A second order Goldbach conjecture?

Paul Blain Levy p.b.levy at bham.ac.uk
Sat Jul 31 05:19:56 EDT 2021


The Goldbach conjecture is a famous example of a Pi^0_1 sentence that's 
currently an open problem.

I'm looking for a similar example for second order arithmetic.

Let me make this more precise.  Recall that a sentence is said to be

-  arithmetical if its quantifiers range over Nat,

-  Pi^1_1 if it has the form "For every set U of natural numbers, 
phi(U)", where phi is arithmetical.

I'd like an example of a Pi^1_1 sentence R with the following properties.

(a) R is currently an open problem, even in ZFC with the strongest known 
large cardinal hypotheses.

(b) This would still be the case even if we knew the truth value of 
every arithmetical sentence.  In other words, for every arithmetical 
sentence S, either (S => R) is an open problem or ((not S) => R) is an 
open problem.

Condition (b) rules out examples like "For every set of natural numbers, 
the Goldbach conjecture is true."

Any help would be appreciated!


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