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<p>An embedding of the hereditarily finite superstructure over the ordered field of real numbers into the set of reals is constructed which takes Σ-subsets to sets computable by infinite time Blum-Shub-Smale machines (ITBMs). A notion of ITBM-constructivizable structure is introduced. It is proved that constructivizability of an arbitrary algebraic structure over the ordered field of real numbers implies its ITBM-constructivizability. We obtain a theorem on the existence of ITBM-constructivizable models of the cardinality of the continuum for countable consistent theories with infinite models.</p>

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On Two Ways of Representation of Uncountable Structures

  • 期刊论文

  • 发表日期：

    2025年11月1日

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  • A. S. Morozov

摘要

An embedding of the hereditarily finite superstructure over the ordered field of real numbers into the set of reals is constructed which takes Σ-subsets to sets computable by infinite time Blum-Shub-Smale machines (ITBMs). A notion of ITBM-constructivizable structure is introduced. It is proved that constructivizability of an arbitrary algebraic structure over the ordered field of real numbers implies its ITBM-constructivizability. We obtain a theorem on the existence of ITBM-constructivizable models of the cardinality of the continuum for countable consistent theories with infinite models.