This post originates from a question I read on stackexchange. If we assume that the structuralist thesis is true, saying that mathematics is really the study of patterns and nothing else, where does this leave set theory? We have some intuition about number theory studying the pattern of natural numbers, real analysis about the reals and so on, but *prima facie* it seems that set theory doesn’t fit patterns in the same way. One might start off with the naive answer that set theory studies the pattern of sets – but then we encounter the classical paradoxes. If we instead fix ourselves on ZFC, how come set theory studies (almost exclusively) objects independent of ZFC?

# Author: Dan Saattrup Nielsen

# All things ultra

I’ve personally always been fascinated by the notion of an *ultraproduct*. Before I came to know of the construction, I guess it could’ve been because it kept popping up and was “unknown territory” – and the cool sounding name, I must admit. However, after actually knowing what the construction is all about, it doesn’t fascinate me any less, which is largely due to its powerful applications. I’ll here try to define the notion of *ultrafilters*, *ultraproducts* and *ultrapowers* as well as showing a particular application, namely a rigorous account of infinitesimals in a branch of mathematics called non-standard analysis. Continue reading

# DLO’s III – from rational to real

From the last two posts we managed to prove both the existence and the uniqueness of the rationals, which could be done by explicit definitions using known facts about the natural numbers. However, we also showed that the property that distinguished the real numbers from the rationals, namely completeness, couldn’t be defined in first-order logic. This also means that we can’t proceed to prove the existence of the reals in a direct matter as the rationals, and we will do so by indirect means. Continue reading

# DLO’s II – uniqueness of rationals

Now that we’ve proven that the rationals exist, we would like to know if it would make a difference had we picked another representative of the countable DLO’s without endpoints as our rationals. Would any candidate have been suitable or is our choice of special? Continue reading

# DLO’s I – existence of the rationals

One of the very first things we learn at higher math education is the different kinds of number sets, e.g. the naturals, rationals and reals. These are however just taken for granted, and we merely assume that these actually exist and are well-defined. I’ll spend the next few blog posts to characterize the rationals and the reals from facts known from the natural numbers alone. This post will be dedicated to proving the existence of the rationals Continue reading

# Is mathematics a posteriori?

The general consensus among mathematical philosophers (or philosophical mathematicians, for that matter) is that mathematics is *a priori* – independent of experience. However, a flourishing contemporary branch within the philosophy of mathematics is that of *structuralism*, which I will explain in detail in a little bit. Advocates of the structuralist approach are e.g. Shapiro [1] and Resnik [2], who argue how this approach answers a lot of important philosophical questions, including Resnik’s way of answering how mathematics can have applications in the physical world. There is a catch however: he assumes that mathematics is *a posteriori* – dependent of experience – contrary to popular belief.

# The Baire space II – the Baire Category Theorem

Having established the Baire space, we will here state and prove a version of the Baire Category Theorem, which is an important tool used within functional analysis, as well as also having applications in other parts of analysis and topology. After proving the theorem, we will state a few results in which the proofs rely on the theorem, to get a feeling of its power. Continue reading