Equiconsistency

	Equiconsistency In mathematical logic, two theory (mathematical logic), theories are equiconsistent if the consistency of one theory implies the consistency of the other theory, and Vice-versa, 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. David Hilbert, Hilbert proposed a Hilbert's program, program at the beginning of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Second-order Arithmetic In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation of mathematics, foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book ''Grundlagen der Mathematik''. The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order logic, first-order counterpart Peano_axioms#Peano_arithmetic_as_first-order_theory, Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows Quantification (logic), quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite set, infinite) sets of natural numbers in well-known ways, and because second-order arithmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mathematical Logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Computable Set In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every natural number in a finite number of steps. A set is noncomputable (or undecidable) if it is not computable. Definition A subset S of the natural numbers is computable if there exists a total computable function f such that: :f(x)=1 if x\in S :f(x)=0 if x\notin S. In other words, the set S is computable if and only if the indicator function \mathbb_ is computable. Examples Every recursive language is a computable. Every finite or cofinite subset of the natural numbers is computable. The empty set is computable. The entire set of natural numbers is computable. *Every natural number is computable. The subset of prime numbers is computable. The set of Gödel numbers is computable. Non-examples The set of Turing machines that halt is not computable. The set of pairs of homeomorphic finite simplicial complexes ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Akihiro Kanamori is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinals, '' The Higher Infinite''. He has written several essays on the history of mathematics, especially set theory. Kanamori graduated from California Institute of Technology and earned a Ph.D. from the University of Cambridge ( King's College), and is a professor of mathematics at Boston University Boston University (BU) is a Private university, private research university in Boston, Massachusetts, United States. BU was founded in 1839 by a group of Boston Methodism, Methodists with its original campus in Newbury (town), Vermont, Newbur .... With Matthew Foreman, Kanamori is the editor of the ''Handbook of Set Theory'' (2010). Selected publications * A. Kanamori, M. MagidorThe evolution of large cardinal axioms in set theory in: ''Higher set theory'' (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Mathematics, 669, Spring ... [...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 ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Weakly Compact Cardinal In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function ''f'': � 2 → there is a set of cardinality κ that is homogeneous for ''f''. In this context, � 2 means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f'' if and only if either all of 'S''sup>2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. Equivalent formulations The following are equivalent for any uncountable cardinal κ: # κ is weakly compact. # for every λn → � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Mahlo Cardinal In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by . As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consistent). A cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ... \kappa is called strongly Mahlo if \kappa is strongly inaccessible and the set U = \ is stationary in \kappa. In other words, \kappa is strongly inaccessible, and for any unbounded set S\subseteq\kappa of cardinals, there is a strongly inaccessible cardinal \lambda < \kappa which is a limit of members of $S$ . A cardinal $\kappa$ is called weakly Mahlo if $\kappa$ is weakly inaccessible and the set of weakly inaccessible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Aronszajn Tree In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree of height ''κ'' in which all levels have size less than ''κ'' and all branches have height less than ''κ'' (so Aronszajn trees are the same as \aleph_1-Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by . A cardinal ''κ'' for which no ''κ''-Aronszajn trees exist is said to have the tree property (sometimes the condition that ''κ'' is regular and uncountable is included). Existence of κ-Aronszajn trees Kőnig's lemma states that \aleph_0-Aronszajn trees do not exist. The existence of Aronszajn trees (=\aleph_1-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Inaccessible Cardinal In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it has meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced by . Strongly inaccessible cardinals were introduced by and ; in the latter they were referred to along with \aleph_0 as ''Grenzzahlen'' ( English "limit numbers"). Every strongly inaccessible cardinal is a weakly inaccessible cardinal. The generalized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well. The two notions of an inaccessible cardinal \kappa describe a cardinality \kappa which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Kurepa Tree In set theory, a Kurepa tree is a tree (''T'', <) of height ω₁, each of whose levels is countable , and has at least ℵ₂ many branches. This concept was introduced by . The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms of ZFC: Solovay ... [...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]
	Axiom Of Determinacy In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of a certain type is determined; that is, one of the two players has a winning strategy. Steinhaus and Mycielski's motivation for AD was its interesting consequences, and suggested that AD could be true in the smallest natural model L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD followed from theorems proved earlier by Stefan Banach and Stanisław Mazur, and Morton Davis. Mycielski and Stanisław Świerczkowski contributed another one: AD implies that all sets of real numbers are Lebesgue measurable. Later Donald A. Martin and others proved more important consequences, especially in descriptive set theory. In 1988, John R. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]