[FOM] Micro Set Theory

[Joe Shipman]

If V_0 is the empty set and V_(i+1) is the power set of V_i, then the elements of V_7 have infeasible size and can code objects smaller than 2^65536, so that an oracle for the theory of V_7 would allow answers to any mathematical question we care about (for example "Does the Riemann Hypothesis have a proof from ZF + your favorite large cardinal axiom of length < 10^1000 ?"), thereby rendering mathematicians obsolete.

V_4 is trivially uninteresting; it has only 16 elements so brute force programming will let us answer any question about it involving up to 10 or so quantifiers.

An oracle for V_6 would be very helpful since it would (I think but need to verify) let us solve instances of PSPACE-complete problems of length in the thousands of bits, for example calculating Ramsey numbers, but would it make mathematicians almost as obsolete as an oracle for V_7 would? 

Would an oracle for V5 tell us anything interesting at all?

-- JS

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