The theory of hereditarily bounded sets

Abstract

We show that for any k∈ω$k\in \omega$, the structure false⟨Hk,∈false⟩$\langle H_k,{\in }\rangle$ of sets that are hereditarily of size at most k is decidable. We provide a transparent complete axiomatization of its theory, a quantifier elimination result, and tight bounds on its computational complexity. This stands in stark contrast to the structure Vω=⋃kHk$V_\omega =\bigcup _kH_k$ of hereditarily finite sets, which is well known to be bi‐interpretable with the standard model of arithmetic false⟨double-struckN,+,·false⟩$\langle \mathbb {N},+,\cdot \rangle$.

Unprovability of Strong Complexity Lower Bounds in Bounded Arithmetic

“…(For a definition of theory PV, see [Kra95] or the equivalent presentation from [Jeř06].) Jeřábek [Jeř04, Jeř05, Jeř07a] developed a sophisticated (but intuitive) framework for approximate counting in APC 1 built on an elegant formalisation of the Nisan-Wigderson PRG [NW94] in this theory.…”