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Shelah Cardinal
In axiomatic set theory, Shelah cardinals are a kind of large cardinals. A cardinal \kappa is called Shelah iff for every f:\kappa\rightarrow\kappa, there exists a transitive class In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Sim ... N and an elementary embedding j:V\rightarrow N with critical point \kappa; and V_\subset N. A Shelah cardinal has a normal ultrafilter containing the set of weakly hyper-Woodin cardinals below it. References * Ernest Schimmerling, ''Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model'', Proceedings of the American Mathematical Society 130/11, pp. 3385-3391, 2002online {{settheory-stub Large cardinals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiomatic 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 system fo ... [...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] |
Cardinal Number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The '' transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph (aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Class
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. Similarly, a class M is transitive if every element of M is a subset of M. Examples Using the definition of ordinal numbers suggested by John von Neumann, ordinal numbers are defined as hereditarily transitive sets: an ordinal number is a transitive set whose members are also transitive (and thus ordinals). The class of all ordinals is a transitive class. Any of the stages V_\alpha and L_\alpha leading to the construction of the von Neumann universe V and Gödel's constructible universe L are transitive sets. The universes V and L themselves are transitive classes. This is a complete list of all finite transitive sets with up to 20 brackets: * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, * \, ... [...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 pa ... [...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 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrafilter (set Theory)
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (such filters are called ) and that is also "maximal" in that there does not exist any other proper filter on X that contains it as a proper subset. Said differently, a proper filter U is called an ultrafilter if there exists proper filter that contains it as a subset, that proper filter (necessarily) being U itself. More formally, an ultrafilter U on X is a proper filter that is also a maximal filter on X with respect to set inclusion, meaning that there does not exist any proper filter on X that contains U as a proper subset. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weakly Hyper-Woodin Cardinal
In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions :f : \lambda \to \lambda there exists a cardinal \kappa < \lambda with : and an elementary embedding : from the Von Neumann universe into a transitive with critical point and : An equivalent definition is this: is Woodin |
