Re: Equal Sets and Identical Sets

On Mar 11, 7:34 am, apoorv <sudhir...@xxxxxxxxxxx> wrote:
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?

I didn't say anything about undefined or non-existent.

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.

In mathematics, "there is only one x such that Px" is taken to mean
"there is an x such that Px and if Px and Py then x=y". And that
follows precisely from E!xPx. Moreover, from this, we can easily prove
that if z is a set such all its members are empty, then the
cardinality of z is 1, which is just another way of saying that "there
is only one x such that x is empty".

Come on, please, don't waste our time with utterly INCORRECT nits.

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.

'equal' and 'identical' are English language words that we use in the
context of set theory and in this context we take them as synonymous.
Again, your quibble has no import.

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?

First, again, as primitive, '=' is always for the identity relation
on the domain. This is a feature of our semantics for first order
languages. For '=' as defined in Z set theory (with a suitably
adjusted axiom of extensionality), we PROVE that in Z set theory '='
is such that the identity axioms are satisfied. That is provable since
Z set theory has a finite number of predicate symbols (in general, for
a language with an infinite number of predicate symbols, this would
not be provable). That is to say that we prove BOTH directions of the
Leibniz principle: the identity of indsicernibles and the
indiscerniblity of identicals. Then an interpretation must map to {<x
x> | xeU}, which is the identity relation on the universe.

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.

Apparently your religion is quite interfering with your mathematical
interests.

MoeBlee


.

Relevant Pages

  • 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 ...
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  • 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 ...
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  • Re: Equal Sets and Identical Sets
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  • Re: Skolems Paradox and why is math the way it is?
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