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 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

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

The Baire space I – getting off the ground

An area in mathematics concerning a special kind of topological spaces called Baire spaces is an example of a notion that intersects three branches of mathematics, namely topology, analysis and set theory – this intersection in particular is called descriptive set theory. We start off by defining the space, then exploring its properties and ending up looking at its applications throughout analysis and topology. Continue reading