Joan R. Moschovakis | Searcharxiv

arXiv2022

The strong Spector-Gandy Theorem for the higher analytical pointclasses

Assuming projective determinacy, we extend Spector's strong version of the Spector-Gandy Theorem to all odd levels of the projective hierarchy: Theorem. For every space $X$ which is a finite product of the natural numbers $N$ and Baire space $N^N$ and for every n, if $P$ is a $Π^1_{2n+1}$ subset of $X$, then there is a $Π^1_{2n}$ set $Q$ such that $P(x) \Longleftrightarrow (\exists!α)Q(x,α) \Longleftrightarrow (\existsα\inΔ^1_{2n+1}(x))Q(x,α)$.

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arXiv2020

Intuitionistic Mathematics and Logic

The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincaré, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan Brouwer, who taught mathematics at the University of Amsterdam from 1909 until 1951. By proving powerful theorems on topological invariants and fixed points of continuous mappings, Brouwer quickly build a mathematical reputation strong enough to support his revolutionary ideas about the nature of mathematical activity. These ideas influenced Hilbert and Gödel and established intuitionistic logic and mathematics as subjects worthy of independent study. Our aim is to describe the development of Brouwer's intuitionism, from his rejection of the classical law of excluded middle to his controversial theory of the continuum, with fundamental consequences for logic and mathematics. We borrow Kleene's formal axiomatic systems (incorporating earlier attempts by Kolmogorov, Glivenko, Heyting and Peano) for intuitionistic logic and arithmetic as subtheories of the corresponding classical theories, and sketch his use of gödel numbers of recursive functions to realize sentences of intuitionistic arithmetic including a form of Church's Thesis. Finally, we present Kleene and Vesley's axiomatic treatment of Brouwer's continuum, with the function-realizability interpretation which establishes its consistency.

math.LO↗

arXiv2018

Note on $Π^0_{n+1}$-LEM, $Σ^0_{n+1}$-LEM and $Σ^0_{n+1}$-DNE

We show that results of Akama, Berardi, Hayashi and Kohlenbach, on the relative independence of certain arithmetical principles over intuitionistic arithmetic HA, hold also over Kleene and Vesley's system FIM of intuitionistic analysis, which extends HA and is consistent with classical arithmetic but not with classical analysis. The double negations of the universal closures of these principles are also considered, and shown to behave differently with respect to HA and FIM. Various elementary questions are left open.

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