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STRONG MEASURE ZERO SETS ON
$2^\kappa $ FOR 
$\kappa $ INACCESSIBLE
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 03 January 2024, pp. 1-31
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- Article
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We investigate the notion of strong measure zero sets in the context of the higher Cantor space
$2^\kappa $ for 
$\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC.
$$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$
