✍ Scribed by Christine Gaßner
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The present article deals with the power of the axiom of choice (AC) within the second‐order predicate logic. We investigate the relationship between several variants of AC and some other statements, known as equivalent to AC within the set theory of Zermelo and Fraenkel with atoms, in Henkin models of the one‐sorted second‐order predicate logic with identity without operation variables. The construction of models follows the ideas of Fraenkel and Mostowski. It is e. g. shown that the well‐ordering theorem for unary predicates is independent from AC for binary predicates and from the trichotomy law for unary predicates. Moreover, we show that the AC for binary predicates follows neither from the trichotomy law for unary predicates nor from Zorn's lemma for unary predicates nor from the formalization of the axiom of choice for disjoint families of sets for binary predicates, and that the trichotomy law for unary predicates does not follow from AC for binary predicates.
Mathematics Subject Classification: 03B15, 03E25, 04A25.
📜 SIMILAR VOLUMES
## Abstract It is known that – assuming the axiom of choice – for subsets __A__ of ℝ the following hold: (a) __A__ is compact iff it is sequentially compact, (b) __A__ is complete iff it is closed in ℝ, (c) ℝ is a sequential space. We will show that these assertions are not provable in the absence
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