Re: Cantor's circular "proof" that evens = integers



george wrote:
On May 16, herbzet wrote:

If ZFC has any models at all, then con(ZFC) cannot
be true in all of them. So either it does not have a fixed
truth-value in all models, or it has a fixed truth-value of FALSE
in all models.

I'm not sure I get _your_ point

MY point is that since the predicate "consistent"
DOES NOT OCCUR in the language of ZFC
(ZFC's ONLY predicate is membership), we must therefore --
AS with the predicate "infinite" in the thread under
which we still labor with Phil -- STIPULATE how WE want
to ENCODE our natural-language concept
of "consistent" IN ZFC. ZFC itself doesn't contain any
restrictions on what we might mean BY "consistent",
OR by "infinite". We could take ANY definable predicate
and ALLEGE that its "meaning" was "consistent" or "infinite"
IF WE WANTED TO. In order to make the whole process
defensible, we have to assign these definitions in ways that
do in fact capture the intended natural language concepts
to the extent possible.

Right.

In the case of ZFC and "infinite",
this extent is totally broad and easy
because the C collapses all the competing
definitions of "infinite" into equivalence.

OK.

In the case of
ZFC (or even PA) and "consistent", it is IMPOSSIBLE
because neither of these theories can prove
the UNprovability (from its own TOTAL set of axioms)
of ANYthing, LET ALONE of its own consistency statement.

Assuming ZFC (or PA) is consistent, it is true that
cannot prove the unprovabiliy of anything, by G2.

Why this should mean that if ZFC (or PA) is consistent
then no ZFC (or PA) formula can _express_ (capture
the natlang concept of) "consistent" is unclear.

EVERY unprovability statement has the property that
SOME model must get it wrong.

If not all of them, e.g. a statement that some ZFC theorem
is unprovable would, I suppose, be false in all models.

My point is that in order to defend THE APPROPRIATENESS
OF YOUR STIPULATED TRANSLATIONS (via your stipulated
DEFINITIONS), you have to find AT LEAST ONE model that
assigns the SAME truth-value to the translated statement
AS its natural-language truth-value.

This sounds good, but try as I might, I just don't see it.

I don't see why a formula intended to _express_ the consistency
(or inconsistency) of ZFC (or PA) must be _true_ in some model in
order for it to mean what we would like it to mean.

THAT model is the
"intended interpretation" or "standard" model (or would be,
if there were just one, or one obviously simplest one;
which, in the case of PA, there is).

So if we have somehow arrived at a "standard" model within
which to interpret the formulae of ZFC, then con(ZFC) must
be true in that model for it to be an appropriate translation
of "ZFC is consistent"?

My point is you cannot truthfully, reasonably, or correctly
claim to be translating a natlang sentence known to have
a natlang truth-value of true into a formal-language sentence
known to have a truth-value of false in ALL models.

This does not imply that ZFC does not prove ~con(ZFC).
Rather, this implies that IF it proved it, then con(ZFC)
would not MEAN "ZFC is consistent".

Of course, I have argued earlier&elsewhere that the mere
fact that ZFC+~con(ZFC) is consistent implies that con(ZFC)
does not mean "ZFC is consistent" -- the non-standard
models in this case are refuting themSELVES by EXISTING,
since a theory is consistent iff it has a model.
But in the case of PA, you have the dodge that
"you have to interpret it in the standard model".
In the case of ZFC, clarifying what makes a model
standard is much harder (not that that won't stop those
who think they can clarify it from alleging that *I* am the
stupid one for refusing to concede their clarity).

Well, I thank you for trying to explain your view to me.
Sometimes it just takes time for things to sink in.

--
hz
.

Relevant Pages

  • Re: Cantors circular "proof" that evens = integers
    ... truth-value in all models, or it has a fixed truth-value of FALSE ... MY point is that since the predicate "consistent" ... DOES NOT OCCUR in the language of ZFC ...
    (sci.logic)
  • Re: Cantors circular "proof" that evens = integers
    ... ZFC proved this, then it would also have to be "possible" ... ENCODES "ZFC is consistent" if conhas the WRONG ... in all the contexts in which it can have a truth-value. ... OTHER Than "ZFC is inconsistent". ...
    (sci.logic)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... asking whether the cartesian product of nonempty sets is nonempty. ... If ZF is consistent then there are models of ZF in which AC is false ... things have to be taken in context. ... Let's take "ZFC is consistent" for comparison. ...
    (comp.theory)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... asking whether the cartesian product of nonempty sets is nonempty. ... If ZF is consistent then there are models of ZF in which AC is false ... things have to be taken in context. ... Let's take "ZFC is consistent" for comparison. ...
    (sci.math)
  • Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
    ... asking whether the cartesian product of nonempty sets is nonempty. ... If ZF is consistent then there are models of ZF in which AC is false ... things have to be taken in context. ... Let's take "ZFC is consistent" for comparison. ...
    (sci.logic)