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Binary choice games and arithmetical comprehension.

J P Aguilera1, T Kouptchinsky1

  • 1Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria.

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|June 11, 2026
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Summary
This summary is machine-generated.

We prove Arithmetical Comprehension is equivalent to the determinacy of specific clopen integer games. This finding connects foundational logic principles with game theory concepts in mathematics.

Keywords:
DeterminacyReverse mathematicsWeak König’s lemma

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Area of Science:

  • Set Theory
  • Mathematical Logic
  • Foundations of Mathematics

Background:

  • Arithmetical Comprehension is a key principle in mathematical logic.
  • Game theory provides frameworks for analyzing strategic interactions.
  • Clopen integer games are a specific type of mathematical game.

Purpose of the Study:

  • To establish an equivalence between Arithmetical Comprehension and game determinacy.
  • To explore the relationship between logical principles and game-theoretic properties.
  • To characterize the complexity of Arithmetical Comprehension through a game-theoretic lens.

Main Methods:

  • Formal proof construction.
  • Axiomatic set theory.
  • Analysis of infinite games on integers.
  • Demonstration of mutual reduction between Arithmetical Comprehension and game determinacy.

Main Results:

  • Arithmetical Comprehension is proven to be equivalent to the determinacy of all clopen integer games.
  • Each player in these games has at most two moves per turn.
  • This equivalence provides a new perspective on the strength of Arithmetical Comprehension.

Conclusions:

  • The determinacy of restricted clopen integer games is a strong logical principle.
  • This result deepens the understanding of the relationship between logic and computability.
  • Foundational mathematical principles can be illuminated through the study of abstract games.