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La logica matematica ha sviluppato strumenti in grado di rendere precise affermazioni del tipo «il teorema A è più forte del teorema B». In particolare sono stati ottenuti un consistente numero di risultati che stabiliscono la forza assiomatica di molti teoremi in diversi settori della matematica. I risultati in questione hanno dato origine ad un programma di ricerca noto con il nome di reverse mathematics. Nel presente articolo evidenziamo gli «antenati» della reverse mathematics, descriviamo lo...

We study some natural sets arising in the theory of ordinary differential equations in one variable from the point of view of descriptive set theory and in particular classify them within the Borel hierarchy. We prove that the set of Cauchy problems for ordinary differential equations which have a unique solution is ${\prod }_2^0$-complete and that the set of Cauchy problems which locally have a unique solution is ${\sum }_3^0$-complete. We prove that the set of Cauchy problems which have a global solution is ${\sum }_0^4$-complete...

We show that if $X$ is a ${\Sigma }_1^1$ separable metrizable space which is not $\sigma$-compact then $C_p^{*}\left(X\right)$, the space of bounded real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_1^1$-complete. Assuming projective determinacy we show that if $X$ is projective not $\sigma$-compact and $n$ is least such that $X$ is ${\Sigma }_n^1$ then $C_p\left(X\right)$, the space of real-valued continuous functions on $X$ with the topology of pointwise convergence, is Borel-${\Pi }_n^1$-complete. We also prove a simultaneous improvement of theorems of Christensen...

We show that each of the classes of hereditarily locally connected, finitely Suslinian, and Suslinian continua is Π₁¹-complete, while the class of regular continua is Π₀⁴-complete.