# Index [[Axiom of choice]] [Axiom of extension](Axiom%20of%20extension) [Axiom of pairing](Axiom%20of%20pairing) [Axiom of Powers](Axiom%20of%20Powers.md) [Axiom of separation](Axiom%20of%20separation) [[Axiom of sums]] [[Axioms of set theory]] [Axiom of union](Axiom%20of%20union) [[Axiomatic definition of a set]] [[Binary operations]] [Cantor's Paradox](Cantor's%20Paradox) [[Cardinality]] [Cartesian product](Cartesian%20product.md) [[closed set]] [[Classes]] [[Closure under an operation]] [[Complement of a set]] [[completeness]] [[Conservative extensions of ZFC]] [[countably infinite set]] [[Domain]] [[Disjoint union]] [[Equivalence class]] [[Equivalence relations]] [[Finite sets]] [[Greatest element]] [[Image]] [[Infinite set]] [[infinum]] [[Intersection of sets]] [[Lower bound]] [[Least element]] [[Maps]] [[Maximal]] [[Minimal]] [[Multisets]] [[von Neumann-Bernays-Gödel set theory]] [[one-to-one]] [[onto]] [[open set]] [Ordered pairs](Ordered%20pairs.md) [[Ordering]] [[Paradoxes in naive set theory]] [Partial ordering](Partial%20ordering.md) [[Power set]] [[Quotient sets]] [Relations on sets](Relations%20on%20sets.md) [[Rule of assignment]] [Russell's Paradox](Russell's%20Paradox.md) [[Sets]] [[Sets of sets]] [[Set difference]] [[smooth map]] [smoothness](smoothness) [[Spaces]] [[Subsets]] [[Supremum]] [[The null set]] [[Total ordering]] [[uncountably infinite set]] [Union of sets](Union%20of%20sets.md) [[Upper bound]] # Proofs and examples [[Proof that all equivalence classes are also partitions of sets and vice versa]] --- # Basic concepts Here we cover a topic that is at the foundation of other areas of mathematics. One approach (and perhaps the only approach) to establishing a foundation for all of mathematics is through the study of [sets](Sets%20(index).md#Sets)^[%%There needs to be a note here about reformulations of math as categories.%%]. The conclusions from set theory are implied to be true also in mathematics where it isn't invoked since set theoy is required to prove a lot of what may be treated as underlying assumptions in many conexts. This is merely a convenience rather than a reality. Thus, at some point, one can't ignore set theory when studying any of the other branches of mathematics. In the Quantum Well, these branches are namely [algebra,](Algebra%20(index).md) [geometry](Geometry%20(index).md) [analysis,](Analysis%20(index).md) [numbers,](Numbers%20(index).md) [statistics, and probability.](Statistics%20and%20Probability%20(index).md) They all assume some (even if rudimentary) familiarity of set theory. However, even when this is the case, it is also usually possible (and preferable) to omit [non-naive set theories.](Sets%20(index).md#Naive%20and%20axiomatic%20set%20theories) ^e0a7ce ## Naive and axiomatic set theories There are two approaches to set theory, a _naive_ set theory and _[axiomatic](Axioms%20of%20set%20theory.md)_ set theory. The former relies on natural language and is found to lead to [paradoxes.](Paradoxes%20in%20naive%20set%20theory.md) Despite this, naive set theory is sufficient in many applications since both approaches lead to the same conclusions. %%Do they always lead to the same conclusio?%% ## Sets In our (multi)[uni]verse we have collections of things we refer to as _[[Sets]]_. This is a _[[Primitive notion]]_, meaning that it does not follow from any prior definition or axiom. An additional primitive notion is the idea that an object may be a _member_ or _an element_ of a set. What these primitive notions mean mathematically are as follows:   --- # Bibliography [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) [Aluffi, P., _Algebra: Chapter 0_, American Mathematical Society, 2009](Aluffi,%20P.,%20Algebra%20Chapter%200,%20American%20Mathematical%20Society,%202009.md) [Axler S., Gerhing F.W., Ribet K.A. _Linear Algebra Done Right_, Springer, 2nd edition, 1997](Axler%20S.,%20Gerhing%20F.W.,%20Ribet%20K.A.%20Linear%20Algebra%20Done%20Right,%20Springer,%202nd%20edition,%201997.md) [Barrington, D. M., _A Mathematical Foundation for Computer Science_, Kendall Hunt Publishing Company, Preliminary edition, 2019.](Barrington,%20D.%20M.,%20A%20Mathematical%20Foundation%20for%20Computer%20Science,%20Kendall%20Hunt%20Publishing%20Company,%20Preliminary%20edition,%202019..md) [Jech, T., _Set Theory_, Springer, 3rd edition, 2002.](Jech,%20T.,%20Set%20Theory,%20Springer,%203rd%20edition,%202002..md) [Jänich K., _Topology_. Translated by Silvio Levy, Springer-Verlag 1984](Jänich%20K.,%20Topology.%20Translated%20by%20Silvio%20Levy,%20Springer-Verlag,%201984.md) Halmos P. R. _Naive Set Theory_ %20-%20libgen.lc.pdf) Suppes P. _Axiomatic Set Theory_ %20-%20libgen.lc.pdf) [Marcolli M., The notion of space in mathematics, general audience lecture, sponsored by Revolution Books, Berkeley, 2009.](Marcolli%20M.,%20The%20notion%20of%20space%20in%20mathematics,%20general%20audience%20lecture,%20sponsored%20by%20Revolution%20Books,%20Berkeley,%202009..md) [Munkres, J. R., _Topology_, Prentice Hall, 2nd Edition, 2000.](Munkres,%20J.%20R.,%20Topology,%20Prentice%20Hall,%202nd%20Edition,%202000..md) [Weisstein, Eric W. "Space."" From MathWorld--A Wolfram Web Resource.](Weisstein,%20Eric%20W.%20Space.%20From%20MathWorld--A%20Wolfram%20Web%20Resource..md) #MathematicalFoundations #MathematicalFoundations/SetTheory #Bibliography #index