Information: - In mathematical logic, suppose "T" is a theory in the language - Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. - In mathematics, in set theory, the constructible universe (or Godel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets. It was introduced by Kurt Godel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this, he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result. - Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. These areas share basic results on logic, particularly first-order logic, and definability. In computer science (particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those. - In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the Cartesian product of a collection of non-empty sets is non-empty". It states that for every indexed family formula_1 of nonempty sets there exists an indexed family formula_2 of elements such that formula_3 for every formula_4. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. - Kurt Friedrich Godel (April 28, 1906 January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Godel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, A. N. Whitehead, and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor. - In set theory , a branch of mathematics , the condensation lemma is a result about sets in the constructible universe . It states that if X is a transitive set and is an elementary submodel of some level of the constructible hierarchy L , that is , ( X , \ in ) \ prec ( L -- \ alpha , \ in ) , then in fact there is some ordinal \ beta \ leq \ alpha such that X = L -- \ beta . More can be said : If X is not transitive , then its transitive collapse is equal to some L -- \ beta , and the hypothesis of elementarity can be weakened to elementarity only for formulas which are \ Sigma_1 in the Levy hierarchy . Also , the assumption that X be transitive automatically holds when \ alpha = \ omega_1 . The lemma was formulated and proved by Kurt Godel in his proof that the axiom of constructibility implies GCH . - A mathematical object is an abstract object arising in mathematics. The concept is studied in philosophy of mathematics. - An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek "axima" 'that which is thought worthy or fit' or 'that which commends itself as evident.' - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Given the information, choose the subject and object entities that have the relation of 'instance of'.
The answer to this question is:
condensation lemma , theorem