It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy which is built on a start function like the successor function. In this paper we study for a given start function g iteration hierarchies with a sub-linear modulus h of iteration. In terms of g and h we classify the phase transition for the resulting diagonal function from being primitive recursive to being Ackermannian.

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$$\end{document} in question, d > 1, we fix a natural extension of Peano Arithmetic, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T \supseteq \sf{PA}}$$\end{document}, that proves the corresponding second-order sentence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{WPO}\left({\rm (...) S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$$\end{document}. Having this we consider the following parametrized first-order slow well-partial-ordering sentence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| 1}$$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$$\end{document} is not provable in T.Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta _{1}^{1}\sf{CA}}$$\end{document}, if d = 2 and d = 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$$\end{document} and prove analogous theorems. (In the sequel we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}_{+}}$$\end{document} the set of EFA-provably computable positive reals). In the basic case T = PA we strengthen the basic phase transition result by adding the following static threshold clause\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$$\end{document} is still provable in T = PA (actually in EFA). Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1α given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$$\end{document} being the familiar fast growing Hardy function and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$$\end{document} the corresponding slowly growing inversion. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha < \varepsilon _{0}}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$$\end{document} is provable in T = PA. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$$\end{document} is not provable in T = PA. We conjecture that this pattern is characteristic for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T\supseteq \sf{PA}}$$\end{document} under consideration and their proof-theoretical ordinals o (T ), instead of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varepsilon _{0}}$$\end{document}. (shrink)

We give a survey of current research on Gödel’s incompleteness theorems from the following three aspects: classifications of different proofs of Gödel’s incompleteness theorems, the limit of the applicability of Gödel’s first incompleteness theorem, and the limit of the applicability of Gödel’s second incompleteness theorem.

In this paper we prove general exact unprovability results that show how a threshold between provability and unprovability of a finite well-quasi-orderedness assertion of a combinatorial class is transformed by the sequence-construction, multiset-construction, cycle-construction and labeled-tree-construction. Provability proofs use the asymptotic pigeonhole principle, unprovability proofs use Weiermann-style compression techniques and results from analytic combinatorics.