Approximating Beppo Levi's \textit{principio di approssimazione} (Q2925330)

In the paper under review, the authors undertake a notable effort to provide possible interpretations (using mostly the language of contemporary mathematics) of the Approximation Principle (AP) conceived by Beppo Levi in 1918, who tried -- via his principle -- to suggest a ``constructive'' alternative to Zermelo's axiom of choice (AC) suitable for mathematical analysis. Levi strongly objected to AC and the principle AP was his contribution and role in the debate and the strong criticism against AC and its non-constructive nature.NEWLINENEWLINETo the best of our knowledge, the paper under review is the first paper that attempts to shed some light to the admittedly obscure formulation of AP (and to the meaning of some notions introduced in AP) as this was given in \textit{B. Levi}'s paper [in: Scritti matematici offerti ad Enrico d'Ovidio. Torino: Fratelli Bocca. 305--324 (1918; JFM 46.0313.01)], and also reproduced in the current paper (Section 2.3, p. 155). It should be noted here that \textit{G. H. Moore}, in his book [Zermelo's axiom of choice. Its origins, development and influence. Berlin: Springer-Verlag (1982; Zbl 0497.01005)], also provides, on page 244, an interpretation of AP, as well as, a discussion on the principle and its deductive strength and connection with weak choice principles. In particular, Moore mentions (among other facts, on page 244) that AP is deducible from the axiom of countable choice (i.e., every countable family of non-empty sets has a choice function; this axiom is referred to as ``the denumerable axiom'' in Moore's book), and that Levi had observed that his principle was not equivalent to AC.NEWLINENEWLINEThe current paper is mainly organized as follows: Firstly, in Section 2.2, the authors supply various formulations of AP (which, although they have substantial differences from Levi's original formulation of AP, they provide possible interpretations and clarifications of notions involved in Levi's AP) and discuss their mutual relationships, as well as, their relationship with AC. Secondly, in Section 2.3, they present Levi's original formulation of AP and discuss its relation with their versions. Thirdly, in Section 3, the authors illustrate how (their versions of) AP can be applied in order to establish certain statements that concern metric spaces, aiming in this way to reflect the role and implementation of AP in an area of mathematics where weak choice principles like the axiom of countable choice and the principle of dependent choice hold a prominent position. The latter effort of the authors is consistent with Levi's goal to present AP as an alternative to AC suitable for analysis. As Moore mentions on page 244 of his book, Levi proved that AP implies Lindelöf's covering theorem: Every family of open intervals covering a subset \(A\) of \(\mathbb{R}\) has a countable subfamily covering \(A\).NEWLINENEWLINEThe authors' versions of AP (given in Section 2.2) are \(\mathrm{AP}^{1}\), \(\mathrm{AP}^{+}\), \(\mathrm{AP}^{C}\), along with their pointwise versions \(\mathrm{AP}_{P}^{1}\), \(\mathrm{AP}_{P}^{+}\), and \(\mathrm{AP}_{P}^{C}\), respectively. They prove that all these principles are equivalent to the full AC (Corollary 2.10). Furthermore, the authors provide a (seemingly) weaker version of \(\mathrm{AP}^{C}\), namely \(\mathrm{AP}^{C}(\mathbb{N})\) (given on page 153 of the paper), which is \(\mathrm{AP}^{C}\) restricted to deductive domains which contain only countable sets, where (according to the terminology in the paper) a deductive domain is a pair \((\mathcal E,f)\), where \(\mathcal E\) is a family of non-empty sets (non-empty sets are referred to as `prime aggregates' (or `inhabited sets') in the paper) and \(f\) is a choice function of \(\mathcal E\) (Definition 2.1). In Proposition 2.11, they establish that under the assumption of the axiom of countable choice, denoted by \(\mathrm{AC}_{\mathbb{N}}\) in the paper, \(\mathrm{AP}^{C}(\mathbb{N})\) implies \(\mathrm{AP}^{C}\) (hence, by Corollary 2.10 of the paper, \(\mathrm{AP}^{C}(\mathbb{N})\) implies AC). (Although, the formulation of \(\mathrm{AC}_{\mathbb{N}}\) is not explicitly given in the paper, it follows from the context of the paragraph prior to Proposition 2.11 and the context of Section 3, that the authors mean, as \(\mathrm{AC}_{\mathbb{N}}\), the principle: Every countable family of non-empty sets has a choice function.)NEWLINENEWLINEHowever, we believe that the proof of Proposition 2.11 is incorrect, since the authors seem to apply \(\mathrm{AC}_{\mathbb{N}}\) (first paragraph of the proof of Proposition 2.11) for each \(A\in\mathcal D\) (in order to obtain a certain countable subset \(C_A\) of \(A\) for each \(A\in\mathcal D\)), where \(\mathcal D\) is a suitable subfamily of a given arbitrary deductive domain \((\mathcal E,f)\), and \(\mathcal D\) is not necessarily countable (hence, \(\mathrm{AC}_{\mathbb N}\) cannot be applied uniformly). According to our opinion, \(\mathrm{AP}^{C}(\mathbb{N})\) implies AC (thus, implies -- by Corollary 2.10 -- \(\mathrm{AP}^{C}\)), without invoking the axiom of countable choice \(\mathrm{AC}_{\mathbb N}\). Here is a sketchy outline of a proof (which is similar to the proof of Proposition 2.9.1 that \(\mathrm{AP}_{P}^{C}\) implies AC): Let \(X,Y\) be two sets and let \(F:X\rightsquigarrow Y\) be an MV-function (i.e., \(\forall x\in X\), \(F(x)\) is a non-empty subset of \(Y\) (so that \(F\) is a function from \(X\) to \(\mathcal P(Y)\setminus\{\emptyset\}\)); the latter notation and terminology is according to the paper's Definition 2.3) be as in the formulation of AC given on page 148 of the paper. One seeks for a function \(g:X\to Y\) such that \(g(x)\in F(x)\) for all \(x\in X\). To this end, let \(\mathcal E=\{\{x\}\times\omega:x\in X\}\) (where \(\omega\) denotes, as usual, the set of natural numbers) and let \(f:\mathcal E\to\bigcup\mathcal E\) be defined by \(f(\{x\}\times\omega)=(x,0)\) for all \(x\in X\). Then \((\mathcal E,f)\) is a deductive domain and \(\mathcal E\) is comprised only of countable sets. Let \(G:\mathcal E\rightsquigarrow Y\) be defined by \(G(\{x\}\times\omega)=F(x)\) for all \(x\in X\) and also let \(d:\mathcal P(Y)\times\mathcal P(Y)\to\mathbb{Q}_{0}^{+}\) be defined by \(d(Z,Z')=0\) if \(Z=Z'\) and \(d(Z,Z')=1\) if \(Z\neq Z'\). It is fairly easy to show (due to the definition of \(\mathcal E\)) that \(\forall A\in\mathcal E\), \(\psi(A,G,d)\) is true (where \(\psi(A,G,d)\) is defined on page 149 of the paper), thus letting \(\mathcal D=\{A\in\mathcal E:\psi(A,G,d)\}\), we may conclude that \(\mathcal D=\mathcal E\). By \(\mathrm{AP}^{C}(\mathbb{N})\), there is a function \(f^{*}:\mathcal D\to Y\) such that \(f^{*}(\{x\}\times\omega)\in G(\{x\}\times\omega)\;(=F(x))\) for all \(x\in X\). Define a map \(g:X\to Y\) by \(g(x)=f^{*}(\{x\}\times\omega)\) for all \(x\in X\). Then \(g\) is the required function. A similar argument can be employed in order to show that the pointwise version \(\mathrm{AP}_{P}^{C}(\mathbb{N})\) of \(\mathrm{AP}^{C}(\mathbb{N})\) also implies AC. In the light of the latter implication, the assumption of \(\mathrm{AC}_{\mathbb N}\) in Theorem 3.4 (i.e., \(\mathrm{AP}_{P}^{C}(\mathbb{N})\) and \(\mathrm{AC}_{\mathbb N}\) prove that every open and bounded subset of \(\mathbb{R}^{2}\) is countably coverable) is redundant (however, as it appears from the discussion before Theorem 3.4, the authors' intention is to recast Levi's related result). In addition, \(\mathrm{AP}_{P}^{C}(\mathbb{N})\), or \(\mathrm{AP}^{C}(\mathbb{N})\), could be suitably used as an assumption in order to obtain the stronger (than Theorem 3.4) result of Proposition 3.10.NEWLINENEWLINEWe would also like to point out that the authors' remark, in the paragraph prior to Proposition 2.11, that ``\dots a proof of the fact that a countable union of finite sets is countable is already possible in ZF alone, hence it does not require \(\mathrm{AC}_{\mathbb N}\).'' is erroneous, since there are models of ZF in which the aforementioned statement is false (e.g., the second Cohen model \(\mathcal M7\) in [\textit{P. Howard} and \textit{J. E. Rubin}, Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)].NEWLINENEWLINEConcluding this review, it is worth mentioning that the paper is rich in historical remarks and philosophical comments on the approximation principle, and also rich in bibliographical entries; in particular, the authors provide 49 related references. Furthermore, as the authors mention in (the first paragraph of) their last Section 4, it would be important to classify the strength of AP and to find its connections with known principles of choice. In our opinion, the current paper has provided a fertile ground for further consideration and research on Beppo Levi's AP.