Sets & operations
Define membership, subsets, unions, intersections, products and power sets as the basic operations on collections.
Side 210Side Studies / Research
Subject
Set Theory
Purpose
Collections, membership, functions, cardinality, ordinals and axiomatic foundations studied as a language for modern mathematics.
Structure
05 movesFormal systemV0
Definitions → structures → relations → proof → application
Build mathematical objects from membership and axioms.
Set theory supplies both a working language for collections and a foundational theory whose paradoxes force precision about what sets are allowed to exist.
Functions & relations
Represent mappings, equivalence and order through sets of ordered pairs and structural conditions.
Cardinality
Compare sizes through bijections, including the distinction between finite, countably infinite and uncountable sets.
Ordinals & transfinite construction
Use well-ordering and transfinite recursion to extend counting and induction beyond finite stages.
Axioms & independence
Study why axioms such as separation, replacement and choice are needed, and how some statements cannot be settled from standard axioms alone.
Keep the boundaries visible.
membership ≠ subset
cardinality ≠ measure
paradox ≠ inconsistency of every set theory
Questions that organize the Side.
Why is unrestricted set formation inconsistent?
How can two infinite sets have the same cardinality when one seems contained in the other?
What does it mean for a statement to be independent of an axiom system?
What should carry weight here?
Proofs should name the axioms or set-theoretic facts being used; diagrams and examples illustrate but do not establish claims about arbitrary sets.