What is a Higher Level Set?

Structuralist foundations of mathematics aim for an “invariant” conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. In this paper I argue in favor of the former over the latter. First, I explain why to pick between them we need to ask the question of what is the correct “categorified” version of a set. Second, I argue in favor of groupoids over categories as “categorified” sets by introducing a pre-formal understanding of groupoids as abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics

What is a Higher Level Set?

Structuralist foundations of mathematics aim for an “invariant” conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. In this paper I argue in favor of the former over the latter. First, I explain why to pick between them we need to ask the question of what is the correct “categorified” version of a set. Second, I argue in favor of groupoids over categories as “categorified” sets by introducing a pre-formal understanding of groupoids as abstract shapes. This conclusion lends further support to the perspective taken by the Univalent Foundations of mathematics

Fermat’s last theorem proved in Hilbert arithmetic. I. From the proof by induction to the viewpoint of Hilbert arithmetic

In a previous paper, an elementary and thoroughly arithmetical proof of Fermat’s last theorem by induction has been demonstrated if the case for “n = 3” is granted as proved only arithmetically (which is a fact a long time ago), furthermore in a way accessible to Fermat himself though without being absolutely and precisely correct. The present paper elucidates the contemporary mathematical background, from which an inductive proof of FLT can be inferred since its proof for the case for “n = 3” has been known for a long time. It needs “Hilbert mathematics”, which is inherently complete unlike the usual “Gödel mathematics”, and based on “Hilbert arithmetic” to generalize Peano arithmetic in a way to unify it with the qubit Hilbert space of quantum information. An “epoché to infinity” (similar to Husserl’s “epoché to reality”) is necessary to map Hilbert arithmetic into Peano arithmetic in order to be relevant to Fermat’s age. Furthermore, the two linked semigroups originating from addition and multiplication and from the Peano axioms in the final analysis can be postulated algebraically as independent of each other in a “Hamilton” modification of arithmetic supposedly equivalent to Peano arithmetic. The inductive proof of FLT can be deduced absolutely precisely in that Hamilton arithmetic and the pransfered as a corollary in the standard Peano arithmetic furthermore in a way accessible in Fermat’s epoch and thus, to himself in principle. A future, second part of the paper is outlined, getting directed to an eventual proof of the case “n=3” based on the qubit Hilbert space and the Kochen-Specker theorem inferable from it