natural numbers as sets
The natural numbers can be explicitly constructed as sets. Zero is defined to be the empty set, and each n > 0 is defined to be the set of natural numbers less than n. This is an example of how set theory serves as a powerful foundation for much of mathematics.
This concept has the prerequisites:
- set operations
- equivalence relations (One needs to define equality.)
- order relations (One needs to define the comparison operators.)
- Peano axioms (One must show that the construction satisfies the Peano axioms.)
- Construct the natural numbers in terms of sets.
- Show that the constructed number system satisfies the Peano postulates.
- Define equality and comparison operators, and show that these are an equivalence relation and order relations, respectively.
- Define the addition and multiplication operators.
- Show that these operators are commutative and associative.
Core resources (read/watch one of the following)
→ Elements of Set Theory
An introductory textbook on axiomatic set theory.
Location: Chapter 4, "Natural numbers," pages 66-88
Supplemental resources (the following are optional, but you may find them useful)
→ Notes on Set Theory (2013)
Lecture notes for a course on axiomatic set theory.
→ An Introduction to Set Theory (2008)
Lecture notes on axiomatic set theory.
- Zermelo-Frankl axioms
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