Statements true in L - AbsoluteAstronomy.com

Here is a list of propositions that hold in the constructible universe

Constructible universe

In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...

(denoted L):

The generalized continuum hypothesis and as a consequence
- The axiom of choice
Diamondsuit
Diamondsuit
In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by that holds in the constructible universe and that implies the continuum hypothesis...
- Clubsuit
  Clubsuit
  In mathematics, and particularly in axiomatic set theory, ♣S is a family of combinatorial principles that are weaker version of the corresponding ◊S; it was introduced in 1975 by A...
Global square
The existence of morasses
Morass (set theory)
In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations...
The negation of the Souslin conjecture
The non-existence of 0^#
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...

and as a consequence
- The non existence of all large cardinals which imply the existence of a measurable cardinal
  Measurable cardinal
  - Measurable :Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ...
The truth of Whitehead's conjecture
Whitehead problem
In group theory, a branch of abstract algebra, the Whitehead problem is the following question:Shelah proved that Whitehead's problem is undecidable within standard ZFC set theory.-Refinement:...

that every abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

A with Ext
Ext functor
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...

¹(A, Z) = 0 is a free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

.
The existence of a definable well-order
Well-order
In mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...

of all sets (the formula for which can be given explicitly). In particular, L satisfies V=HOD
Ordinal definable set
In mathematical set theory, a set S is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first order formula. Ordinal definable sets were introduced by ....

.

Accepting the axiom of constructibility

Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...

(which asserts that every set is constructible

Constructible set

In mathematics, constructible set may refer to either:* a notion in Gödel's constructible universe.* a union of locally closed set in a topological space. See constructible set ....

) these propositions also hold in the von Neumann universe

Von Neumann universe

In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...

, resolving many propositions in set theory and some interesting questions in analysis.

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