Mathematics First Thesis Concept

My thesis in mathematics will first have an introduction that is divided in two chapters:

• 1. Introduction to Set Theory & Ultrafilters: Overleaf: Introduction to Ultrafilters
• 2. Introduction to Model Theory & Ultraproducts: Overleaf: Introduction to Ultraproducts

I will then have presented the sufficient background to introduce the reader to the ultraproducts:

• 3. Introduction to Ultraproducts

Some construction like the one presented below will follow; I expect to edit and delete some of the present construction, and add new ones as time goes on and intuition comes, therefore the following are to be considered as the very first constructions. I am both using the knowledge acquired in Model Theory (Lecture) and Models and Ultraproducts, J. L. Bell, A. B. Slomson.

A different approach using the same tools is to first take a possible world and, given a sequence of possible worlds, consider it as an ultraproduct, I expand the possible results coming out of this approach in Algebra on Worlds.

Colouring Ultraproduct

The colouring ultraproduct has two main features: its ultrafilter is non-principal and any two models in the sequence differ only in the mapping of a function of the form $f:\{c\}\to D$ for $c$ a constant and $D$ an infinite set. The name comes from the easy-to-picture example where $f$ is the function mapping an object to its colour. The question begging for a mathematical proof is what is the colour of that object in the ultraproduct of that sequence of models? On one hand $c$ must have a colour (since it has one in every model of the sequence), on the other hand  it cannot have a specific colour, since that would be true for one model in the sequence only (by non-principality of the ultrafilter and Łos Theorem, that cannot be the case). Here I first analyse the mathematical structure of the ultraproduct and then point to philosophical consequences of considering such an ultraproduct on possible worlds.

This ultraproduct has a similar structure to ${\prod }_{p\in \mathbb{P}}\overline{{\mathbb{F}}_p}/\mathcal{U}$, where the colour function is the characteristic of the field (see more here).

To see the latest details on the construction, visit Overleaf: Colouring Ultraproduct, else download the last published version: Colour_Ultraproducts.pdf.

Mirror Ultraproduct

What do a mathematician at work and a kid playing a videogame have in common? Perhaps the fact that both are creating some worlds inside the actual one. Consider the notion of a subworld as being a set of complete and consistent sentences listed inside another world. One might see the actual world as being a proper metaphysical world and the videogame as a linguistic construction, a mathematical model. Though, since we are considering possible worlds as being nothing formalised into mathematical models (see: Possible Worlds as Mathematical Models), the video game and the actual world are of the same sort. Furthermore since we are assuming modal realism (see Modal Realism) the video game is in no way less real than the actual world. We might then ask, is it possible to have a videogame just like the actual world? Better said, is it possible to have a possible world that contains itself? Directly defining a model that contains itself clearly ends in circularity issues hence, to give a positive answer to the question, ultraproducts may help. 

To see the latest details on the construction, visit Overleaf: Mirror Ultraproduct, else download the last published version: Mirror_Ultraproduct.pdf.

Co-Ultraproduct

I give a generalisation of this section in Algebra on Worlds.

Is there a way to define the opposite of a world? Clearly negating every proposition, would give rise to some internal contradictions (see Impossible Worlds), we could in fact not call a world possible where tautologies are negated. On the other hand, there certainly are some propositions that can be negated, I want therefore to construct an ultraproduct that negates all that can be negated by another ultraproduct, in a word: its opposite, the co-ultraproduct.

In a more general sense, the mathematical investigation will aim to find any sort of algebraic structure on generators of ultrafilters, so that, for instance, the negation of an ultrafilter will be the ultrafilter generated by the negation operation on the generator of the first ultrafilter. I will then investigate any sort of operations on generators of ultrafilters, so that those operations can be thereby defined on ultrafilters and hence on ultraproducts. Once interpreted a possible world as an ultraproduct, I will then be able to define any such sort of operations on possible worlds. Note that the relation between ultraproduct and possible worlds is here handled differently from the approach used above: instead of constructing an ultraproduct out of a sequence of possible worlds, I first consider a possible world and treat it as an ultraproduct generated by some (perhaps unknown) sequence of possible worlds.

Once further work shows that such algebraic structures on generators of ultrafilters exist, I will let this section become the heart of the mathematical thesis, if that were not the case, I will continue with the formerly presented aproaches.

To see the latest details on the construction, visit Overleaf: Co-Ultraproduct, else download the last published version: Co_Ultraproduct.pdf.