Why is "0# exists" independent?

[Harvey Friedman]

*regarding *
*martdowd *
*Fri Jun 24 19:50:07 EDT 2022*
https://cs.nyu.edu/pipermail/fom/2022-June/023453.html

What I think you are saying is that in the usual large cardinal hierarchy
there is a huge jump to get to the spot where "0# exists" is provable.

I'm not sure that this is true. The jump is from

there exists kappa such that for all countable alpha, kappa arrows alpha

to

there exists kappa such that kappa arrows omega_1

which doesn't seem like much of a jump.

Also the following are provably equivalent to the existence of 0#:

there exists kappa such that kappa arrows omega_1 for constructible
partitions

there exists kappa such that kappa arrows kappa for constructible
partitions.

Harvey Friedman
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