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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In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s.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] |
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] |
Robert M
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be use ... [...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] |
Large Cardinal Property
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] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...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 para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel 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 cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on P. If X is an arbitrary set, its power set \wp(X), ordered by set inclusion, is always a Boolean algebra and hence a poset, and ultrafilters on \wp(X) are usually called X.If X happens to be partially ordered, too, particular care is needed to understand from the context whether an (ultra)filter on \wp(X) or an (ultra)filter just on X is meant; both kinds of (ultra)filters are quite different. Some authors use "(ultra)filter" ''of'' a partial ordered set" vs. "''on'' an arbitrary set"; i.e. they write "(ultra)filter on X" to abbreviate "(ultra)filter of \wp(X)". An ultrafilter on a set X may be considered as a finitely additive measure on X. In this view, every subset of X is either considered "almost everything" (has measure 1) or "almos ... [...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 on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivision of all of its subsets into large and small sets such that itself is large, and all singletons are small, complements of small sets are large and vice versa. The intersection of fewer than 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 Stanislaw 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 on the power set of ''κ''. (Here the term ''κ-additive'' means that, for any sequence ''A''''α'', α<λ of cardinality '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indiscernibles
In mathematical logic, indiscernibles are objects that cannot be distinguished by any property or relation defined by a formula. Usually only first-order formulas are considered. Examples If ''a'', ''b'', and ''c'' are distinct and is a set of indiscernibles, then, for example, for each binary formula \beta , we must have : \beta (a, b) \land \beta (b, a) \land \beta (a, c) \land \beta (c, a) \land \beta (b, c) \land \beta (c, b) \lor \lnot \beta (a, b) \land \lnot \beta (b, a) \land \lnot \beta(a, c) \land \lnot \beta (c, a) \land \lnot \beta (b, c) \land \lnot \beta (c, b) \,. Historically, the identity of indiscernibles was one of the laws of thought of Gottfried Leibniz. Generalizations In some contexts one considers the more general notion of order-indiscernibles, and the term sequence of indiscernibles often refers implicitly to this weaker notion. In our example of binary formulas, to say that the triple (''a'', ''b'', ''c'') of distinct elements is a sequence of ind ... [...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] |
