FOM: CH and 2nd-order validity
Robert Black
Robert.Black at nottingham.ac.uk
Tue Oct 17 16:37:58 EDT 2000
> One thing some set theorists say is "V is not equal to L". (They are
>actually more interested in stronger, more useful assertions which imply
>V not equal L.) This is false if we interpret "set" to mean the same as
>"set in L", and epsilon to stand for membership.
Of course if your set theory contains an axiom like 'there is a measurable
cardinal' then you can no longer be interpreted as meaning 'constructible
set' by 'set'. But arguments of this general type are still gong to apply.
The point is that whatever your axioms are, for any sentence not decided by
those axioms you can be (mis?)interpreted so that the sentence comes out as
either true or false, and if the only constraint on correct interpretation
is that the axioms should come out as true, the undecidable sentence will
lack truth-value.
> The basic line of the Skolemites, that one cannot assert
>that there is an uncountable set, is best refuted in the style of
>Dr. Johnson's kicking the stone. In fact, we assert there are
>uncountable sets all the time. Your theory of meaning is just wrong if it
>implies that no one can assert that there are uncountable sets, or sets
>not in L.
I agree with that entirely; the argument must be wrong because the
conclusion is false. But Dr Johnson's response, though healthy, falls a bit
short of philosophical diagnosis.
> I haven't read Putnam's "Models and Reality" in a long time
>(was that the paper you were referring to?), but as I remember it
>basically just fell into some Skolemite error.
Yes, I did mean 'Models and Reality', though I also haven't read it for
some time and was reconstructing (perhaps not alltogether faithfully) the
argument from memory. Identifying just where the error lies is, I think,
not all that easy. It's *not* just the error of thinking that formal
semantics fixes meaning.
Robert
Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD
tel. 0115-951 5845
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