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Mintys.MatematikosIšsiaiškinimoBūdai istorija
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Pakeistos 146 eilutės iš
ZermeloFraenkel axioms of set theory
 Axiom of Extensionality. Two sets are the same set if they have the same elements.
 Axiom of Regularity. Every nonempty 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.
 WellOrdering Theorem. For any set X, there is a binary relation R which wellorders 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?
 Wellordering theorem. Each vertex is related by edges to the other vertices. Established by the qweight.
 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 nonempty set x contains a member y such that x and y are disjoint sets.
 Wellordering theorem. For any set X, there is a binary relation R which wellorders 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. WellOrdering 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
į:
Pakeistos 3741 eilutės iš
 Evolution. Tree of variations.
 Atlas. Adjacency graph.
 Handbook. Total order.
 Chronicle. Powerset lattice. WellOrdering Theorem.
 Catalog. Decomposition.
į:
 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. WellOrdering Theorem.
 Chronicle. Powerset lattice. Axiom of Power Set.
 Catalog. Decomposition. Axiom of Union.
Pakeistos 4246 eilutės iš
 Tour. Directed graph. Axiom of Regularity.
į:
 Tour. Directed graph. Axiom of Regularity.
Relate to multiplication
http://www.selflearners.net/uploads/multiplicationmodels.png
Pakeistos 3742 eilutės iš
 Evolution.
 Atlas.
 Handbook.
 Chronicle. Total order.
 Catalog.
 Tour. Directed graph.
į:
 Evolution. Tree of variations.
 Atlas. Adjacency graph.
 Handbook. Total order.
 Chronicle. Powerset lattice. WellOrdering Theorem.
 Catalog. Decomposition.
 Tour. Directed graph. Axiom of Regularity.
Pridėtos 3542 eilutės:
Reorganizings
 Evolution.
 Atlas.
 Handbook.
 Chronicle. Total order.
 Catalog.
 Tour. Directed graph.
Pakeistos 713 eilutės iš
 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
 Wellordering theorem. For any set X, there is a binary relation R which wellorders X.
į:
 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.
 WellOrdering Theorem. For any set X, there is a binary relation R which wellorders X.
Pakeista 17 eilutė iš:
Implicit math: Sets are simplexes.
į:
Implicit math: Sets are simplexes. Are simplexes defined by their subsimplexes as well?
Pakeistos 2134 eilutės iš
į:
 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 nonempty set x contains a member y such that x and y are disjoint sets.
 Wellordering theorem. For any set X, there is a binary relation R which wellorders 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.
Pakeistos 1112 eilutės iš
 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.
į:
 Axiom of power set. For any set x, there is a set y that contains every subset of x
 Wellordering theorem. For any set X, there is a binary relation R which wellorders 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.
 Wellordering theorem. Each vertex is related by edges to the other vertices. Established by the qweight.
 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.
Pridėtos 112 eilutės:
ZermeloFraenkel axioms of set theory
 Axiom of Extensionality. Two sets are the same set if they have the same elements.
 Axiom of Regularity. Every nonempty 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.
 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.

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