|
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] |
Large Cardinals
In the mathematical field of set theory, a large cardinal property is a certain kind of property of Transfinite number, 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 am ... [...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] |
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 philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Recursive Arithmetic
Primitive recursive arithmetic (PRA) is a quantifier-free formalization of the natural numbers. It was first proposed by Norwegian mathematician , reprinted in translation in as a formalization of his finitist conception of the foundations of arithmetic, and it is widely agreed that all reasoning of PRA is finitist. Many also believe that all of finitism is captured by PRA, but others believe finitism can be extended to forms of recursion beyond primitive recursion, up to ε0, which is the proof-theoretic ordinal of Peano arithmetic. PRA's proof theoretic ordinal is ωω, where ω is the smallest transfinite ordinal. PRA is sometimes called Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including the operations of addition, multiplication, and exponentiation. PRA cannot explicitly quantify over the domain of natural numbers. PRA is often taken as the basic metamathematical form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Higher Infinite
''The Higher Infinite: Large Cardinals in Set Theory from their Beginnings'' is a monograph in set theory by Akihiro Kanamori, concerning the history and theory of large cardinals, infinite sets characterized by such strong properties that their existence cannot be proven in Zermelo–Fraenkel set theory (ZFC). This book was published in 1994 by Springer-Verlag in their series Perspectives in Mathematical Logic, with a second edition in 2003 in their Springer Monographs in Mathematics series, and a paperback reprint of the second edition in 2009 (). Topics Not counting introductory material and appendices, there are six chapters in ''The Higher Infinite'', arranged roughly in chronological order by the history of the development of the subject. The author writes that he chose this ordering "both because it provides the most coherent exposition of the mathematics and because it holds the key to any epistemological concerns". In the first chapter, "Beginnings", the material includes ... [...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 cardinal property, 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, Cambridge, King's College). He is a professor of mathematics at Boston University. With Matthew Foreman he is the editor of the ''Handbook of Set Theory'' (2010). Selected publications * A. Kanamori, Menachem Magidor, 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, Springer, 99–275. * Robert Solovay, R. M. Solovay, W. N. Reinhardt, A. KanamoriStrong axioms of infinity and elementary embeddings ''Annals of Mathematical Logic'', 13(1978), 73–116. * A. Kan ... [...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'': º 2 → there is a set of cardinality κ that is homogeneous for ''f''. In this context, º 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. Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact ... [...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 \kappa is called strongly Mahlo if \kappa is strongly inaccessible and the set U = \ is stationary in κ. A cardinal \kappa is called weakly Mahlo if \kappa is weakly inaccessible and the set of weakly inaccessible cardinals less than \kappa is stationary in \kappa. The term "Mahlo cardinal" now usually means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals. Minimal condition sufficient for a Mahlo cardinal * If κ is a limit ''ordinal'' and the set of regular ordinals less than κ is stationary in κ, then κ is weakly Mahlo. The main difficulty in proving this is to show that κ is regular. We will suppose that it is not regular and construct a ... [...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 KÅ‘nig'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 KÅ‘nig's lemma does not hold for uncountable trees. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inaccessible Cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of fewer than cardinals smaller than , and \alpha < \kappa implies . The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). Some authors do not require weakly and strongly ... [...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 at most 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 showed in unpublished work that there are Kurepa trees in Gödel's |
.jpg "Click for more on -> Akihiro Kanamori")