Working with Nonstandard Models
Most of the research in foundations of mathematics that I do in some way or another involves the use of nonstandard models. I will give a few examples, and indicate what is involved.
Selection for Borel Relations
We present several selection theorems for Borel relations, involving only Borel sets and functions, all of which can be obtained as consequences of closely related theorems proved in [3], [4], [5], [6], involving coanalytic sets. The relevant proofs given there use substantial set theoretic methods, which were also shown to be necessary. We show that none of our Borel consequences can be proved without substantial set theoretic methods. The results are established for Baire space. We give equivalents of some of the main results for the reals.
Time and Heritage
As a historian studying what can be considered a form of intellectual history, I have slowly come to adopt Michel de Certeau’s observation made at the end of the 1980s that ‘objectifying the past, for the last three centuries, has undoubtedly left unconsidered time within a discipline that has continued to use it as a taxonomic instrument’. To a certain degree, time has become commonplace for the historian who has preserved or instrumentalized it. It is not considered because it is inconceivable, but because we do not think of it or, more simply, we do not think about it. As a historian attempting to pay attention to the time I’m living in, I have thus, like many others, noticed the swift development of the category of the present until it has become obvious that the present is omnipresent. This is what I refer to here as ‘presentism’.