Non-discrete metrics in and some notions of finiteness (Q2827951)

In the paper under review, the author introduces two notions of finite, namely the notions of \textit{metrically finite} and \textit{real finite}, and studies in set theory without the axiom of choice (\(\mathsf{AC}\)) the relationship between these and certain known notions of finite. (There are several possible definitions of finite which are equivalent under \(\mathsf{AC}\), but they are not provably equivalent without \(\mathsf{AC}\). A summary of the known definitions of finite and the relationships between them in set theory without \(\mathsf{AC}\) can be found for example in Note 94 of \textit{P. Howard} and \textit{J. E. Rubin} [Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)].NEWLINENEWLINEA set \(X\) is \textit{metrically finite} if every metric on \(X\) is discrete (in other words, if every metric on \(X\) induces the discrete topology on \(X\)); otherwise \(X\) is \textit{metrically infinite}. \(X\) is \textit{real finite} if every subset of \(X\) which is equipollent with a subset of \(\mathbb{R}\) (the real numbers) is finite; otherwise \(X\) is \textit{real infinite}. (\(X\) is \textit{finite} if there is a bijection between \(X\) and a natural number \(n\); otherwise \(X\) is \textit{infinite}.)NEWLINENEWLINEThe author proves -- among several results -- the following ones: {\parindent=0.6cm\begin{itemize}\item[--] it is relatively consistent with \(\mathsf{ZF}\) (i.e., Zermelo-Fraenkel set theory without \(\mathsf{AC}\)) that there are infinite metrically finite sets; \item[--] the notion of real infinite is strictly stronger than that of metrically infinite; \item[--] given a set \(X\), the following are equivalent: {\parindent=0.8cm\begin{itemize}\item[{\(\bullet\)}] \(X\) is metrically infinite; \item[{\(\bullet\)}] \(X\) is weakly Dedekind infinite; \item[{\(\bullet\)}] the cardinality of the set of all metrically finite subsets of \(X\) is strictly less than the cardinality of \(\wp(X)\) (the power set of \(X\)). NEWLINENEWLINE\end{itemize}} NEWLINENEWLINE\end{itemize}} We recall that a set \(X\) is \textit{weakly Dedekind finite} if \(\wp(X)\) is \textit{Dedekind finite}, that is, if there is no injection \(f:\omega \rightarrow \wp(X)\). Otherwise, \(X\) is \textit{weakly Dedekind infinite}. We also recall that \(|X|<|Y|\) (i.e., the cardinality of \(X\) is strictly less than the cardinality of \(Y\)) if there is an injection \(f:X\rightarrow Y\), but there is no injection \(g:Y\rightarrow X\).NEWLINENEWLINEThis reviewer should like to point out that the author's result of item (3.) above adds information to the known result of the equivalence between (b) and (c). In particular, the latter equivalence (with `weakly Dedekind finite' in place of `metrically finite' in (c)) has been established by \textit{P. Vejjajiva} and \textit{S. Panasawatwong} [Notre Dame J. Formal Logic 55, No. 3, 413--417 (2014; Zbl 1338.03097)], where furthermore it is shown that each one of (b) and (c) (and hence (a)) is equivalent to \(|\mathbb{R}|\leq|\wp(X)|\). (\(|X|\leq |Y|\) if there is an injection \(f:X\rightarrow Y\).)