[FOM] Difficult?
Harvey Friedman
friedman at math.ohio-state.edu
Sun Jul 13 20:45:36 EDT 2003
Previously on the FOM I made the following conjectures.
CONJECTURE. sin(2^2^2^2^2^2^2^2) > 0 cannot be proved or refuted in
ZFC even with large cardinal axioms, without using at least 2^2^100
symbols, even if abbreviations are allowed.
CONJECTURE. "The number of primes less than 2^2^2^2^2^2^2^2 is even"
cannot be proved or refuted in ZFC even with large cardinal axioms,
without using at least 2^2^100 symbols, even if abbreviations are
allowed.
Note that both of these statements can be proved or refuted in ZFC.
Here is another conjecture, this time, involving outright independence.
PROPOSITION A. The number of distinct sets that are the unique
solution to a formula of set theory (epsilon,=) with exactly one
free variable and at most 1000 occurrences of variables, is even.
Proposition A is a statement in class theory.
CONJECTURE. Proposition A cannot be proved nor refuted in MK + GC,
even with the addition of any of the well studied large cardinal
axioms, assuming that this addition does not result in inconsistency.
Also Proposition A cannot be proved nor refuted in MK + V = L.
Here MK is Morse Kelley class theory.
Now here is a related problem that perhaps might be tractable.
CONJECTURE. The set of all n for which the number of distinct sets
that are the unique solution to a formula of set theory (epsilon,=)
with exactly one free variable and at most n occurrences of
variables, is even, has density 1/2.
Harvey Friedman
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