# Counting and Pairing Infinity

von Leunbach, Rosa Nan & Julie Lundbak Kofod

Studenteropgave: Semesterprojekt

## Abstrakt

This study investigates the concept of infinity. The aim is to unravel the mathematical origin of Cantor's transfinite cardinals, focusing on the underlying concept of pairing. A brief account of potential infinity in the Antiquity is given, to contrast nineteenth century mathematics and its plurality of actual infinities. The fundamental concepts of set theory are presented in the context of Hume's principle, proving counting to be a special case of pairing. Bijective mappings between infinite sets elucidates the somewhat puzzling concept of countability and the following leap to transfinite cardinals. A thorough and critical investigation of Cantor's two diagonal proofs of respectively the real numbers and power sets, reveals an ingenious method of 'flipping' which exemplifies how pairing acts beyond countability. Even though Cantor regarded his cardinals actual compared to the limits of contemporary calculus, it is argued that his transfinite set theory is characterized by the potentiality perceived in Antiquity. The continuum hypothesis is the open end in Cantor's theory and it relates the study to modern axiomatic set theory. It is concluded that diagonalization, the method of making a flip-function dependant on the bijection to be contradicted, works in virtue of the validity of interminably changing a number, and the sequence in which it is done. Diagonalization is the key to mathematically extending the finite, and it renews the discussion of potential and actual infinities.

Uddannelser Filosofi og Videnskabsteori, (Bachelor/kandidatuddannelse) Bachelor el. kandidat Engelsk 15 jan. 2015 Patrick Rowan Blackburn

• Infinity
• Zeno
• Cantor
• set-theory