Re: Equal Sets and Identical Sets

On Mar 15, 7:58 am, Aatu Koskensilta <aatu.koskensi...@xxxxxxxxx>
wrote:
On 2008-03-14, in sci.logic, MoeBlee wrote:

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"?

Can you give any example of a context where this ambiguity would
result in confusion or mathematical difficulties?

I don't have a particular example in mind. The ambiguity is general,
in this sense:

Suppose two different situations:

(1) Someone (such as Quine, or Takeuti & Zaring, and probably others)
takes '=' as defined and not primtive. Then this person says, "Let <S
R> be a model of ZF". Well, that allows that when we go on to take
models for the extended language that includes '=', we have models in
which '=' does not map to the identity relation on S.

(2) Someone (such as probably most people) takes '=' as primitive from
identity theory and with the fixed semantics for '=' and says 'Let <S
R> be model of ZF". In that case, '=' maps to the identity relation on
S.

So, unless we know which method for '=' the person takes, the
expression "<S R> is a model of ZF" is ambiguous, since it could mean
either that '=' (if the model were expanded for the language extended
by definitions) may map to other than the identity relation on S or it
may mean that '=' must map to the identity relation on S.

MoeBlee
.

Relevant Pages

  • Re: Set theory and identity theory
    ... identity relation on the universe. ... treated as from identity theory and with the ordinary fixed semantics ... so that the model must map '=' to the identity relation on the universe ...
    (sci.logic)
  • Re: Equal Sets and Identical Sets
    ... theory" we mean one in which '=' must map to the identity relation ... identity relation on the universe. ... Can you give any example of a context where this ambiguity would ...
    (sci.logic)
  • Re: Equal Sets and Identical Sets
    ... conistent first order theory has a model in which '=' maps to some ... relation on the universe other than the identity relation. ...
    (sci.logic)
  • Re: Equal Sets and Identical Sets
    ... conistent first order theory has a model in which '=' maps to some ... relation on the universe other than the identity relation. ... exceptions. ...
    (sci.logic)