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First-order logic in the Medvedev lattice

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Rutger Kuyper

Logic · Submitted 22 Aug 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally, we consider Heyting arithmetic in the Medvedev hyperdoctrine and prove an analogue of Tennenbaum's theorem on computable models of arithmetic.

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arXiv ID1408.5311 (version 2)
Pith IDYSRORUNF, Pith's permanent record of this paper and its review
CitationRutger Kuyper. "First-order logic in the Medvedev lattice." arXiv:1408.5311 (2014). https://pith.science/paper/YSRORUNF
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What the abstract itself states, as probabilities: claims first 0.24 · states limitation 0.16 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.86 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic using the notion of a first-order hyperdoctrine from categorical logic to a structure which we will call the Medvedev hyperdoctrine. We also study the intermediate logic that the Medvedev hyperdoctrine gives us, where we focus in particular on subintervals of the Medvedev hyperdoctrine in an attempt to obtain an analogue of Skvortsova's result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. Finally, we consider Heyting arithmetic in the Medvedev hyperdoctrine and prove an analogue of Tennenbaum's theorem on computable models of arithmetic.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. background 1.00 0.02 0.02 asserted 1.00
s1 This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. background 1.00 0.03 0.02 asserted 1.00
s2 However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. gap 0.97 0.04 0.03 asserted 1.00
s3 We extend the work of Medvedev to first-order logic using the notion of a first-order hyperdoctrine from categorical logic to a structure which we will call the Medvedev hyperdoctrine. contribution 0.92 0.95 0.28 asserted 1.00
s4 We also study the intermediate logic that the Medvedev hyperdoctrine gives us, where we focus in particular on subintervals of the Medvedev hyperdoctrine in an attempt to obtain an analogue of Skvortsova's result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. contribution 1.00 0.93 0.13 asserted 0.88
s5 Finally, we consider Heyting arithmetic in the Medvedev hyperdoctrine and prove an analogue of Tennenbaum's theorem on computable models of arithmetic. result 0.78 0.96 0.29 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic using the notion of a first-order hyperdoctrine from categorical logic to a structure which we will call the Medvedev hyperdoctrine. We also study the intermediate logic that the Medvedev hyperdoctrine gives us, where we focus in particular on subintervals of the Medvedev hyperdoctrine in an attempt to obtain an analogue of Skvortsova's result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. Finally, we consider Heyting arithmetic in the Medvedev hyperdoctrine and prove an analogue of Tennenbaum's theorem on computable models of arithmetic.

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