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A BOREL MAXIMAL COFINITARY GROUP
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 11 December 2023, pp. 1-14
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CONSTRUCTING MAXIMAL COFINITARY GROUPS
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- Journal:
- Nagoya Mathematical Journal / Volume 251 / September 2023
- Published online by Cambridge University Press:
- 30 January 2023, pp. 622-651
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- September 2023
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Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in
$\mathsf {ZF}$, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.
A CO-ANALYTIC COHEN-INDESTRUCTIBLE MAXIMAL COFINITARY GROUP
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- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 2 / June 2017
- Published online by Cambridge University Press:
- 19 June 2017, pp. 629-647
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- June 2017
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Assuming that every set is constructible, we find a
${\text{\Pi }}_1^1 $ maximal cofinitary group of permutations of 
$\mathbb{N}$ which is indestructible by Cohen forcing. Thus we show that the existence of such groups is consistent with arbitrarily large continuum. Our method also gives a new proof, inspired by the forcing method, of Kastermans’ result that there exists a 
${\text{\Pi }}_1^1 $ maximal cofinitary group in L.
