Hierarchical Incompleteness Results for Arithmetically Definable Extensions of Fragments of Arithmetic

Abstract

There has been a recent interest in hierarchical generalizations of classic incompleteness results. This paper provides evidence that such generalizations are readily obtainable from suitably formulated hierarchical versions of the principles used in the original proofs. By collecting such principles, we prove hierarchical versions of Mostowski’s theorem on independent formulae, Kripke’s theorem on flexible formulae, Woodin’s theorem on the universal algorithm, and a few related results. As a corollary, we obtain the expected result that the formula expressing “ $\mathrm {T}$ is $\Sigma _n$ -ill” is a canonical example of a $\Sigma _{n+1}$ formula that is $\Pi _{n+1}$ -conservative over $\mathrm {T}$ .

The Modal Logic of Set-Theoretic Potentialism and the Potentialist Maximality Principles

“…We may similarly use the universal sequence to derive the "flexible" formula result of Kripke [Kri62]. This was also observed by Rasmus Blanck [Bla18]. Here, I also achieve a uniform version of the result in statement (2).…”

On Guaspari's problem about partially conservative sentences

“…Let ϕ be an arbitrary sentence and γ any Γ sentence. Proposition 3] for n = 2, or Blanck [3,Corollary 4.32]). For any theory T , Pr Σ n T ( 0 = 1 ) ∈ Cons(Π n , T ).…”

Where Pigeonhole Principles meet König Lemmas

“…The second proof of Theorem 5.2 originates from a construction devised by Theodore A. Slaman in around 2011; see Haken [14,Chapter 3]. What allows us to improve on Slaman's construction is the following recent theorem from Blanck [3,Theorem 5]. Here Π n -Tr denotes the set of all (standard and nonstandard) Π n sentences that are declared true by the usual satisfaction predicate for Π n formulas.…”

“…We thank Leszek Ko lodziejczyk for introducing to us the references relevant to the ∆ 0 pigeonhole principle. We thank Ali Enayat for bringing to our attention Blanck's recent preprint [3].…”