A question on indiscernibles

martdowd at aol.com martdowd at aol.com
Tue Sep 1 17:54:16 EDT 2020

For an additional remark to those of
Gabriel Goldberg has noted (private email) that II is equivalent to a
Pi-1-2 statement in the second order language of arithmetic.
It follows by Shoenfield absoluteness that II is equivalent to II^L.
A write-up of his proof follows.

For another remark, by theorem 18.1 of [Jech] "0# exists" implies II,
so a large cardinal hypothesis is unnecessary.  If V=L then "0# exists"
is false, but this is not known for II.

- Martin Dowd

In [Dowd16] the notation II is used to denote the statement that
for some limit ordinal \alpha, L_\alpha has an infinite set of
indiscernibles.  It is observed that \alpha may, w.l.g.,
be taken to be countable.  Letting A denote this latter statement,
it may be seen to be \Sigma_1 over HC, the hereditarily countable sets.
Indeed, A may be written as
 "\exists\alpha\exists S\exists A
  \alpha is an ordinal and S\subseteq \alpha and S is infinite
  and A is the set of ordered finite sequences from S
  and S is a set of indiscernibles for L_\alpha"
The matrix is readily seen (using A in the last conjunct) to be Sigma_1
over HC.  By lemma 25.25 of [Jech], there is a $\Sigma^1_2$ formula of
the second order language of arithmetic, which is equivalent to A; call
this B.  By theorem 25.20 of [Jech], B is equivalent to B^L.  Since L is
a model of ZFC, B^L is equivalent to II^L.

[Dowd16]   https://www.researchgate.net/publication/313588697
[Jech]     https://www.springer.com/gp/book/9783540440857

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