[FOM] Concerning Impossible Counting

[Harvey Friedman]

Taranovsky writes:

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.

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The way I go about this, I don't get your (a) nor do I get your  (b).
My main struggle was just to do as well as I could squeezing down to
12 for ZFC.

I think I have a little more wiggle room to squeeze out 11 or maybe
10. I need to think about this some more to explore the major
surrounding issues such as what you raise.

There is enough wiggle room that I have confidence this can be
extended to all of the usual ZFC + LC, such as V(lambda + 1) into
V(lambda + 1), and to ZF + j:V into V, and more, as claimed. There is
something like (b) but much weaker, and I am not ready to formulate it
precisely.

Harvey Friedman