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ON COMPACTNESS OF WEAK SQUARE AT SINGULARS OF UNCOUNTABLE COFINALITY
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 04 January 2024, pp. 1-11
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Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at
$\aleph _\omega $, meaning that it is consistent that 
$\square _{\aleph _n}$ holds for all 
$n<\omega $ while 
$\square _{\aleph _\omega }$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild 
${{\mathsf {PCF}}}$-theoretic hypotheses, the weak square principle 
$\square _\kappa ^*$ is in fact compact at singulars of uncountable cofinality.
