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. 1. General algebra and measurable cardinals. An unexpectedly direct connection. 2. Borel selection and higher set theory. A descriptive set theoretic context extensively pursued by some functional analysts. 3. Equational Boolean relation theory and Mahlo cardinals. A discrete mathematical context. 4. We conclude with an adaptation of 1 to weak second order logic. Some innocent looking statements in general algebra turn out to be equivalent to the existence of a measurable cardinal. In fact, the first measurable cardinal turns out to be clearly identifiable in a basic context in general algebra. Here an algebra is just a relational structure based on finitely many constant and function symbols and no relation symbols. PROPOSITION 1.1. Every algebra with a sufficiently large domain has a proper extension with the same countable subalgebras up to isomorphism. PROPOSITION 1.2. Every algebra with a sufficiently large domain has a proper extension with the same finitely generated subalgebras up to isomorphism.