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Preview of Relating Type Theories and Set Theories: A Study of Their Strength and Interpretation
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Summary

From Impredicative Proofs to Constructive Strength

Peter Aczel explores the relationship between type theories (like Martin-Lof's) implemented in proof assistants like Lego and Coq, and classical set theories (like ZF). His work aims to determine the theoretical strength of these type systems by mapping them onto corresponding set theories.

Key Insights:

Obvious Type-as-Set Interpretation: Aczel leverages a "obvious" interpretation where types are represented as sets. This allows for direct correspondence between the impredicative types (like propositions) and strongly inaccessible cardinal numbers in classical set theory.

Two Approaches to Reduction:

Classical Approach: Uses the Law of Excluded Middle (EM) to translate Martin-Lof type theories into ZF set theory, demonstrating a reduction from MLW+EM to ZF.
 Constructive Approach: Replaces classical logic with intuitionistic logic and uses constructively defined sets (in CZF +). This leads to a reduction from MLW to CZF+.

Infinite Hierarchies of Types: Aczel extends his analysis to type theories with infinite hierarchies of types, showing that each hierarchy corresponds to a set theory of the same strength.

Key Results:

Type theories like MLWU (a variant of MLW) have the same theoretical strength as CZF+, a constructive set theory.

* A new handle on the problem: While the specific set theory used might be unfamiliar, this approach offers a novel perspective on comparing type theories and set theories.

Future Directions: Aczel plans to explore further properties of the constructed set theories, particularly focusing on their unique characteristics arising from the type system they represent.

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Description

Peter Aczel explores the theoretical strength of type theories implemented in development systems like Lego and Coq, combining impredicative types from construction calculus with inductive types and hierarchies from Martin-Lof's constructive type theory. He suggests using 'obvious' types-as-sets interpretation within classical axiomatic set theory for a straightforward upper bound assessment.

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  • File Format: PDF
  • File Size: 274 KB
  • Pages: 20
  • Language: EN
  • Total Downloads: 49
  • Last Updated: 3 weeks ago

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