Abstract
Using standard methods of analytic combinatorics we elaborate critical points (thresholds) of phase transitions from provability to unprovability of arithmetical well-partial-ordering assertions in several familiar theories occurring in the reverse mathematics program.
Highlights
Consider a familiar arithmetical theory T and a sufficiently complicated arithmetical assertion A(r) with computable real parameter r > 0
Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs have been obtained in the seventies-eighties by Paris and Friedman [16] who produced mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory, showing definitely that Godel’s incompleteness matters to mathematics
Lev Gordeev and Andreas Weiermann for example Tauberian theory, analytic combinatorics and even the theory of the Riemann zeta function and L-function theory, played an intrinsic role, whereas on the logical side proof theory and the theory of subrecursive hierarchies have been used to a great extent [2, 17, 18, 20, 21]
Summary
Consider a familiar (presumably consistent) arithmetical theory T and a sufficiently complicated arithmetical assertion A(r) with computable real parameter r > 0. Lev Gordeev and Andreas Weiermann for example Tauberian theory, analytic combinatorics and even the theory of the Riemann zeta function and L-function theory, played an intrinsic role, whereas on the logical side proof theory and the theory of subrecursive hierarchies have been used to a great extent [2, 17, 18, 20, 21] We continue this program by investigating T -(un)provability of assertions regarding well partial-orderedness of sets of noniterated and iterated finite sequences and trees, where T is not restricted to Peano Arithmetic, while ranging over subsystems of second-order arithmetic featuring in Friedman-Simpson’s reverse mathematics program [5, 6, 15]; these theories can be characterized (formalized) by familiar principles (theorems) of the ordinary mathematics, rather than abstract set theory only. The results presented in the paper provide a useful and highly nontrivial applications of classical analytic methods (e.g., the enumeration of certain combinatorial classes) to traditional “discrete” proof theory and the reverse mathematics categorizing the strengths of classical logical frameworks
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