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Kunen's Inconsistency Theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: *There is no non-trivial elementary embedding of the universe ''V'' into itself. In other words, there is no Reinhardt cardinal. *If ''j'' is an elementary embedding of the universe ''V'' into an inner model ''M'', and λ is the smallest fixed point of ''j'' above the critical point κ of ''j'', then ''M'' does not contain the set ''j'' "λ (the image of ''j'' restricted to λ). *There is no ω-huge cardinal. *There is no non-trivial elementary embedding of ''V''λ+2 into itself. It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though showed that there is no definable elementary embedding from ''V'' into ''V''. That is there is no formula ''J'' in the language of set theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of '' naive set theory''. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is commonly employed as a foundational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more". There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct phil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term ''satisfiable'' is used instead. The syntactic definition states a theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when \varphi, \lnot \varphi \in \langle A \rangle for no formula \varphi. If there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets, there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. In many cases, a set arising from choosing elements arbitrarily can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available – some distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Equivalence
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one often needs a stronger condition. In this case ''N'' is called an elementary substructure of ''M'' if every first-order ''σ''-formula ''φ''(''a''1, …, ''a''''n'') with parameters ''a''1, …, ''a''''n'' from ''N'' is true in ''N'' if and only if it is true in ''M''. If ''N'' is an elementary substructure of ''M'', then ''M'' is called an elementary extension of ''N''. An embedding ''h'': ''N'' → ''M'' is called an elementary embedding of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''. A substructure ''N'' of ''M'' is elementary if and only if it passes the Tarski–Vaught test: every first-order formula ''φ''(''x'', ''b''1, …, ''b''''n'') with p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reinhardt Cardinal
In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the Axiom of Choice). They were suggested by American mathematician William Nelson Reinhardt (1939–1998). Definition A Reinhardt cardinal is the critical point of a non-trivial elementary embedding j:V\to V of '' V'' into itself. This definition refers explicitly to the proper class j. In standard ZF, classes are of the form \ for some set a and formula \phi. But it was shown in that no such class is an elementary embedding j:V\to V. So Reinhardt cardinals are inconsistent with this notion of class. There are other formulations of Reinhardt cardinals which are not known to be inconsistent. One is to add a new function symbol j to the language of ZF, together with axioms stating that j is an elementary embedding of V, and Separation and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Point (set Theory)
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself. p. 323 Suppose that j: N \to M is an elementary embedding where N and M are transitive classes and j is definable in N by a formula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(\omega) = \omega. If j(\alpha) = \alpha for all \alpha \kappa, then \kappa is said to be the critical point of j. If N is '' V'', then \kappa (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number ''κ'' such that there exists a \kappa-complete, non-principal ultrafilter over \kappa. Specifically, one may take the filter to be \. Generally, there will be many other <''κ''-complete, non-principal ultrafilters over . However, might be different from the |
ω-huge Cardinal
In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and :^M \subset M.\! Here, ''αM'' is the class of all sequences of length α whose elements are in M. Huge cardinals were introduced by . Variants In what follows, j''n'' refers to the ''n''-th iterate of the elementary embedding j, that is, j composed with itself ''n'' times, for a finite ordinal ''n''. Also, ''<αM'' is the class of all sequences of length less than α whose elements are in M. Notice that for the "super" versions, γ should be less than j(κ), not . κ is almost n-huge if and only if there is ''j'' : ''V'' → ''M'' with critical point κ and :^M \subset M.\! κ is super almost n-huge if and only if for every ordinal γ there is ''j'' : ''V'' → ''M'' with critical point κ, γ<j(κ), and :^M \subset M.\! κ is n-huge if and only if there is ''j'' : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse–Kelley Set Theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG). While von Neumann–Bernays–Gödel set theory restricts the bound variables in the schematic formula appearing in the axiom schema of Class Comprehension to range over sets alone, Morse–Kelley set theory allows these bound variables to range over proper classes as well as sets, as first suggested by Quine in 1940 for his system ML. Morse–Kelley set theory is named after mathematicians John L. Kelley and Anthony Morse and was first set out by and later in an appendix to Kelley's textbook ''General Topology'' (1955), a graduate level introduction to topology. Kelley said the system in his book was a variant of the systems due to Thoralf Skolem and Morse. Morse's ow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It is often encoded as a subset of the integers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number. Its existence is unprovable in ZFC, the standard form of axiomatic set theory, but follows from a suitable large cardinal axiom. It was first introduced as a set of formulae in Silver's 1966 thesis, later published as , where it was denoted by Σ, and rediscovered by , who considered it as a subset of the natural numbers and introduced the notation O# (with a capital letter O; this later changed to the numeral '0'). Roughly speaking, if 0# exists then the universe ''V'' of sets is much larger than the universe ''L'' of constructible sets, while if it does not exist then the universe of all sets is closely approximated by the constructible sets. Definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), 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. What is can be thought of as being built in "stages" resembling the const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < λ is one of the elements of the set Vλ of the .) *Axiom I3: There is a nontrivial of Vλ into itself. *Axiom I2: There is a nontrivial elementary embedding of V into a transitive class M that includes Vλ where λ is the first fixed point above the critical point. *Axiom I1: There is a nontrivial elementary embedd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
