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# Zero sharp

In the mathematical discipline of set theory, 0# (zero sharp, also 0#) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. [1]

## Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.

## Baire space (set theory)

In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology.

## Cardinal number

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.

## Cardinality

In mathematics, the cardinality of a set is a measure of the "number of elements of the set".

## Chang's conjecture

In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by, states that every model of type (ω2,ω1) for a countable language has an elementary submodel of type (ω1, ω).

## Cofinal (mathematics)

In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition: This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”.

## Cofinality

In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member.

## Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets.

## Donald A. Martin

Donald A. Martin (born December 24, 1940), also known as Tony Martin, is an American set theorist and philosopher of mathematics at UCLA, where he is a member of the faculty of mathematics and philosophy.

## Erdős cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by.

## Forcing (mathematics)

In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results.

## Gödel numbering

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number.

## Generic filter

In the mathematical field of set theory, a generic filter is a kind of object used in the theory of forcing, a technique used for many purposes, but especially to establish the independence of certain propositions from certain formal theories, such as ZFC.

## Hereditarily finite set

In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets.

## Indiscernibles

In mathematical logic, indiscernibles are objects which cannot be distinguished by any property or relation defined by a formula.

## Ineffable cardinal

In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by.

## Jack Silver

Jack Howard Silver (23 April 1942 – 22 December 2016) was a set theorist and logician at the University of California, Berkeley.

## Jensen's covering theorem

In set theory, Jensen's covering theorem states that if 0# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality.

## Large cardinal

In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.

## Leo Harrington

Leo Anthony Harrington (born May 17, 1946) is a professor of mathematics at the University of California, Berkeley who works in recursion theory, model theory, and set theory.

## Lightface analytic game

In descriptive set theory, a lightface analytic game is a game whose payoff set A is a \Sigma^1_1 subset of Baire space; that is, there is a tree T on \omega\times\omega which is a computable subset of (\omega\times\omega)^.

## Measurable cardinal

In mathematics, a measurable cardinal is a certain kind of large cardinal number.

## Ramsey cardinal

In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey.

## Regular cardinal

In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality.

## Robert M. Solovay

Robert Martin Solovay (born December 15, 1938) is an American mathematician specializing in set theory.

## Ronald Jensen

Ronald Björn Jensen (born April 1, 1936) is an American mathematician active in Europe, primarily known for his work in mathematical logic and set theory.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Tarski's undefinability theorem

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics.

## Transactions of the American Mathematical Society

The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.

## Turing degree

In computer science and mathematical logic the Turing degree (named after Alan Turing) or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set.

## Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

## Well-formed formula

In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.

## Zermelo–Fraenkel set theory

In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

## Zero dagger

In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.

## References

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