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Almost sure OTM-realizability

Merlin Carl

TL;DR

It is shown that, in contrast to the classical case, almost sure realizability differs from plain realizability, while closure under intuitionistic predicate logic and realizability of Kripke-Platek set theory continue to hold.

Abstract

Combining the approaches made in works with Galeotti and Passmann, we define and study a notion of "almost sure" realizability with parameter-free ordinal Turing machines (OTMs). In particular, we show that, in contrast to the classical case, almost sure realizability differs from plain realizability, while closure under intuitionistic predicate logic and realizability of Kripke-Platek set theory continue to hold.

Almost sure OTM-realizability

TL;DR

It is shown that, in contrast to the classical case, almost sure realizability differs from plain realizability, while closure under intuitionistic predicate logic and realizability of Kripke-Platek set theory continue to hold.

Abstract

Combining the approaches made in works with Galeotti and Passmann, we define and study a notion of "almost sure" realizability with parameter-free ordinal Turing machines (OTMs). In particular, we show that, in contrast to the classical case, almost sure realizability differs from plain realizability, while closure under intuitionistic predicate logic and realizability of Kripke-Platek set theory continue to hold.
Paper Structure (6 sections, 22 theorems, 2 equations)

This paper contains 6 sections, 22 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic results
  3. Deduction Rules
  4. Axioms of Set Theory
  5. Conclusion and further work
  6. Appendix: Proofs

Key Result

lemma 1

For each $\in$-formula $\varphi$ and each tuple $\vec{p}$ of real numbers, the following are true:

Theorems & Definitions (46)

  • definition 1
  • definition 2
  • lemma 1
  • proof
  • lemma 2
  • theorem 1
  • proof
  • definition 3
  • lemma 3
  • proof
  • ...and 36 more