FOM: CH and 2nd-order validity

Robert Black Robert.Black at
Tue Oct 17 19:21:41 EDT 2000

For any set of people, the axioms
>explicitly stated and agreed to by those people are at worst a recursive
>set, so their theory T is axiomatizable.  Must the truth value of Con(T)
>be undetermined by their usage of the language of T?
>          Is there any ambiguity in the language of Peano arithmetic?
>John Steel
Again I couldn't agree more. If the argument works it ought to still work
when transferred to arithmetic. And now take any pi_1 sentence independent
not only of the axioms we accept, but of all the axioms we would accept
(like Cons(PA), Cons(PA+Cons(PA)), Cons (ZF) or whatever). For all we know,
Goldbach's conjecture might be such a sentence. Then the argument leads to
the conclusion that that sentence lacks truth-value, whereas obviously any
undecidable pi_1 sentence is true. However, that still only shows (given
certain plausible but ultimately contestable premises) that the argument
has a false conclusion, it doesn't identify where the fallacy lies.
Personally, I think that every sentence of arithmetic has  a truth-value,
and that CH has a truth-value too. but the problem is to persuade

Robert Black
Dept of Philosophy
University of Nottingham
Nottingham NG7 2RD

tel. 0115-951 5845

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