily identity. How does the 'always' come in?

It's been years that you've been posting about such subjects. I really
don't understand why you don't get a few good books on the subject so
that you could grab all this information right away, and in a
systematic way too.

-:) This one variant of Russell's paradox I don't find in the text
books
'God serves those (and only those ) who do not serve themselves'
But then God does'nt ''does'nt exist''; He is divine.
-apoorv

.

Relevant Pages

  • Re: Equal Sets and Identical Sets
    ... membership and doesn't work well at all for 'e' interpreted as certain ... Only if you assume that equal  sets (as defined in set theory) are ... one that reads roughly as "equals can be substituted for equals) from ... Then an interpretation must map to {<x ...
    (sci.logic)
  • Re: Equal Sets and Identical Sets
    ... "Interpreting 'it' as an UNDEFINED membership relation on ... Only if you assume that equal sets (as defined in set theory) are ... one that reads roughly as "equals can be substituted for equals) from ... don't understand why you don't get a few good books on the subject so ...
    (sci.logic)
  • Re: Equal Sets and Identical Sets
    ... the axiom of extensionality works fine for 'e' interpreted as set ... By interpreting it as the membership relation on a domain of sets. ... Now, that is not a matter of set theory alone, but rather of identity ... one that reads roughly as "equals can be substituted for equals) from ...
    (sci.logic)
  • Re: Can someone summarize the Cantor Confusion thread
    ... You'll need to get your rudiments of set theory mastered to do that. ... membership predicate, married through the definition of the von Neumann ... vehicle in motion and that the pistons are only needed to make the ... I POSTED for you the axiom of infinity in the PRIMITIVE language, ...
    (sci.math)
  • Re: Is {} and element of all sets?
    ... empty set is an element of every set. ... In no set theory is an element of itself, so at worst it can only be ... Quigley tells us "Intuitively, sets ... This leads me to ask what is meant by "capable of membership". ...
    (sci.math)
Loading