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29 relations: Akihiro Kanamori, Arithmetic, Aronszajn tree, Axiom of determinacy, Consistency, Continuum hypothesis, David Hilbert, Forcing (mathematics), Gödel's incompleteness theorems, Hilbert's program, Inaccessible cardinal, Kurepa tree, Kurt Gödel, Large cardinal, List of Latin phrases (V), Mahlo cardinal, Mathematical logic, Mathematical object, Metamathematics, Peano axioms, Primitive recursive arithmetic, Recursively enumerable set, Reverse mathematics, Robinson arithmetic, Second-order arithmetic, Set theory, Theory (mathematical logic), Weakly compact cardinal, Zermelo–Fraenkel set theory.
Akihiro Kanamori
is a Japanese-born American mathematician.
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Arithmetic
Arithmetic (from the Greek ἀριθμός arithmos, "number") is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them—addition, subtraction, multiplication and division.
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Aronszajn tree
In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels.
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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.
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Consistency
In classical deductive logic, a consistent theory is one that does not contain a contradiction.
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Continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.
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David Hilbert
David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.
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Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results.
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Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic.
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Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies.
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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.
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Kurepa tree
In set theory, a Kurepa tree is a tree (T, 1, 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 constructible universe. More precisely, the existence of Kurepa trees follows from the diamond plus principle, which holds in the constructible universe. On the other hand, showed that if a strongly inaccessible cardinal is Lévy collapsed to ω2 then, in the resulting model, there are no Kurepa trees. The existence of an inaccessible cardinal is in fact equiconsistent with the failure of the Kurepa hypothesis, because if the Kurepa hypothesis is false then the cardinal ω2 is inaccessible in the constructible universe. A Kurepa tree with fewer than 2ℵ1 branches is known as a Jech–Kunen tree. More generally if κ is an infinite cardinal, then a κ-Kurepa tree is a tree of height κ with more than κ branches but at most |α| elements of each infinite level α1, and results in a tree with exactly ℵ1 branches.
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Kurt Gödel
Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.
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Large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers.
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List of Latin phrases (V)
Additional references.
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Mahlo cardinal
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number.
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Mathematical logic
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.
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Mathematical object
A mathematical object is an abstract object arising in mathematics.
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Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods.
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Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.
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Primitive recursive arithmetic
Primitive recursive arithmetic, or PRA, is a quantifier-free formalization of the natural numbers.
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Recursively enumerable set
In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if.
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Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.
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Robinson arithmetic
In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out in R. M. Robinson (1950).
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Second-order arithmetic
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Theory (mathematical logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language.
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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.
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Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
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Consistency strength, Equiconsistent.
References
[1] https://en.wikipedia.org/wiki/Equiconsistency
