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**1 - 4**of**4**### UNAVOIDABLE SEQUENCES IN CONSTRUCTIVE ANALYSIS

"... Kleene’s formalization FIM of intuitionistic analysis ([3] and [2]) includes bar induction, countable and continuous choice, but is consistent with the statement that there are no non-recursive functions ([5]). Veldman ([12]) showed that in FIM the constructive analytical hierarchy collapses at Σ 1 ..."

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Kleene’s formalization FIM of intuitionistic analysis ([3] and [2]) includes bar induction, countable and continuous choice, but is consistent with the statement that there are no non-recursive functions ([5]). Veldman ([12]) showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions

### Oberwolfach Proof Theory and Constructive Math

"... Vesley [1965], as extended by Kleene [1969]) includes bar induction, countable and continuous choice, but cannot prove that the constructive arithmetical hierarchy is proper. Veldman showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions to interp ..."

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Vesley [1965], as extended by Kleene [1969]) includes bar induction, countable and continuous choice, but cannot prove that the constructive arithmetical hierarchy is proper. Veldman showed that in FIM the constructive analytical hierarchy collapses at Σ 1 2. These are serious obstructions to interpreting the constructive content of classical analysis, just as the collapse of the arithmetical hierarchy at Σ 0 3 in HA + MP0 + ECT0 limits the scope and effectiveness of recursive analysis. Question: Can we do better by working within classical extensions of nonclassical theories, or within classically correct theories obeying e.g. Church’s Rule or Brouwer’s Rule? We work in a two-sorted language L with variables over numbers and one-place number-theoretic functions (choice sequences). Our base theory M – the minimal theory used by Kleene [1969] to formalize the theory of recursive partial functionals, function

### Note on Π 0 n+1-LEM, Σ0 n+1-LEM and Σ0 n+1-DNE⋆

"... Abstract. In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ 0 n+1-DNE from Π 0 n+1-LEM over HA, and hence the independence of Σ 0 n+1-LEM from Π 0 n+1-LEM over HA, for all n ≥ 0. We show that the same relat ..."

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Abstract. In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ 0 n+1-DNE from Π 0 n+1-LEM over HA, and hence the independence of Σ 0 n+1-LEM from Π 0 n+1-LEM over HA, for all n ≥ 0. We show that the same relative independence results hold for these arithmetical principles over Kleene and Vesley’s system FIM of intuitionistic analysis [3], which extends HA and is consistent with PA but not with classical analysis. 1 The double negations of the closures of Σ 0 n+1-LEM, Σ 0 n+1-DNE and Π 0 n+1-LEM are also considered, and shown to behave differently with respect to HA and FIM. Various elementary questions remain to be answered.