Re: Equal Sets and Identical Sets

On Fri, 14 Mar 2008 06:58:31 -0700 (PDT), MoeBlee <jazzmobe@xxxxxxxxxxx>
wrote:


Suppose we take '=' as defined in our set theory, not primitive from
identity theory. Then in our interpretation we only have to interpret
'e', since '=' is a defined symbol. But then '=' might get mapped to
some relation other than the identity relation on the universe. But
usually, in casual discussion, we're not so specific about whether we
took '=' from identity theory or whether we took it as defined in our
set theory. So in such a context, we don't know whether by "a model
for set theory" we mean one in which '=' must map to the identity
relation on the universe or one in which '=' very well might not map
to the identity relation on the universe. That ambiguity now is
puzzling me. How do I know which of the two options the author intends
when he says something like "consider a model for ZF"?

Just a wild (and uneducated) guess. Maybe that difference is not
"essential". Maybe those models are "isomorphic" (in a sense).
Or wait... equivalence classes come to mind... In the model for ZFC with
defined "=" we might consider equivalence classes of all those objects
in the universe for which Az(z e a <-> z e b) holds. Then there should
be an isomorphism between the set of these equivalence classes and, etc.


F.

--

E-mail: info<at>simple-line<dot>de
.

Relevant Pages

  • Re: The Power Set Paradox
    ... itself, infinity and zero, the origin and ... The universe is infinite, infinite sets are equivalent, etcetera. ... Set theory is useful, its early adopters in formalization found ... maybe after a couple years investigating the perceived paradoxes ...
    (sci.logic)
  • Re: Set theory and consistence.
    ... use of English but just about the terminology in the subject matter ... set theory would'nt represent "e" by the membership relation. ... A universe of a model for set theory usually IS a set. ...
    (sci.math)
  • Re: Physics upgrade.
    ... > Basically, given Axiom 1, no two objects in the physical universe are ... Set theory deals mainly with collections which qualify as sets. ... but the physical universe is not abstract. ... > Axiom 1 is an extremely general statement which can be made about real ...
    (sci.math)
  • Re: Uncountable sets
    ... the existence of anything bigger. ... theory (with a universe), the Mengenlehre, is not ZF, the Zermelo- ... A primary difference between naive set theory, ... Where each of the other axioms of ZF set theory was enlisted to enrich ...
    (sci.math)
  • Re: Uncountable sets
    ... the existence of anything bigger. ... theory (with a universe), the Mengenlehre, is not ZF, the Zermelo- ... A primary difference between naive set theory, ... Where each of the other axioms of ZF set theory was enlisted to enrich ...
    (sci.math)