[FOM] 579: Impossible Counting

[Dmytro Taranovsky]

Dear Harvey Friedman,

On 05/26/2015 08:58 PM, Harvey Friedman wrote:
> HOW MANY BINARY OPERATIONS ARE THERE UP TO 12-SIMILARITY? - i.e., they
> have the same restrictions to 12 element sets up to isomorphism.
>
> ZFC, even ZFC augmented with standard large cardinal hypotheses, is
> insufficient to get an exact count.

Can you clarify which of the following you mean:
(a) There is a hierarchy of large cardinal axioms beyond I1 (nontrivial 
elementary j:V_{lambda+1}-->V_{lambda+1}) that decides this (with the 
axioms having a reasonable number of symbols), or
(b) no consistent extension of ZFC with less than, say, 1 million 
symbols decides this.

Given the large number of classes of relations, (b) seems plausible.

Sincerely,
Dmytro Taranovsky