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\mathbb {r}  \mathbb  compact  definable  equivalent  lebesgue  prove  subset \mathbb  subset  {r} lebesgue  {r}  σn definable  σn 
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Preview: Logic Journal of IGPL - Advance Access

Logic Journal of the IGPL Advance Access





Published: Fri, 02 Feb 2018 00:00:00 GMT

Last Build Date: Sat, 03 Feb 2018 00:49:45 GMT

 



Erratum

Fri, 02 Feb 2018 00:00:00 GMT

Dialectica categories, cardinalities of the continuum and combinatorics of ideals



Compactness, colocatedness, measurability and ED

Mon, 08 Jan 2018 00:00:00 GMT

Abstract
In classical analysis, every compact subset of $\mathbb {R}$ is Lebesgue measurable, but it is not true in constructive analysis. In this paper, we prove that the statement ‘every compact set K in a locally compact space X is integrable with respect to a positive measure μ’ is equivalent to LPO, over Bishop’s constructive analysis. We also prove that the existence of a compact subset of $\mathbb {R}$ which is not Lebesgue integrable is equivalent to the schema ED, which asserts that ‘there exists an intuitionistically enumerable subset of $\mathbb {N}$ which is not intuitionistically decidable’. Moreover, classically, every open subset of $\mathbb {R}$ is Lebesgue measurable, but it is not true constructively. We show that Lebesgue integrability of a significant group of open sets, i.e. colocated sets, in [0, 1] is equivalent to LPO. We also prove that the existence of a colocated subset of [0, 1] which is not Lebesgue integrable is equivalent to the schema ED.



Gödel’s second incompleteness theorem for Σn-definable theories

Mon, 08 Jan 2018 00:00:00 GMT

Abstract
Gödel’s second incompleteness theorem is generalized by showing that any Σn+1-definable and Σn-sound extension of Peano arithmetic (PA) cannot prove its own Σn-soundness. The optimality of the generalization is shown by presenting a Σn+1-definable and Σn−1-sound extension of PA that proves its own Σn−1-soundness.