Abstract
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 \({\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}\) in question, d > 1, we fix a natural extension of Peano Arithmetic, \({T \supseteq \sf{PA}}\) , that proves the corresponding second-order sentence \({\sf{WPO}\left({\rm S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }\) . Having this we consider the following parametrized first-order slow well-partial-ordering sentence \({\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}\)
for a natural additive Seqd-norm |·| and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for IΔ 0 + exp. We show that the following basic phase transition clauses hold with respect to \({T = \Pi_{1}^{0}\sf{CA}_{ < \varphi ^{_{\left( d-1\right) }} \left(0\right) }}\) and the threshold point1.
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1.
If r < 1 then \({\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }\) is provable in T.
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2.
If \({r > 1}\) then \({\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }\) is not provable in T.
Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA 0 and \({\Delta _{1}^{1}\sf{CA}}\) , if d = 2 and d = 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions \({f: \mathbb{N} \rightarrow \mathbb{R}_{+}}\) and prove analogous theorems. (In the sequel we denote by \({\mathbb{R}_{+}}\) 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
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3.
\({\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}\) 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 1 α given by \({1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}\) being the familiar fast growing Hardy function and \({H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}\) the corresponding slowly growing inversion.
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4.
If \({\alpha < \varepsilon _{0}}\) , then \({\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}\) is provable in T = PA.
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5.
\({\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}\) is not provable in T = PA.
We conjecture that this pattern is characteristic for all \({T\supseteq \sf{PA}}\) under consideration and their proof-theoretical ordinals o (T ), instead of \({\varepsilon _{0}}\) .
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Gordeev, L., Weiermann, A. Phase transitions of iterated Higman-style well-partial-orderings. Arch. Math. Logic 51, 127–161 (2012). https://doi.org/10.1007/s00153-011-0258-3
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DOI: https://doi.org/10.1007/s00153-011-0258-3
Keywords
- Mathematical logic
- Foundations of mathematics
- Proof theory
- Ordinal notations
- Well-partial-orderings
- Asymptotic combinatorics