The foundations of mathematics in the theory of sets
The author's core problem is to give a set-theoretic foundation for the whole of (modern) mathematics. To reach this goal he first reflects about the problem of foundations, and argues that such a foundation cannot be given via an elementary theory formalized within first-order logic. This is, because neither (the metatheory of) first-order logic as a tool to axiomatically describe mathematical structures, nor even the syntax of the formalized language of (first-order) logic can be understood without a previous understanding of the notion of set. Therefore he comes back to the ``traditional'' understanding of the notion of an axiom as some statement which is obvious by itself. NEWLINENEWLINENEWLINEThe axioms he chooses for his purpose form a weak version of the standard ZF axioms (without infinity). The crucial point is that the author does not allow unrestricted, ``global'' quantifications over the universe of sets. More formally: \(\Sigma_0\)-formulas are allowed for comprehension, but \(\Sigma_1\)- formulas are not. That means, restricted quantifications are considered as unproblematic, unrestricted ones, however, as problematic. NEWLINENEWLINENEWLINEMoreover, he argues that the usual laws of classical logic hold true only for restricted formulas, and fail (to some extent) for ``global'' formulas. NEWLINENEWLINENEWLINEAt all, the author adheres to a static understanding of mathematics. The notion of an effective procedure, i.e.\ the notion of a construction, is not an essentially mathematical notion for him (cf.~pp. 15/16). NEWLINENEWLINENEWLINEA very basic observation is that the author reads the Greek notion of number, that means the Greek notion of ``arithmoi'', which means the ``multitudes composed of units'', essentially as finite sets. And argues then that the existence of such numbers is more or less obvious from our experience. Furthermore, number words like ``five'' from his point of view get a meaning like words as ``horse''. And these ``arithmoi'' are ``authentic inhabitants of the world, independent of human beings and their mental activity.'' NEWLINENEWLINENEWLINEOur common numbers then are ``symbol generated abstractions'': abstractions coming through the use of certain symbols to denote certain situations. And this is the very basic notion underlying all the understanding of foundations for mathematical abstractions. NEWLINENEWLINENEWLINEHowever, the ``arithmoi'' are more fundamental as the (natural) numbers in their modern understanding. NEWLINENEWLINENEWLINEHe uses ``finite'' in its essential and original -- non-technical -- sense as ``limited (or definite, or bounded) in size''. However, ``finitenes'' remains a (primitive) notion which cannot be defined but only explained. NEWLINENEWLINENEWLINEThen there is a distinction: the ``Euclidean'' approach takes for granted that ``the whole is greater than the part'', and the ``Cantorian'' approach allows for wholes which have the same size as some of their proper parts. NEWLINENEWLINENEWLINEIn Chapter 5 on ``Cantorian Finitism'' the standard axiom of infinity is added to his previous axioms, as well as the axiom of choice. NEWLINENEWLINENEWLINEFurthermore the author adds other principles which, e.g., guarantee the existence of the transitive hull for each set, and which (finally) allow to prove the (unrestricted) principle of transfinite induction/recursion along the whole class of ordinals. NEWLINENEWLINENEWLINEChapter 6 discusses the axiomatic method, including a particular discussion about the nature of mathematical objects. Regarding the problem of the nature of mathematical objects, the author's main claim is, that -- besides ``arithmoi'', i.e.\ sets -- for the objects of axiomatic theories this question is unsubstantiated: for all isomorphic models are on equal footing, and there is nothing like a ``standard model'' for such theories. Instead he prefers to look, in the framework of second-order logic, for categorical theories. So again all reduces to the set-theoretic foundations, also for second-order logic. NEWLINENEWLINENEWLINEThe following Chapter 7 discusses axiomatic set-theories, essentially ZF in a first-order and in a second-order version. NEWLINENEWLINENEWLINEBeginning with Chapter 8 he studies his ``Euclidean Finitism'', i.e.\ the standard axiom of infinity is replaced by an axiom which states that each set is finite (in the standard sense of this word), i.e.\ does not have a bijection to any of its proper subsets. NEWLINENEWLINENEWLINEThe axioms are ZF-like as before, again with an additional one which guarantes the existence of the transitive hull for each set. The resulting set theory is developed to such extent that it becomes obvious that a very interesting theory results. NEWLINENEWLINENEWLINEFirst he derives the principle of choice, as well as principles for inductive proofs, and recursive definitions -- through all sets. Furthermore he derives a kind of \(g\)-ary representation of (the sizes of) sets. NEWLINENEWLINENEWLINENow it becomes impossible, however, to prove that any two simply infinite systems are isomorphic, i.e.\ it becomes impossible to define a unique natural number sequence (up to isomorphism, of course). Nevertheless it can be proven that also for simply infinite systems one has the principles for inductive proofs and of recursive definitions, of course for each system (separately). And this, e.g., allows to have the arithmetical operations in each simply infinite system. NEWLINENEWLINENEWLINEHowever, as different simply infinite systems may be non-isomorphic, one does not have one distinguished among them, which, e.g., means that unlimited counting, i.e.\ counting ``for ever'', becomes an unavailable idea. NEWLINENEWLINENEWLINEFinally, it remains undecided which of the two approaches should get preference. But both approaches are, mathematically as well as philosophically, sound and interesting ones.
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