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Inner Model Theory
In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe ''V'', or sometimes of a generic extension of ''V''. Inner model theory studies the relationships of these models to determinacy, large cardinals, and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model theory. Examples *The class of all sets is an inner model containing all other inner models. *The first non-trivial example of an inner model was the constructible universe ''L'' developed by Kurt Gödel. Every model ''M'' of ZF has an inner model ''L''M satisfying the axiom of constructibility, and this will be the smallest inner model of ''M'' containing all the ordinals of ''M''. Regardless of the properties of the original model, ''L''''M'' will satisfy the generalized continuum hypothesis and combinatorial axioms such as the ... [...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] |
Generalized Continuum Hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers. History Cantor believed the continuum hypothesis to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core Model
In set theory, the core model is a definable inner model of the von Neumann universe, universe of all Set (mathematics), sets. Even though set theorists refer to "the core model", it is not a uniquely identified mathematical object. Rather, it is a class of inner models that under the right set-theoretic assumptions have very special properties, most notably covering lemma, covering properties. Intuitively, the core model is "the largest canonical inner model there is" (Ernest Schimmerling and John R. Steel) and is typically associated with a large cardinal notion. If Φ is a large cardinal notion, then the phrase "core model below Φ" refers to the definable inner model that exhibits the special properties under the assumption that there does ''not'' exist a cardinal satisfying Φ. The core model program seeks to analyze large cardinal axioms by determining the core models below them. History The first core model was Kurt Gödel's constructible universe L. Ronald Jensen proved the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...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] |
Consistency Strength
In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and vice versa. In this case, they are, roughly speaking, "as consistent as each other". In general, it is not possible to prove the absolute consistency of a theory ''T''. Instead we usually take a theory ''S'', believed to be consistent, and try to prove the weaker statement that if ''S'' is consistent then ''T'' must also be consistent—if we can do this we say that ''T'' is ''consistent relative to S''. If ''S'' is also consistent relative to ''T'' then we say that ''S'' and ''T'' are equiconsistent. Consistency In mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enough to model different mathematical objects, it is natural to wonder about their own consistency. Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematical methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Cardinal Axiom
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 philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistent
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 ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The definition is a bit awkward, because there might be ''no'' set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0† does not exist" is consistent. ZFC + "0† exists" is not known to be inconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows: :0† exists if and only if there exists a non-trivial elementary embedding ''j'' : ''L ' → ''L ' for the relativized Gödel constructible universe ''L ', where ''U'' is an ultrafilter witnessing that some cardinal κ is measurable. If 0† exists, then a careful analysis of the embeddings of ''L ' into itself reve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L(R)
In set theory, L(R) (pronounced L of R) is the smallest transitive inner model of ZF containing all the ordinals and all the reals. Construction It can be constructed in a manner analogous to the construction of L (that is, Gödel's constructible universe), by adding in all the reals at the start, and then iterating the definable powerset operation through all the ordinals. Assumptions In general, the study of L(R) assumes a wide array of large cardinal axioms, since without these axioms one cannot show even that L(R) is distinct from L. But given that sufficient large cardinals exist, L(R) does not satisfy the axiom of choice, but rather the axiom of determinacy. However, L(R) will still satisfy the axiom of dependent choice, given only that the von Neumann universe, ''V'', also satisfies that axiom. Results Given the assumptions above, some additional results of the theory are: * Every projective set of reals – and therefore every analytic set and every Borel set of reals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solovay's Theorem
In the mathematical field of set theory, the Solovay model is a model constructed by in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal. In this way Solovay showed that the axiom of choice is essential to the proof of the existence of a non-measurable set, at least granted that the existence of an inaccessible cardinal is consistent with ZFC, the axioms of Zermelo–Fraenkel set theory including the axiom of choice. Statement ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension ''V'' 'G''such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property. Construction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Definable
In mathematical set theory, a set ''S'' is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first-order formula. Ordinal definable sets were introduced by . A drawback to this informal definition is that it requires quantification over all first-order formulas, which cannot be formalized in the language of set theory. However there is a different way of stating the definition that can be so formalized. In this approach, a set ''S'' is formally defined to be ordinal definable if there is some collection of ordinals ''α''1, ..., ''α''''n'' such that S \isin V_ and S can be defined as an element of V_ by a first-order formula φ taking α2, ..., α''n'' as parameters. Here V_ denotes the set indexed by the ordinal ''α''1 in the von Neumann hierarchy. In other words, ''S'' is the unique object such that φ(''S'', α2...α''n'') holds with its quantifiers ranging over V_. The class of all ordinal definable sets is denoted O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
