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We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if the norm of f(x) is small, then x is close to a zero of f. Working in Bishops constructive setting, we first study pointwi
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We show that numerous distinctive concepts of constructive mathematics arise automatically from an antithesis translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically constructivize classical definitions, handling the resulting bookkeeping automatically.
We generalise sheaf models of intuitionistic logic to univalent type theory over a small category with a Grothendieck topology. We use in a crucial way that we have constructive models of univalence, that can then be relativized to any presheaf models, and these sheaf models are obtained by localisation for a left exact modality. We provide first an abstract notion of descent data which can be thought of as a higher version of the notion of prenucleus on frames, from which can be generated a nucleus (left exact modality) by transfinite iteration. We then provide several examples.
We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.
The variety generated by the Brandt semigroup ${bf B}_2$ can be defined within the variety generated by the semigroup ${bf A}_2$ by the single identity $x^2y^2approx y^2x^2$. Edmond Lee asked whether or not the same is true for the monoids ${bf B}_2^1$ and ${bf A}_2^1$. We employ an encoding of the homomorphism theory of hypergraphs to show that there is in fact a continuum of distinct subvarieties of ${bf A}_2^1$ that satisfy $x^2y^2approx y^2x^2$ and contain ${bf B}_2^1$. A further consequence is that the variety of ${bf B}_2^1$ cannot be defined within the variety of ${bf A}_2^1$ by any finite system of identities. Continuing downward, we then turn to subvarieties of ${bf B}_2^1$. We resolve part of a further question of Lee by showing that there is a continuum of distinct subvarieties all satisfying the stronger identity $x^2yapprox yx^2$ and containing the monoid $M({bf z}_infty)$, where ${bf z}_infty$ denotes the infinite limit of the Zimin words ${bf z}_0=x_0$, ${bf z}_{n+1}={bf z}_n x_{n+1}{bf z}_n$.
These are the lecture notes of an introductory course on ordinal analysis. Our selection of topics is guided by the aim to give a complete and direct proof of a mathematical independence result: Kruskals theorem for binary trees is unprovable in conservative extensions of Peano arithmetic (note that much stronger results of this type are due to Harvey Friedman). Concerning prerequisites, we assume a solid introduction to mathematical logic but no specialized knowledge of proof theory. The material in these notes is intended for 14 lectures and 7 exercise sessions of 90 minutes each.
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