a question on indiscernibles

[martdowd at aol.com]

FOM,                                                                            

I would like to elaborate on some comments I made in
 https://cs.nyu.edu/pipermail/fom/2020-August/022316.html
It is a question of considerable interest whether,
for some countable $\alpha$ there is a set
$S\in L_{\aleph_1}$ of indiscernibles for $L_\alpha$.

If V=L and an $\omega$-Erdos cardinal exists then
the answer is yes.

However, the question is entirely decided in $$L_{\aleph_1}$,
and it is of interest whether it can be decided either way,
without appealing to any large cardinal hypothesis.
 https://www.researchgate.net/publication/313588697
contains partial results concerning whether indiscernibles
can be constructed in L.  This research should be pursued further.
If, as I suspect, they can't be, this would seem to cast doubt
on the existence of $\omega$-Erdos cardinals.

It is of interest whether models can be constructed of ZFC + V=L,
which either do contain, or do not contain, indiscernibles.

- Martin Dowd

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