Equiconsistency of Peano arithmetic with Zermelo-Fraenkel set theory without the axiom of infinity

The purpose of this thesis is to prove the equiconsistenchy of Peano Arithmetic with Zermelo-Fraenkel set theory without the axiom of infinity using the Ackermann encoding. The first chapter is dedicated to an informal introduction to the agruments, while the second chapter consists of the formal introduction of either Zarmelo Fraenkel set theory and Peano Arithmetic. In chapter 2, a modified version of both theories is also defined to simplify subsequent demonstrations. Chapters 3 and 4 consist of the proof of the equiconsiststency of ZF-Inf and PA, obtained using Ackermann encoding. The last chapter consist of a comment about the role of axiom of choice and the final conslusion.

The purpose of this thesis is to prove the equiconsistenchy of Peano Arithmetic with Zermelo-Fraenkel set theory without the axiom of infinity using the Ackermann encoding. The first chapter is dedicated to an informal introduction to the agruments, while the second chapter consists of the formal introduction of either Zarmelo Fraenkel set theory and Peano Arithmetic. In chapter 2, a modified version of both theories is also defined to simplify subsequent demonstrations. Chapters 3 and 4 consist of the proof of the equiconsiststency of ZF-Inf and PA, obtained using Ackermann encoding. The last chapter consist of a comment about the role of axiom of choice and the final conslusions.