Re: Moore on Skolem's Paradox

Ullrich:
>> The "paradox" has to do with the existence of models of
>> set theory with certain "paradoxical" properties.
[Ockham]
>If you read Moore's article, you will see he does not state that. "
Ullrich:
> You say he doesn't state that
> this arises from the existence of certain models of set
> theory.

I note you use the expression "arises from" in your restatement of what
you said, whereas in you first statement you say "has to do with".

Certainly, Moore _would_ agree that the paradox "arises from" the
existence of certain models of set theory, i.e. that their possible
existence is the root cause of it. I'm sure he would NOT agree that
these models themselves have "paradoxical" properties. He says the
paradox is that any statement of the "relativism" in our claim that
P(w) is uncountable will have to be understood within a framework that
casts it as a straightforward error. This is perfectly consistent with
the claim that the models in question have in themselves no
"paradoxical properties".

Ullrich
> He doesn't use the word "paradox" in the paragraph above.
> He does use the word later. You say he doesn't state that
> this arises from the existence of certain models of set
> theory. Did you see the sentence "...will still be true under
> an interpretation which results from the intended interpretation
> by the elimination of all but countably many of its sets."

And then he says right afterwards "not that this is paradoxical, nor
does it constitute our difficulty". This is where I say "Jeez".

Me:
> Is what he says above incorrect, then? He later says, in the version
>which I did quote, what the "problem" alluded to above really is. I
>shall quote it again.

>" Our description of P(w) as uncountable, even though correct, must be
>understood relative to our own current point of view. From another
>point of view this very set may be countable.

> And now explain to me exactly what he might mean about talking
> about this and that from this point of view versus that point
> of view, _except_ that this or that is true in one model and
> false in another.
> Jeez.

Read to the end. The whole passage, from which you snipped the crucial
part, is:

" Our description of P(w) as uncountable, even though correct, must be
understood relative to our own current point of view. From another
point of view this very set may be countable. But I want to argue that
such relativism, compelling though it is, is subject to the by now
familiar predicament that any statement of it, if it is to be
intelligible at all, will have to be understood within a framework that
casts it as a straightforward error. ****It is this which I take to be
Skolem's paradox****."

The asterixes are there from my last posting. The predicament is NOT
that some statement is true in one model and false in another. It is
how we, from our point of view, can state the relativistic statement "S
is true in one model and false in another" in a language whose model is
the one in which S is true (S being the statement that P(w) is
uncountable).

.

Relevant Pages

  • Re: Moore on Skolems Paradox
    ... >Skolem's Paradox really is a paradox after all. ... >seems to be that notions such as countablility and uncountability are ... >want to argue that such relativism, compelling though it is, is subject ... >legitimate concepts of countability and uncountability which do involve ...
    (sci.logic)
  • Re: Moores account of Skolems Paradox
    ... |The article from which this extract is taken appeared as Moore, ... |to be that notions such as countablility and uncountability are ... this relativism. ...
    (sci.logic)
  • Re: The Smullyan paradox
    ... This paradox only arises if we insist that there is absolute truth, ... T1 proves the existence of a positive number A with an explicit ... T2 are inconsistent theories (by the NAFL yardstick). ... NAFL requires that each of the disjuncts in or can ...
    (sci.logic)
  • Moores account of Skolems Paradox
    ... where it is claimed that Skolem's Paradox really is a ... to be that notions such as countablility and uncountability are ... want to argue that such relativism, compelling though it is, is subject ... legitimate concepts of countability and uncountability which do involve ...
    (sci.logic)
  • Moore on Skolems Paradox
    ... Skolem's Paradox really is a paradox after all. ... seems to be that notions such as countablility and uncountability are ... want to argue that such relativism, compelling though it is, is subject ... legitimate concepts of countability and uncountability which do involve ...
    (sci.logic)