Spontansets

Spontansets, also known as Spontaneous Sets, are a type of set in mathematics that are defined by their ability to be constructed without the use of the axiom of choice or any other non-constructive principles. They were introduced by Paul Cohen in his work on the independence of the continuum hypothesis. Spontaneous sets are typically constructed using a method known as forcing, which allows for the addition of new sets to a given model of set theory while preserving the truth of certain statements.

One of the key properties of spontaneous sets is that they are often used to construct models of set theory that satisfy certain desirable properties, such as the failure of the axiom of choice or the existence of non-measurable cardinals. For example, in the model obtained by forcing with a spontaneous set, the axiom of choice may fail, and the continuum hypothesis may be independent of the axioms of Zermelo-Fraenkel set theory.

Spontaneous sets have also been used in the study of large cardinals, which are infinite cardinal numbers that satisfy certain strong properties. In particular, spontaneous sets have been used to construct models of set theory that contain large cardinals, such as measurable cardinals and weakly compact cardinals.

In summary, spontaneous sets are a type of set in mathematics that are constructed using forcing and have the property of being spontaneous, meaning that they can be constructed without the use of non-constructive principles. They have been used in the study of the independence of the continuum hypothesis, the failure of the axiom of choice, and the existence of large cardinals.