How can set theory be used as a foundation for arithmetic and real numbers?

Set theory serves as a foundational framework for mathematics, allowing for the rigorous construction of number systems from basic logical principles.1

Construction of Natural Numbers

In Zermelo-Fraenkel set theory, natural numbers are defined as sets. Starting with the empty set ($\varnothing$) as 0, each subsequent number is defined as the set containing all preceding numbers. For example:
* 0 = $\varnothing$
* 1 = {0} = {$\varnothing$}
* 2 = {0, 1} = {$\varnothing$, {$\varnothing$}}
* n = {0, 1, ..., n-1}

This approach, utilizing the successor function, allows for the derivation of Peano axioms and the definition of arithmetic operations like addition and multiplication.2

Real Numbers and Continuity

To extend this to real numbers, mathematicians use set-theoretic constructions to fill the "gaps" in the rational numbers. One primary method is the Dedekind cut, which partitions the set of rational numbers into two subsets. An irrational number is defined by a cut where the lower set has no largest member and the upper set has no least member, effectively defining the real number as a specific set of rationals. Through these methods, set theory provides a consistent, axiomatic basis for the entirety of arithmetic and analysis.314