*Natural, rational, and real arithmetic in a finitary theory of finite sets*

Supervisor: John Mayberry

In the first part of this thesis, I describe a finitary theory of finite sets called Euclidean Arithmetic, which was introduced by Mayberry in his The Foundations of Mathematics in the Theory of Sets. In the second part, I develop the theory of simply infinite systems in this theory. Since Dedekind’s Isomorphism Theorem for Simply Infinite Systems does not hold in Euclidean Arithmetic, there is a vast and varied fauna of simply infinite systems with different and surprising properties. I describe many of these systems; I detail the properties they have and the relations that hold between them; and I give recipes by means of which new systems may be defined from old, and from which systems with desired properties may be defined. In the final part of this thesis, I describe a novel version of infinitesimal analysis in a intuitionistic extension of the basic theory of Euclidean Arithmetic. I prove analogues to many of the standard elementary theorems of real analysis and the calculus.