Eilenberg and Mac Lane [1] explained the notion of a 'natural' embedding by giving a categorical definition. Starting from their examples, we argue that one could equally well explain natural as meaning 'uniformly definable in set theory'. But do the categorically natural embeddings coincide with the uniformly definable ones? This is partly a technical question about whether certain well-known ...

Throughout, we will fix an algebraically closed field K. Classical algebraic geometry studies varieties defined over K, where in this proposal, a variety means a solution set of some polynomial equation system over K. Therefore, the study of a variety V is equivalent to the study of the maximal spectrum of its coordinate ring A(V ). Grothendieck realized that a relative version of the concept o...

The development of the memory bus has studied spreadsheets, and current trends suggest that the evaluation of consistent hashing will soon emerge. Given the current status of random epistemologies, security experts famously desire the evaluation of Web services. In order to solve this grand challenge, we motivate a flexible tool for simulating thin clients (FersCong), disconfirming that red-bla...

We identify precisely which singular ordinals are definable by monadic second order formulae over the ordinals, assuming knowledge of the definable regular cardi-

A compression algorithm takes a finite structure of a class K as input and produces a finite structure of a different class K’ as output. Given a property P on the class K defined in a logicL, we study the definability of property P on the class K’. We consider two compression schemas on unary ordered structures (words), compression by runlength encoding and the classical Lempel-Ziv. First-orde...

In this survey paper we consider presentations of structures in admissible sets and some effective reducibilities between structures and their degrees of presentability. Degrees of Σ-definability of structures, as well as degrees of presentability with respect to different effective reducibilities, are natural measures of complexity which are total, i.e. defined for any structure. We also consi...