Research

For the non-specialist, my research is best described as follows:

Georg Cantor's famous work identifies different magnitudes of infinity, the smallest infinity being represented by the set of natural numbers, i.e. the countable. The latter has served as a wonderful tool for the logical study of ordinary mathematics and (sequential) computation. My main research interest is in

the study of the uncountable in ordinary mathematics and computation, including Reverse Mathematics, a program that aims to identify the minimal axioms needed to prove theorems of ordinary mathematics.

A recent introductory talk may be found on Youtube: https://youtu.be/xQD3yX2r1Ng

In more technical detail, I am actively working on the following topics:

Beyond Hilbert's program and the Gödel hierarchy (jww Dag Normann)

Hilbert's famous list of open problems, presented at the Paris ICM in 1900, contains a number of foundational problems. For instance, the second problem pertains to the consistency of mathematics, i.e. the fact that no false theorems can be proved. Hilbert later developed this problem into Hilbert's program for the foundations of mathematics. However, Gödel's famous incompleteness theorems show that Hilbert's program is impossible. On a positive note, the notion of consistency gave rise to the Gödel hierarchy, a linear order that is said to capture all natural and significant logical systems (See [1]), including those that are part of the program Reverse Mathematics ([2,3]). The Gödel hierarchy is a central object of study to which all major sub-fields of logic contribute.

In recent work, Dag Norman and I identify a number of basic theorems of uncountable mathematics that fall far outside of the Gödel hierarchy. Some examples are as follows:

a) Cousin's lemma, i.e. the Heine-Borel compactness of the unit interval.

b) Lindelöf's lemma, i.e. the Lindelöf property for the real line.

c) Other covering lemmas due to Vitali, Besicovitch, Rademacher, Young, etc.

d) Basic properties of the gauge integral. The latter is a generalisation of the Lebesgue and improper Riemann integral, and the only direct formalisation of Feynman's path integral.

e) Basic theorems from topology pertaining to dimension and paracompactness (See here).

f) Uniform theorems (See here); these are theorems in which the objects claimed to exist only depend on few parameters.

g) Many other theorems; see my Papers section.

In terms of (higher-order) comprehension axioms, these theorems require full second-order arithmetic for a proof. Note that these results do not depend on the axiom of choice: e.g. Cousin's lemma is provable without it.

Finally, as in 'second-order' Reverse Mathematics, this topic is intimately connected with higher-order computability theory, which is Dag Normann's area of expertise.

References

[1] Stephen G. Simpson, The Gödel hierarchy and reverse mathematics, in: Kurt Gödel: Essays for His Centennial, Association for Symbolic Logic, 2010.

[2] Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, 2009.

[3] John Stillwell, Reverse mathematics, proofs from the inside out, Princeton Univ. Press, 2018.

The computational content of Nonstandard Analysis (jww Dag Normann).

Nonstandard Analysis, as pioneered by Abraham Robinson, provides a natural formalisation of the well-known intuitive infinitesimal calculus from physics and (historically) mathematics. Based on work by Benno van den Berg et al (See here), I have shown that large parts of Nonstandard Analysis have computational content. A number of papers on this topic may be found in my Papers section.

These results are based on the syntactic approach to Nonstandard Analysis, in particular Edward Nelson's internal set theory. I have a general interest in topics pertaining to Nonstandard Analysis.

The role of computability in formal semantics (jww Kristina Liefke)

Together with Kristina Liefke, I have applied my expertise in computability theory in the study of the formal semantics of natural languages, a sub-discipline of linguistics. We show that computable techniques provide novel and elegant solutions to known problems in natural language semantics such as Partee’s temperature puzzle. See my Papers section. We are currently investigating other fruitful problems using computability theory.