Non-well-founded set theoryNon-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom.
Martin's axiomIn the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consistent with ZFC and the negation of the continuum hypothesis. Informally, it says that all cardinals less than the cardinality of the continuum, , behave roughly like . The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma.
Suslin's problemIn mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.
Reverse mathematicsReverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory.
Set-builder notationIn set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Defining sets by properties is also known as set comprehension, set abstraction or as defining a set's intension. Set (mathematics)#Roster notation A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples: is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
Paradoxes of set theoryThis article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers.
Extension by definitionsIn mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol for the set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant and the new axiom , meaning "for all x, x is not a member of ". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition.
Whitehead problemIn group theory, a branch of abstract algebra, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group? Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory. Assume that A is an abelian group such that every short exact sequence must split if B is also abelian. The Whitehead problem then asks: must A be free? This splitting requirement is equivalent to the condition Ext1(A, Z) = 0.
Uniformization (set theory)In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if is a subset of , where and are Polish spaces, then there is a subset of that is a partial function from to , and whose domain (the set of all such that exists) equals Such a function is called a uniformizing function for , or a uniformization of . To see the relationship with the axiom of choice, observe that can be thought of as associating, to each element of , a subset of .
Thoralf SkolemThoralf Albert Skolem (ˈtùːrɑɫf ˈskùːlɛm; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania (later renamed Oslo), passing the university entrance examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology and botany.
