|
Kenneth Kunen
Herbert Kenneth Kunen (August 2, 1943August 14, 2020) was a professor of mathematics at the University of Wisconsin–Madison who worked in set theory and its applications to various areas of mathematics, such as set-theoretic topology and measure theory. He also worked on non-associative algebraic systems, such as loops, and used computer software, such as the Otter theorem prover, to derive theorems in these areas. Personal life Kunen was born in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive w ... in 1943 and died in 2020. He lived in Madison, Wisconsin, with his wife Anne, with whom he had two sons, Isaac and Adam. Education Kunen completed his undergraduate degree at the California Institute of Technology and received his Ph.D. in 1968 from Stanford ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hearst Mining Building
The Hearst Memorial Mining Building at the University of California, Berkeley, is home to the university's materials science, Materials Science and Engineering Department, with research and teaching spaces for the subdisciplines of biomaterials; chemical and electrochemical materials; computational materials; electronic, magnetic, and optical materials; and structural materials. The Beaux-Arts architecture, Beaux-Arts-style Neoclassical architecture, Classical Revival building is listed in the National Register of Historic Places and is designated as part of California Historical Landmark #946. It was designed by John Galen Howard, with the assistance of architect and Berkeley alumna Julia Morgan and the Dean of the College of Mines at that time, Samuel B. Christy. It was the first building on that campus designed by Howard. Construction began in 1902 as part of the Phoebe Hearst Campus of the University of California, Berkeley, campus development plan. The building was dedicated to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Embedding
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] |
Singular Cardinal
Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singular: Act I'', a 2018 studio album by Sabrina Carpenter *'' Singular: Act II'', a 2019 studio album by Sabrina Carpenter Mathematics * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular matrix, a matrix that is not invertible * Singular measure, a measure or probability distribution whose support has zero Lebesgue (or other) measure * Singular cardinal, an infinite cardinal number that is not a regular cardinal * Singular point of a curve, in geometry See also * Singularity (other) * Singulair Montelukast, sold under the brand name Singulair among others, is a medication used in the maintenance treatment of asthma. It is generally less preferred for this use than inhaled corticosteroids. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin's Axiom
In 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, 𝔠, behave roughly like ℵ0. The intuition behind this can be understood by studying the proof of the Rasiowa–Sikorski lemma. It is a principle that is used to control certain forcing arguments. Statement For a cardinal number ''κ'', define the following statement: ;MA(''κ''): For any partial order ''P'' satisfying the countable chain condition (hereafter ccc) and any set ''D'' = ''i''∈''I'' of dense subsets of ''P'' such that '', D, '' ≤ ''κ'', there is a filter ''F'' on ''P'' such that ''F'' ∩ ''D''''i'' is non- empty for every ''D''''i'' ∈ ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence (mathematical logic), independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe (mathematics), universe V to a larger universe V[G] by introducing a new "generic" object G. Forcing was first used by Paul Cohen (mathematician), Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of mathematical logic such as recursion theory. Descriptive set theory uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in model theory, but it is common in model theory to define generic filter, genericity directly without mention of forcing. Intuition Forcing is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reinhardt Cardinal
In set theory, 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 Collection axioms for all ... [...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 ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let ''L'' = ⟨∈⟩ be the language of set theory. Let ''S'' be a particular set theory, for example the ZFC axioms and let ''T'' (possibly the same as ''S'') also be a theory in ''L.'' If ''M'' is a model for ''S,'' and ''N'' is an such that # ''N'' is a substructure of ''M,'' i.e. the interpretation ∈''N'' of ∈ in ''N'' is ∈''M'' ∩ ''N''2 # ''N'' is a model of ''T'' # the domain of ''N'' is a transitive class of ''M'' # ''N'' contains all ordinals in ''M'' then we say that ''N'' is an inner model of ''T'' (in ''M''). Usually ''T'' will equal (or subsume) ''S'', so that ''N'' is a model for ''S'' 'inside' the model ''M'' of ''S''. If only conditions 1 and 2 hold, ''N'' is called a standard model of ''T'' (in ''M''), a ''standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Compact Cardinal
In set theory, a strongly compact cardinal is a certain kind of large cardinal. An uncountable cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter. Strongly compact cardinals were originally defined in terms of infinitary logic, where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal κ is defined by requiring the number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it ... property of finitary logic. Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure (mathematics), measure on a cardinal ''κ'', or more generally on any set. For a cardinal ''κ'', it can be described as a subdivision of all of its subsets into large and small sets such that ''κ'' itself is large, ∅ and all singleton (mathematics), singletons (with ''α'' ∈ ''κ'') are small, set complement, complements of small sets are large and vice versa. The intersection of fewer than ''κ'' large sets is again large. It turns out that uncountable cardinals endowed with a two-valued measure are large cardinals whose existence cannot be proved from ZFC. The concept of a measurable cardinal was introduced by Stanisław Ulam in 1930. Definition Formally, a measurable cardinal is an uncountable cardinal number ''κ'' such that there exists a ''κ''-additive, non-trivial, 0-1-valued measure (mathematics), measure ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
