Phase transition thresholds for some Friedman‐style independence results

Abstract

We classify the phase transition thresholds from provability to unprovability for certain Friedman-style miniaturizations of Kruskal's Theorem and Higman's Lemma. In addition we prove a new and unexpected phase transition result for ε0. Motivated by renormalization and universality issues from statistical physics we finally state a universality hypothesis.

Some Natural Zero One Laws for Ordinals Below ε 0

“…Using a formula by Lagrange 1775 (cf. Lemma 3.3 in [5]) it can easily be shown that an L zero one law breaks down for many ordinals between ω ω and ε 0 when they are represented by the lexicographic path order over a signature with a binary function symbol and a constant (cf., e.g. the definition of < in [5] p.6).…”

“…Lemma 3.3 in [5]) it can easily be shown that an L zero one law breaks down for many ordinals between ω ω and ε 0 when they are represented by the lexicographic path order over a signature with a binary function symbol and a constant (cf., e.g. the definition of < in [5] p.6). But the expectation is that for any system of ordinal notations published in the literature at least limit laws will hold with respect to L and canonically extended norm functions.…”

Unprovability results involving braids

“…We shall now consider more general sequences, where the entries no longer obey a particular formation law, but only satisfy some growth conditions defined in terms of Garside's complexity, in the spirit of the sentences considered by Friedman [19]. The main result here is that there exists a precise description of the conditions that lead from IΣ IΣ IΣ 1 -provability to IΣ IΣ IΣ 1 -unprovability, thus witnessing a quick phase transition phenomenon analogous to those investigated in [40][41][42].…”

“…What makes this possible is that these two functions cannot be distinguished inside IΣ IΣ IΣ 1 . The general idea of the proof, which is reminiscent of the analysis of phase transition for the Kruskal theorem [40][41][42][43], consists in starting with a long descending sequence, typically a G 3 -sequence (or, equivalently, any sequence witnessing for the principle WO ) and then constructing a new sequence by dilating the original one so as to lower the complexity of the entries. The argument requires that sufficiently many braids of low complexity be available, and this is where the estimate of Corollary 3.9 is crucial.…”

Turchin's Relation and Subsequence Relation in Loop Approximation

“…On the other hand, well-binariness of the homeomorphic embedding is shown to be nonprovable in the Peano arithmetic with the first-order induction scheme [12], and this fact aroused interest of logicians and computer scientists with background in mathematical logic (a thorough study of the proof-theoretical strength of the fact is in [16]). Studies of the homeomorphic embedding as a termination criterion for term rewriting systems ( [11,14]) are located in the middle between these poles of pure theory and practice.…”