Synsets for "inaccessible"
Synset: inaccessible.a.01
Synonyms: inaccessible
Part of Speech: ADJECTIVE
Definition: capable of being reached only with great difficulty or not at all
Examples:
Lemmas: inaccessible unaccessible
Hypernym:
Hyponym:
Antonyms: accessible
Synset: inaccessible.s.02
Synonyms: inaccessible
Part of Speech: ADJECTIVE SATELLITE
Definition: not capable of being obtained
Examples: a rare work, today almost inaccessible | timber is virtually unobtainable in the islands | untouchable resources buried deep within the earth
Lemmas: inaccessible unobtainable unprocurable untouchable
Hypernym:
Hyponym:
Antonyms:
Related Wikipedia Samples:
Article | Related Text |
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Inaccessible cardinal | Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on. |
Inaccessible cardinal | 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 inaccessible cardinals to be uncountable (in which case formula_3 is strongly inaccessible). Weakly inaccessible cardinals were introduced by , and strongly inaccessible ones by and . |
Inaccessible cardinal | The term hyper-inaccessible is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible). Other authors use it to mean that κ is κ-inaccessible. (It can never be κ+1-inaccessible.) It is occasionally used to mean Mahlo cardinal. |
Inaccessible Bay | Inaccessible Bay or Baie Inaccessible is a bay in southwestern New Caledonia. It is known as "Inaccessible" because a narrow peninsula partly prevents access to the inner bay known as Saint Vincent Bay. |
Inaccessible cardinal | ZFC implies that the "V" is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe "L" is a model of ZFC whenever κ is weakly inaccessible. Thus ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal. |
Mahlo cardinal | The term "hyper-inaccessible" is ambiguous. In this section, a cardinal κ is called hyper-inaccessible if it is κ-inaccessible (as opposed to the more common meaning of 1-inaccessible). |
Inaccessible cardinal | a cardinal κ is called α-inaccessible, for α any ordinal, if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal κ is called α-weakly inaccessible if κ is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than κ is unbounded in κ. In this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals. |
Inaccessible Island finch | The Inaccessible Island finch ("Nesospiza acunhae"), also known as the Inaccessible Bunting, is a species of bird in the family Thraupidae (formerly in Emberizidae). |
Gough Island | Gough Island and Inaccessible Island comprise the UNESCO World Heritage Site of Gough and Inaccessible Islands. |
Mahlo cardinal | κ is weakly inaccessible and a strong limit, so it is strongly inaccessible. |
Inaccessible cardinal | An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and formula_3 are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible. |
Inaccessible cardinal | for any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ. |
Inaccessible cardinal | The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from Gödel's second incompleteness theorem, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent. |
Inaccessible cardinal | The α-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. For example, denote by ψ(λ) the λ inaccessible cardinal, then the fixed points of ψ are the 1-inaccessible cardinals. Then letting ψ(λ) be the λ β-inaccessible cardinal, the fixed points of ψ are the (β+1)-inaccessible cardinals (the values ψ(λ)). If α is a limit ordinal, an α-inaccessible is a fixed point of every ψ for β < α (the value ψ(λ) is the λ such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers. |
Inaccessible cardinal | Suppose V is a model of ZFC. Either V contains no strong inaccessible or, taking κ to be the smallest strong inaccessible in V, "V" is a standard model of ZFC which contains no strong inaccessibles. Thus, the consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then "L" is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles". This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals. |
Mahlo cardinal | A cardinal number κ is called "strongly Mahlo" if κ is strongly inaccessible and the set U = {λ < κ: λ is strongly inaccessible} is stationary in κ. |
Mahlo cardinal | A cardinal κ is called "weakly Mahlo" if κ is weakly inaccessible and the set of weakly inaccessible cardinals less than κ is stationary in κ. |
Inaccessible Island | At least three confirmed shipwrecks have occurred off the coast of Inaccessible Island. |
Limit cardinal | formula_1 would be an inaccessible cardinal of both "strengths" except that the definition of inaccessible requires that they be uncountable. Standard Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) cannot even prove the consistency of the existence of an inaccessible cardinal of either kind above formula_1, due to Gödel's Incompleteness Theorem. More specifically, if formula_22 is weakly inaccessible then formula_23. These form the first in a hierarchy of large cardinals. |
Mahlo cardinal | If κ is α-inaccessible, then there are β-inaccessibles (for β < α) arbitrarily close to κ. Consider the set of simultaneous limits of such β-inaccessibles larger than some threshold but less than κ. It is unbounded in κ (imagine rotating through β-inaccessibles for β < α ω-times choosing a larger cardinal each time, then take the limit which is less than κ by regularity (this is what fails if α ≥ κ)). It is closed, so it is club in κ. So, by κ's Mahlo-ness, it contains an inaccessible. That inaccessible is actually an α-inaccessible. So κ is α+1-inaccessible. |