David Booth

Are There Undecidable Propositions

Approximately 18 years ago I showed that in formal systems of a general kind one can specify propositions which are not decidable by means of formal proofs within the systems themselves, but which nevertheless can be decided by virtue of their conceptual content (see Finsler [1926a]). A formal proof was considered to be admissible for the purpose of this argument only if its interpretation constituted a logically unobjectionable proof.

Intrinsic Analysis of Antinomies and Self-Reference

This essay proposes a reconstruction of the ideas lying behind Finsler’s analysis of antinomical situations. We do not think that it is necessary to give a literal account of his arguments: They can be easily followed in his papers. Instead we sought a basis that the various arguments might have in common. This essay is self-contained. It does not depend on the results of Finsler’s analysis but gives an independent account of antinomical and self-referential situations.

On the Foundations of Set Theory

The first part of this construction of the foundations of set theory had the subtitle Sets and their Axioms. It appeared in 1926 in the Mathematische Zeitschrift [1926b]. The second part was to have treated the number systems. Since the first part met with such a lack of understanding, it seemed preferable to let these investigations, concerning the natural numbers, the continuum and the transfinite ordinal numbers, appear as separate publications ([1933], [1941b], [1954]).

On the Solution of Paradoxes

The antinomies of set theory and similar associated paradoxes have frequently been the subject matter for discussions; but full agreement over the various explanations has never been reached. As these things are of special significance for the foundations of mathematics, it is a pleasure to take the opportunity offered by the publisher of this periodical to comment from a mathematical point of view on the paper of H. Lipps, Die Paradoxien der Mengenlehre [1923], in the Jahrbuch fur Philosophie und Phanomenologische Forschung.

The Existence of the Number Numbers and the Continuum

The absolute consistency of the sequence of natural numbers and of the continuum will be proven here.

Platonism After All

It is particularly pleasing to see it clearly stated, in Wittenberg’s “Why No Platonism?” [1956], that mathematicians would like to be Platonists, if only they were able. This wish can really be fulfilled, however, because the objections against it are not valid.

Formal Proofs and Decidibility

In order to demonstrate the consistency of certain axiom systems, Hilbert makes use of a theory of mathematical proof in which the proof must be thought of as rigorously formalized in concrete symbols (see Hilbert [1922], [1923], [1926], Bernays [1922], Ackermann [1924]). “A proof is an array which must be graphically represented in its entirety” (Hilbert [1923, 152]). He adds: “A formula shall be said to be provable if it is either an axiom, or arises by substitution into an axiom, or is the concluding formula of a proof” (ibid., 152–153). The aim, then, is to show that, in a given axiom system, a contradictory formula (formalized in the same way) can with certainty never be proven. Axiom systems for which this can be demonstrated are said to be “consistent” (ibid., 157 and [1926, 179]). In the following where the formalization is quite general, such systems will be called formally consistent.

The Platonistic Standpoint in Mathematics

In Wittenberg’s interesting paper Uber adaquate Problemstellungen in der Mathematischen Grundlagenforschung [1953] and in the subsequent discussion (Wittenberg et al. [1954]) there is a clear description of a view of mathematics which is called naive or uncritical Platonism, “inhaltlich”, theological, Platonic philosophy, classical philosophy, and by E. Specker [1954] the “an sich” philosophy. It is said, for example, in Wittenberg’s answer that these views embrace the opinion “that this philosophy seizes upon objective relationships, that in itself it describes a factual situation, which as such remains removed from our powers of discretion”. This “Platonic standpoint” is, however, rejected even if with regret. Why? First of all because of the antinomies!

The Infinity of the Number Line

I am happy to be able to speak to your circle here in Basel, with its time honored mathematical tradition, about a time honored theme, namely the natural numbers.

The Combinatorics of Non-well-founded Sets

Even though non-well-founded sets appeared in Mirimanoff [1917a] the most significant early treatment was that of Finsler [1926b]. Since then there have been studies of set theory without an axiom of foundation, but these have usually been relative consistency arguments that fell short of accepting non-well-founded sets themselves. The publication of Aczel [1988] marked a change in this respect: Aczel gave an anti-foundation axiom that replace the axiom of foundation. He also compared Finsler’s axiom of identity to his anti-foundation axiom.