Lightface $\Sigma^1_2$-indescribable cardinals

Publication date: 18 September 2022

Abstract:

S i g m a_{3}^{1}

-absoluteness for ccc forcing means that for any ccc forcing

P

,

{H_{o m e g a_{1}}}^{V} p r e c_{S i g m a_{2}} {H_{o m e g a_{1}}}^{V^{P}}

. "

o m e g a_{1}

inaccessible to reals" means that for any real

r

,

{o m e g a_{1}}^{L [r]} < o m e g a_{1}

. To measure the exact consistency strength of "

S i g m a_{3}^{1}

-absoluteness for ccc forcing and

o m e g a_{1}

is inaccessible to reals", we introduce a weak version of a weakly compact cardinal, namely, a (lightface)

S i g m a_{2}^{1}

-indescribable cardinal;

k a p p a

has this property exactly if it is inaccessible and

H_{k} a p p a p r e c_{S i g m a_{2}} H_{k a p p a^{+}}

.

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