Short Proofs for Slow Consistency

Let $\operatorname{Con}(\mathbf T)\!\restriction\!x$ denote the finite consistency statement "there are no proofs of contradiction in $\mathbf T$ with $\leq x$ symbols". For a large class of natural theories $\mathbf T$, Pudl\'ak has shown that the lengths of the shortest proofs of $\operatorname{Con}(\mathbf T)\!\restriction\!n$ in the theory $\mathbf T$ itself are bounded by a polynomial in $n$. At the same time he conjectures that $\mathbf T$ does not have polynomial proofs of the finite consistency statements $\operatorname{Con}(\mathbf T+\operatorname{Con}(\mathbf T))\!\restriction\!n$. In contrast we show that Peano arithmetic ($\mathbf{PA}$) has polynomial proofs of $\operatorname{Con}(\mathbf{PA}+\operatorname{Con}^*(\mathbf{PA}))\!\restriction\!n$, where $\operatorname{Con}^*(\mathbf{PA})$ is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also obtain a new proof of the result that the usual consistency statement $\operatorname{Con}(\mathbf{PA})$ is equivalent to $\varepsilon_0$ iterations of slow consistency. Our argument is proof-theoretic, while previous investigations of slow consistency relied on non-standard models of arithmetic

Cut elimination for a simple formulation of epsilon calculus

AbstractA simple cut elimination proof for arithmetic with the epsilon symbol is used to establish the termination of a modified epsilon substitution process. This opens a possibility of extension to much stronger systems

An incompleteness theorem via ordinal analysis

We present an analogue of G\"{o}del's second incompleteness theorem for systems of second-order arithmetic. Whereas G\"{o}del showed that sufficiently strong theories that are $\Pi^0_1$-sound and $\Sigma^0_1$-definable do not prove their own $\Pi^0_1$-soundness, we prove that sufficiently strong theories that are $\Pi^1_1$-sound and $\Sigma^1_1$-definable do not prove their own $\Pi^1_1$-soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis

A Finite Semantics of Simply-Typed Lambda Terms for Infinite Runs of<br> Automata

Model checking properties are often described by means of finite automata. Any particular such automaton divides the set of infinite trees into finitely many classes, according to which state has an infinite run. Building the full type hierarchy upon this interpretation of the base type gives a finite semantics for simply-typed lambda-trees. A calculus based on this semantics is proven sound and complete. In particular, for regular infinite lambda-trees it is decidable whether a given automaton has a run or not. As regular lambda-trees are precisely recursion schemes, this decidability result holds for arbitrary recursion schemes of arbitrary level, without any syntactical restriction.Comment: 23 page

Connecting Constructive Notions of Ordinals in Homotopy Type Theory

In classical set theory, there are many equivalent ways to introduce ordinals. In a constructive setting, however, the different notions split apart, with different advantages and disadvantages for each. We consider three different notions of ordinals in homotopy type theory, and show how they relate to each other: A notation system based on Cantor normal forms, a refined notion of Brouwer trees (inductively generated by zero, successor and countable limits), and wellfounded extensional orders. For Cantor normal forms, most properties are decidable, whereas for wellfounded extensional transitive orders, most are undecidable. Formulations for Brouwer trees are usually partially decidable. We demonstrate that all three notions have properties expected of ordinals: their order relations, although defined differently in each case, are all extensional and wellfounded, and the usual arithmetic operations can be defined in each case. We connect these notions by constructing structure preserving embeddings of Cantor normal forms into Brouwer trees, and of these in turn into wellfounded extensional orders. We have formalised most of our results in cubical Agda