Re: Equal Sets and Identical Sets

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.

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.

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.

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.

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.

MoeBlee

.

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: 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: Z.Irregular Set Theory
    ... Axiom of Uniqueness: EQ,Ax ... So x equals Q. Also it is easy to see that x is a singleton. ... In this set theory it can be proved that for some sets, ... infinite number of models. ...
    (sci.math)
  • Re: Equal Sets and Identical Sets
    ... By interpreting it as the membership relation on a domain of sets. ... 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 ...
    (sci.logic)
  • Re: Scott and Georges Teaching Thread
    ... symbol) is a DEFINED 2-place predicate symbol of the language. ... We add that formula to the axioms of our set theory, ... First we could use this axiom: ... formalization of Euclid's "equals for equals" principle (also along ...
    (sci.logic)
Loading