Abstract
In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in mathbf {ZF}, some are shown to be independent of mathbf {ZF}. For independence results, distinct models of mathbf {ZF} and permutation models of mathbf {ZFA} with transfer theorems of Pincus are applied. New symmetric models of mathbf {ZF} are constructed in each of which the power set of mathbb {R} is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube [0, 1]^{mathbb {R}}.
Highlights
2.1 The content of the article in brief mathematicians are aware that a lot of theorems of ZFC that are included in standard textbooks on general topology may fail in ZF and many amazing disasters in topology in ZF have been discovered, new non-trivial results showing significant differences between truth values in ZFC and in ZF of some given propositions can be still surprising
The set of all real numbers is denoted by R and, if it is not stated otherwise, R and every subspace of R are considered with the usual topology and with the metric induced by the standard absolute value on R
Mathematicians are aware that a lot of theorems of ZFC that are included in standard textbooks on general topology may fail in ZF and many amazing disasters in topology in ZF have been discovered, new non-trivial results showing significant differences between truth values in ZFC and in ZF of some given propositions can be still surprising
Summary
The intended context for reasoning and statements of theorems is the Zermelo–Fraenkel set theory ZF without the axiom of choice AC. A well-ordered cardinal number is an initial ordinal number, i.e., an ordinal which is not equipotent to any of its elements. Every well-orderable set is equipotent to a unique well-ordered cardinal number, called the cardinality of the well-orderable set. By transfinite recursion over ordinals α, we define: ω0 = ω (the set of all finite ordinal numbers); ωα+1 = H (ωα); ωα = sup{ωβ : β < α} = {ωβ : β < α} if αis a non-zero limit ordinal, where, for a set A, H (A) is the Hartogs’ number of A, i.e., the least ordinal α which is not equipotent to a subset of A. A set X is called countable if X is equipotent to a subset of ω. The set of all real numbers is denoted by R and, if it is not stated otherwise, R and every subspace of R are considered with the usual topology and with the metric induced by the standard absolute value on R
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