See: Math, 20110402 Math Deep Structure, 20160620 Discovery in Mathematics

**Discovery in Mathematics: A System of Deep Structure**

- Collect math discovery examples.

How does the three-cycle extend our existing mathematical language?

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

Notes

Total order is the same as a labeled simplex.

Extension: 3! + (4 + 4 + 4 + 6) = 4!

3! = 2! + 4 (representations: 2 for edge and 4 for vertex)

We may assign the weight q^(k-1) to the kth vertex and the weights 1/q to each new edge. These weights give each vertex a unique label. The weight of each k-simplex is then the products of the weights of their vertices and edges. The Gaussian binomial coefficients count these weighted k-simplexes. Without the weights the vertices are distinct but there is no way to distinguish them. The symmetry group is the Symmetric group which relabels the vertices.

Matematikos išsiaiškinimo būdai

Parsiųstas iš http://www.ms.lt/sodas/Book/MathDiscovery

Puslapis paskutinį kartą pakeistas 2018 lapkričio 11 d., 16:39