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Zermelo-Fraenkel axioms of set theory

* Axiom of Extensionality. Two sets are the same set if they have the same elements.
* Axiom of Regularity. Every non-empty set x contains a member y such that x and y are disjoint sets. This implies, for example, that no set is an element of itself and that every set has an ordinal rank.
* Axiom Schema of Specification. The subset of a set z obeying a formula ϕ(x) with one free variable x always exists.
* Axiom of Pairing. If x and y are sets, then there exists a set which contains x and y as elements.
* Axiom of Union. The union over the elements of a set exists.
* Axiom Schema of Replacement. The image of a set under any definable function will also fall inside a set.
* Axiom of Power Set. For any set x, there is a set y that contains every subset of x.
* Well-Ordering Theorem. For any set X, there is a binary relation R which well-orders X.

Also:
* Axiom of infinity. Let S(w) abbreviate w ∪ {w}, where w is some set. Then there exists a set X such that the empty set ∅ is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X.

Implicit math: Sets are simplexes. Are simplexes defined by their subsimplexes as well?
* Well-ordering theorem. Each vertex is related by edges to the other vertices. Established by the q-weight.
* Axiom of power set. The power sets are the lattice paths in Pascal's triangle.
* Axiom of union. Simplexes combine to form larger simplexes.
* Axiom of pairing. Simplexes can be "collapsed" or "represented" by a vertex, the highest vertex. Two vertices are linked by an edge.
* Axiom of regularity.
* Axiom of extensionality. Simplexes are defined by their vertices. And the edges?

Eightfold way
* Axiom of Pairing. If x and y are sets, then there exists a set which contains x and y as elements.
* Axiom of Extensionality. Two sets are the same set if they have the same elements.
* Axiom of Union. The union over the elements of a set exists.
* Axiom of Power set. For any set x, there is a set y that contains every subset of x.

* Axiom of Regularity. Every non-empty set x contains a member y such that x and y are disjoint sets.
* Well-ordering theorem. For any set X, there is a binary relation R which well-orders X.
* Axiom Schema of Specification. The subset of a set z obeying a formula ϕ(x) with one free variable x always exists.
* Axiom Schema of Replacement. The image of a set under any definable function will also fall inside a set.

Reorganizings
* Evolution. Tree of variations. Axiom of Pairing (subbranches are part of a branch).
* Atlas. Adjacency graph. Axiom of Extensionality. Two levels of equality.
* Handbook. Total order. Well-Ordering Theorem.
* Chronicle. Powerset lattice. Axiom of Power Set.
* Catalog. Decomposition. Axiom of Union.
* Tour. Directed graph. Axiom of Regularity.

Relate to multiplication

http://www.selflearners.net/uploads/multiplicationmodels.png

* Evolution. Tree of variations.
* Atlas. Adjacency graph.
* Handbook. Total order.
* Chronicle. Powerset lattice. Well-Ordering Theorem.
* Catalog. Decomposition.

* Tour. Directed graph. Axiom of Regularity.

* Evolution.
* Atlas.
* Handbook.
* Chronicle. Total order.
* Catalog.
* Tour. Directed graph.

* Axiom schema of specification. The subset of a set z obeying a formula ϕ(x) with one free variable x always exists.
* Axiom of pairing. If x and y are sets, then there exists a set which contains x and y as elements.
* Axiom of union. The union over the elements of a set exists.
* Axiom schema of replacement. The image of a set under any definable function will also fall inside a set.
* Axiom of power set. For any set x, there is a set y that contains every subset of x
* Well-ordering theorem. For any set X, there is a binary relation R which well-orders X.

Implicit math: Sets are simplexes.

* Axiom of pairing.

* Axiom of power set. For any set x, there is a set y that contains every subset of x.
* Axiom of infinity. Let S(w) abbreviate w ∪ {w}, where w is some set. Then there exists a set X such that the empty set ∅ is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X.