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Article overview
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Descriptive properties of I2-embeddings | | Vincenzo Dimonte
; Martina Iannella
; Philipp Lücke
; | | Date: |
1 Nov 2023 | | Abstract: | We contribute to the study of generalizations of the Perfect Set Property and
the Baire Property to subsets of spaces of higher cardinalities, like the power
set $P(lambda)$ of a singular cardinal $lambda$ of countable cofinality or
products $prod_{i<omega}lambda_i$ for a strictly increasing sequence
$langlelambda_i ~ vert ~ i<omega
angle$ of cardinals. We consider the
question under which large cardinal hypotheses classes of definable subsets of
these spaces possess such regularity properties, focusing on rank-into-rank
axioms and classes of sets definable by $Sigma_1$-formulas with parameters
from various collections of sets. We prove that $omega$-many measurable
cardinals, while sufficient to prove the Perfect Set Property of all
$Sigma_1$-definable sets with parameters in $V_lambdacup{V_lambda}$, are
not enough to prove it if there is a cofinal sequence in $lambda$ in the
parameters. For this conclusion, the existence of an I2-embedding is enough,
but there are parameters in $V_{lambda+1}$ for which I2 is still not enough.
The situation is similar for the Baire Property: under I2 all sets that are
$Sigma_1$-definable using elements of $V_lambda$ and a cofinal sequence as
parameters have the Baire property, but I2 is not enough for some parameter in
$V_{lambda+1}$. Finally, the existence of an I0-embedding implies that all
sets that are $Sigma^1_n$-definable with parameters in $V_{lambda+1}$ have
the Baire property. | | Source: | arXiv, 2311.00376 | | Services: | Forum | Review | PDF | Favorites | |
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