New Infinitary Mathematics is an unusual book and an important contribution to foundations of mathematics. It is the final word of a charismatic and original mathematician and philosopher that covers many years of work on a project that he embarked upon in the 1960s. It is presented in three extensive chapters: I Great Illusion of Twentieth Century Mathematics, II New Theory of Sets and Semisets, and III Infinitesimal Calculus Reaffirmed, and a shorter supplement IV Making Real Numbers Discrete. Much of this material, written in Czech, had been published earlier in four volumes by Karolinum Press. Here, they are put together and edited by Alena Vencovská, who explains in detail how the translations and the editorial process were finalized after Vopěnka’s sudden death in 2015.

In an introduction that is short yet full of interesting biographical details, Vencovská gives a very compelling portrait of Vopěnka. She also explains well and in simple terms the basic features of Vopěnka’s project.

Vopěnka’s early work in the 1970s led him to the creation of what he called Alternative Set Theory (AST). AST is an axiomatic set theory whose primary objects are sets, classes—as in the Gödel-Bernays or Morse set theories—and in addition, a new kind: semisets. Roughly, sets behave like the Cantorian finite sets. For example, if r is a set relation that linearly orders a nonempty set x, then x has an r-least and an r-last element. Every set is a class, and as usual, there are also classes too big to be sets, but there are also proper classes in a new sense. A crucial axiom declares that there is a set that has a subclass that is not a set. Such classes are called semisets.

In AST, natural numbers are defined in the universe of sets just as the von Neumann ordinals are. By the comprehension axiom for classes, they form a class. Since there is no largest number, it is a proper class (not a set), and it can be shown that it cannot be the subclass of a set; hence, it is not a semiset. A set is defined to be finite if all of its subclasses are sets. Finiteness in this sense is definable in AST. Let V be a model of AST, and let $FN$ be the class of finite natural numbers in V. $FN$ with suitably defined addition and multiplication is a model of Peano Arithmetic, which may be either the standard or a nonstandard model. By the prolongation axiom, $FN\subseteq \alpha $ for some natural number $\alpha $ , which shows that $FN$ is a semiset and, therefore, that $\alpha $ is not a finite set.

While AST is no longer a subject in New Infinitary Mathematics, the whole Part II is devoted to a theory of semisets, but, for a reason, no single axiomatic system is established. There is a core collection of postulates about sets, but no fixed comprehension schema for classes. One of the axioms of AST is that there are only two infinite cardinal numbers, but here this is optional. The aim is to provide precise basic concepts and rules for a reconstruction of the classical areas of mathematics, but Vopěnka sees it as a dynamic system in which one is free to change rules according to how mathematics develops, not necessarily in one direction. Still, two postulates are firmly fixed: semisets exist, and every countable class (another technical term) is a subset of a set. Why are they necessary?

In Part I, the author shows how the concept of actual infinity led to confusion and paradoxes throughout the ages and, as a consequence, had been firmly rejected, while mathematics, especially since the introduction of infinitesimal calculus, flourished. Chapter 1, Theological Foundations, is a historical survey of positions on potential and actual infinity. It includes a section The Disputation about Infinity in Baroque Prague, which leads to a more detailed discussion of the contributions of Bernard Bolzano, whose extensive writings on paradoxes of infinity have been Vopěnka’s early inspiration, together with Robinson’s nonstandard analysis and Cohen’s forcing method. Vopěnka credits Bolzano with the first convincing argument that can be considered a proof of the existence of actual infinity. The section Actually Infinite Multitudes (p. 51) ends with the conclusion: Bolzano proved that the multitude of natural numbers is actually infinite (Vopěnka’s bold face).

So what is the problem? Chapters 2 and 3 are about Cantor’s Set Theory and the axioms of ZFC. Chapter 3 begins with the bold statement: Almost all infinitary mathematics of the twentieth century stands and falls with the existence of the set of natural numbers. This is fortified by a quotation from Poincare (p. 103): There is no infinity. The Cantorians forgot this, and so fell into contradiction [… ].

In Chapter 4, Summit and Fall, Vopěnka presents his own work on ultrapower constructions of models of set theory from the 1960s, which eventually led him to the conclusion that If there exists one set of all natural numbers then there exist many different sets of all natural numbers. Consequently, the set of all natural numbers does not exist, for the same reasons as the set of all ordinal numbers [does not exist]. (p. 120).

Semisets to the rescue. Examples such as $FN$ illustrate the theory of semisets, but the actual motivation runs much deeper. In his later work, Vopěnka grounds his mathematical ideas in observations inspired by the phenomenology of Edmund Husserl. The key new concept he introduces is natural infinity.

In a frequently used example, Vopěnka describes a chain that begins with Charlie the monkey and ends with Charles Darwin. Each successor in the chain is a son of its predecessor. The set of individuals in the chain is sharply defined—a concept used to separate sets from proper classes—and this qualifies it to be a set in Vopěnka’s sense. Also, the linear ordering of this chain is a set of ordered pairs. But the collection of monkeys in the chain is not a set. It has a definition, but not a sharp one. It is a proper class. Why? If it were a set, then—as I mentioned earlier—that set would have a last element in the chain, but we would be in trouble trying to find it. In this example, Darwin plays the role of that nonstandard $\alpha $ in a model V of AST, and the class of monkeys in the chain is an isomorphic image of $FN$ with the natural ordering. This is how natural infinity manifests itself in nature. A claim is made that almost everywhere we look, natural infinity shows up. Particularly important examples are finite sequences of objects that we see as disappearing beyond the horizon of our perception. Analyzing such phenomena, Vopěnka comes up with postulates about natural infinity that form a framework for a reconstruction of the number systems and infinitesimal calculus. Unlike in Robinson’s approach, the infinitesimals are not constructed by means of Cantorian set theory: their existence is a built-in feature of the axioms.

New Infinitary Mathematics is a book that should be widely read by mathematicians and philosophers of mathematics. The translators maintained Vopěnka’s distinct writing style. The book is full of interesting mathematics. Geometry and topology play a big role, and the last two parts are devoted to a detailed development of infinitesimal differential calculus.

The translation, enhanced with helpful editorial comments, provides a smooth reading. My only regret is that there is no index. A comprehensive index would help in navigating this rich text, in which familiar terms often have to be understood in their specific modified versions.