Set theory
Branch · 1870s-1930s · Mathematics, stage 2: Map the branches
Cantor started set theory in the 1870s while working on a problem in analysis and found that infinite sets can be compared by size, with the real numbers strictly larger than the whole numbers. The early informal version produced contradictions, so the subject was rebuilt on explicit axioms in the first decades of the twentieth century, giving the system usually called ZFC. It then became the standard foundation, and Gödel and Cohen showed that some natural questions about sets, the continuum hypothesis above all, cannot be settled by those axioms at all.
What it claims
- Every mathematical object can be constructed as a set, so one theory can serve as a foundation for the rest.
- Infinite collections come in different sizes, and there is no largest one.
- The standard axioms leave some plainly stated questions about sets undecided.
Key ideas
People
Sources
- Set theory Wikipedia
- Set Theory Stanford Encyclopedia of Philosophy
- Set theory MacTutor History of Mathematics
