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Regularity in models of arithmetic. (English) Zbl 0584.03044
This paper investigates the quantifier ”there exist unboundedly many” in the context of first-order arithmetic. An alternative axiomatization is found for Peano arithmetic based on an axiom schema of regularity: The union of boundedly many bounded sets is bounded. We also obtain combinatorial equivalents of certain second-order theories associated with cuts in nonstandard models of arithmetic.

MSC:
03H15 Nonstandard models of arithmetic
03F30 First-order arithmetic and fragments
03C80 Logic with extra quantifiers and operators
03C62 Models of arithmetic and set theory
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References:
[1] Logic Colloquium ’77 pp 199– (1978)
[2] Handbook of mathematical logic pp 1133– (1977)
[3] DOI: 10.1007/BFb0067653
[4] Some independence results for Peano arithmetic 43 pp 725– (1978) · Zbl 0408.03048
[5] DOI: 10.1007/BFb0090171
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