When the Canonical Ramsey's Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the Canonical Ramsey's Theorem. We introduce the restriction of Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions, the lambda-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman's Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-omega exponentiation is reducible to this same principle by a uniform computable reduction.
Regressive versions of Hindman’s theorem / Carlucci, Lorenzo; Mainardi, Leonardo. - In: ARCHIVE FOR MATHEMATICAL LOGIC. - ISSN 0933-5846. - 63:3-4(2024), pp. 447-472. [10.1007/s00153-023-00901-6]
Regressive versions of Hindman’s theorem
Carlucci, Lorenzo
;Mainardi, Leonardo
2024
Abstract
When the Canonical Ramsey's Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey's Theorem by Kanamori and McAloon. Taylor proved a "canonical" version of Hindman's Theorem, analogous to the Canonical Ramsey's Theorem. We introduce the restriction of Taylor's Canonical Hindman's Theorem to a subclass of the regressive functions, the lambda-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman's Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-omega exponentiation is reducible to this same principle by a uniform computable reduction.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.