# An induction principle for consequence in arithmetic universes

Research output: Contribution to journal › Article › peer-review

## Standard

**An induction principle for consequence in arithmetic universes.** / Vickers, Steven; Maietti, Maria Emilia.

Research output: Contribution to journal › Article › peer-review

## Harvard

*Journal of Pure and Applied Algebra*, vol. 216, no. 8-9, pp. 2049-2067. https://doi.org/10.1016/j.jpaa.2012.02.040

## APA

*Journal of Pure and Applied Algebra*,

*216*(8-9), 2049-2067. https://doi.org/10.1016/j.jpaa.2012.02.040

## Vancouver

## Author

## Bibtex

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## RIS

TY - JOUR

T1 - An induction principle for consequence in arithmetic universes

AU - Vickers, Steven

AU - Maietti, Maria Emilia

PY - 2012/8/1

Y1 - 2012/8/1

N2 - Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0)→ψ(0) and an induction step that, for generic n, the hypothesis φ(n)→ ψ(n) allows one to deduce φ(n+1)→ ψ(n+1). Then it is already true in that arithmetic universe that (∀ n)(φ(n)→ ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form an exponential φ(n)→ ψ(n).The principle is applied to the question of locatedness of Dedekind sections.The development analyses in some detail a notion of "subspace" of an arithmetic universe, including open or closed subspaces and a boolean algebra generated by them. There is a lattice of subspaces generated by the opens and the closeds, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.

AB - Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0)→ψ(0) and an induction step that, for generic n, the hypothesis φ(n)→ ψ(n) allows one to deduce φ(n+1)→ ψ(n+1). Then it is already true in that arithmetic universe that (∀ n)(φ(n)→ ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form an exponential φ(n)→ ψ(n).The principle is applied to the question of locatedness of Dedekind sections.The development analyses in some detail a notion of "subspace" of an arithmetic universe, including open or closed subspaces and a boolean algebra generated by them. There is a lattice of subspaces generated by the opens and the closeds, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.

U2 - 10.1016/j.jpaa.2012.02.040

DO - 10.1016/j.jpaa.2012.02.040

M3 - Article

VL - 216

SP - 2049

EP - 2067

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 8-9

ER -