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Mathematical concept
Transfinite induction is an extension of mathematical induction to ordinal numbers. Its correctness is a theorem of ZF, and relies on the fact that the
Transfinite_induction
Generalization of "n-th" to infinite cases
of a well-ordered set grounds the principle of transfinite induction, generalizing standard induction by ensuring that if a property fails to hold, there
Ordinal_number
Form of mathematical proof
This is a special case of transfinite induction as described below, although it is no longer equivalent to ordinary induction. In this form the base case
Mathematical_induction
Collection of mathematical objects
P(n).} Transfinite induction is the same, replacing natural numbers by the elements of a well-ordered set. Often, a proof by transfinite induction easier
Set_(mathematics)
Generalization of the real numbers
dyadic fractions; a wider universe is reachable given some form of transfinite induction. There is a generation S0 = { 0 }, in which 0 consists of the single
Surreal_number
Topics referred to by the same term
Transfinite may refer to: Transfinite number, a number larger than all finite numbers, yet not absolutely infinite Transfinite induction, an extension
Transfinite
Kind of transfinite induction
schema of set induction. The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded
Epsilon-induction
Number that is larger than all finite numbers
Infinitesimal Transfinite induction "Definition of transfinite number | Dictionary.com". www.dictionary.com. Retrieved 2019-12-04. "Transfinite Numbers and
Transfinite_number
Mathematical theorem
is also often stated in terms of ordinals. Transfinite recursion is an instance of transfinite induction and the latter works over a well-ordered set
Transfinite_recursion_theorem
Topics referred to by the same term
Strong induction Structural induction Transfinite induction Epsilon-induction Parabolic induction Inductive reasoning, in logic Electromagnetic induction Electrostatic
Induction
Set theory concept
as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe
Von_Neumann_universe
Type of transfinite numbers
smallest epsilon number ε0 appears in many induction proofs, because for many purposes transfinite induction is only required up to ε0 (as in Gentzen's
Epsilon_number
Branch of mathematics that studies sets
soon became known as Cantor's theorem. Cantor developed a theory of transfinite numbers, called cardinals and ordinals, which extended the arithmetic
Set_theory
Class of mathematical orderings
merely admits a well-founded relation), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements
Well-order
Infinite cardinal number
The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements
Aleph_number
Technique for proving independence results
within M {\displaystyle M} , defined by transfinite induction (specifically ∈ {\displaystyle \in } -induction) over the P {\displaystyle \mathbb {P} }
Forcing_(mathematics)
Theorem that every set can be well-ordered
from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique
Well-ordering_theorem
Mathematical logic concept
recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ε0", is neither weaker nor stronger than the system
Gentzen's_consistency_proof
Generalized form of recursion
induction in the same fashion that primitive recursion is related to ordinary induction, or transfinite recursion is related to transfinite induction
Bar_recursion
Possible axiom for set theory
a3, a5, ...⟩). Process all possible strategies of S1 and S2 with transfinite induction on α. For all sequences that are not in A or B after that, decide
Axiom_of_determinacy
Class of mathematical sets
{\displaystyle T_{\delta \sigma }=(T_{\delta })_{\sigma }.} Now define by transfinite induction a sequence G m {\displaystyle G^{m}} , where m {\displaystyle m}
Borel_set
Attempt to formalize all of mathematics, based on a finite set of axioms
was not clearly finitary was a certain transfinite induction up to the ordinal ε0. If this transfinite induction is accepted as a finitary method, then
Hilbert's_program
Mathematician (1845–1918)
interest, a fact of which he was well aware. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused
Georg_Cantor
Axiomatic set theory devised by W.V.O. Quine
{\displaystyle \mathrm {Ord} } can be defined with no problem. Transfinite induction works on stratified statements, which allows one to prove that the
New_Foundations
Axioms for the natural numbers
Gentzen gave a proof of the consistency of Peano's axioms, using transfinite induction up to an ordinal called ε0. Gentzen explained: "The aim of the present
Peano_axioms
Type of binary relation
well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When
Well-founded_relation
Operations on ordinals that extend classical arithmetic
well-ordered set that represents the result of the operation or by using transfinite recursion. In addition to these standard operations for ordinals, there
Ordinal_arithmetic
Measure theory and probability theorem
𝜎-algebra containing G . {\displaystyle G.} It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem. A monotone
Monotone_class_theorem
Ordinals in mathematics and set theory
not show transfinite induction for such large ordinals. For example, the usual first-order Peano axioms do not prove transfinite induction for (or beyond)
Large_countable_ordinal
Mathematical logic hierarchy
the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank. Properties of sets of small finite ranks are important
Borel_hierarchy
Pair of logical equivalences
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
De_Morgan's_laws
Part of ring theory in mathematics
maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact
Krull's_theorem
Paradox in set theory
models can be described as the universe of a cumulative TT in which transfinite types are allowed. (Once an impredicative standpoint is adopted, abandoning
Russell's_paradox
Mathematical technique used in proof theory
{\displaystyle \alpha } and such that T {\displaystyle T} proves transfinite induction of arithmetical statements for R {\displaystyle R} . Some theories
Ordinal_analysis
Generalization of the Riemann integral
\rightarrow 0} . Trying to create a general theory, Denjoy used transfinite induction over the possible types of singularities, which made the definition
Henstock–Kurzweil_integral
Mathematical proposition equivalent to the axiom of choice
such an object by assuming there is no maximal element and using transfinite induction and the assumptions of the situation to get a contradiction. Zorn's
Zorn's_lemma
Size of a possibly infinite set
aleph numbers can be identified with their initial ordinals, they form a transfinite sequence: ℵ 0 = | N | , ℵ 1 , ℵ 2 , … , ℵ α , … . {\displaystyle \aleph
Cardinal_number
Theorem about natural numbers
equivalent to the restricted ordinal theorem (i.e. the claim that transfinite induction below ε0 is valid), and gave a finitist proof for the case where
Goodstein's_theorem
Particular class of sets which can be described entirely in terms of simpler sets
{\displaystyle x} , so is the union y ∪ { y } {\displaystyle y\cup \{y\}} . Transfinite induction can be used to show each ordinal α {\displaystyle \alpha } is in
Constructible_universe
Subfield of mathematics
arithmetic using a finitistic system together with a principle of transfinite induction. Gentzen's result introduced the ideas of cut elimination and proof-theoretic
Mathematical_logic
Hungarian and American mathematician and physicist (1903–1957)
the first strict formulation of principles of definitions by the transfinite induction". Building on the Hausdorff paradox of Felix Hausdorff (1914), Stefan
John_von_Neumann
Standard system of axiomatic set theory
1996. Wolchover 2013. Abian, Alexander (1965). The Theory of Sets and Transfinite Arithmetic. W B Saunders. ———; LaMacchia, Samuel (1978). "On the Consistency
Zermelo–Fraenkel_set_theory
R {\displaystyle R} is a well-order, then we have the schema of transfinite induction over R {\displaystyle R} for arbitrary formulas. Spread (intuitionism)
Bar_induction
Statement of infinite regress
Morgan Teleological argument – Argument for the existence of God Transfinite induction – Mathematical concept Turtle Island (Native American folklore) –
Turtles_all_the_way_down
Set with exactly one element
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Singleton_(mathematics)
Czech mathematician
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Thomas_Jech
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Nested_set_collection
In mathematics, operation on sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Disjoint_union
Diagram that shows all possible logical relations between a collection of sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Venn_diagram
English philosopher and logician (1872–1970)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Bertrand_Russell
Axiom of set theory
earlier ones, we can then easily imagine extending the types into the transfinite—just how far we want to go must necessarily be left open. Now Russell
Axiom_of_regularity
Mathematical set containing no elements
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Empty_set
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
List_of_types_of_sets
Possible axiom of set theory
subsequent Vβ, or equivalently: Vα ⊆ Vβ for α ≤ β. This is proved by transfinite induction on β: β = 0: V0 ⊆ V0. For β+1: By inductive hypothesis, Vα ⊆ Vβ
Axiom_of_limitation_of_size
Set of elements in any of some sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Union_(set_theory)
Infinite Cardinal number
+1}=2^{\beth _{\alpha }},} , and it follows by Cantor's theorem and transfinite induction that the sequence of beth numbers is strictly increasing. ( | A
Beth_number
Large countable ordinal
arithmetic Π 1 1 {\displaystyle \Pi _{1}^{1}} -comprehension + transfinite induction IDω, the system of ω-times iterated inductive definitions Let Ω
Takeuti–Feferman–Buchholz ordinal
Takeuti–Feferman–Buchholz_ordinal
Use of braces for specifying sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Set-builder_notation
Axiom used in set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Axiom_of_extensionality
Paradox in set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Burali-Forti_paradox
Finite ordered list of elements
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Tuple
Mathematician and philosopher (1906–1978)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Kurt_Gödel
One-to-one correspondence
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Bijection
Term in set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Almost
German logician and mathematician (1871–1953)
influence and in 1902 published his first work concerning the addition of transfinite cardinals. By that time he had also discovered the so-called Russell
Ernst_Zermelo
Set whose elements all belong to another set
{\displaystyle [A]^{k}} is also common, especially when k {\displaystyle k} is a transfinite cardinal number. A set A is a subset of B if and only if their intersection
Subset
Type of cardinal number in mathematics
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Regular_cardinal
Property in general topology
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Finite_intersection_property
Problem in set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Suslin's_problem
Sets whose elements have degrees of membership
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Fuzzy_set
Axiomatic set theories based on the principles of mathematical constructivism
-formulation. Set induction in turn enables ordinal arithmetic in this sense. It further allows definitions of class functions by transfinite recursion. The
Constructive_set_theory
Swiss mathematician (1888–1977)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Paul_Bernays
Infinite set not splittable into infinite sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Amorphous_set
American philosopher and logician (1908–2000)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Willard_Van_Orman_Quine
Fixed-point theorem
x_{n}=g(x_{n-1})} . For arbitrary A {\displaystyle A} , we use transfinite recursion or transfinite induction to construct the sequences in a similar way. Now, this
Bourbaki–Witt_theorem
German mathematician (1831–1916)
with Leopold Kronecker, who was philosophically opposed to Cantor's transfinite numbers. Recent findings of past correspondences indicate Cantor plagiarized
Richard_Dedekind
Set of the elements not in a given subset
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Complement_(set_theory)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Generic_filter
Family of subsets representing "large" sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Filter_on_a_set
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Gödel_logic
Sphere-like surface
non-finite projective space the existence of ovoids can be proven using transfinite induction. Any ovoid O {\displaystyle {\mathcal {O}}} in a finite projective
Ovoid_(projective_geometry)
3-volume treatise on mathematics, 1910–1913
counting (or else on non-fundamental and hence questionable methods such as induction). So again Principia depends on everyday techniques, not vice versa. Wittgenstein
Principia_Mathematica
Collection of sets in mathematics that can be defined based on a property of its members
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Class_(set_theory)
Tree in set theory
Uα for all countable α. We construct the countable levels Uα by transfinite induction on α as follows starting with the empty set as U0: If α + 1 is a
Aronszajn_tree
System of mathematical set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Zermelo_set_theory
Type of large transfinite number
common meaning of 1-inaccessible). Suppose κ is Mahlo. We proceed by transfinite induction on α to show that κ is α-inaccessible for any α ≤ κ. Since κ is
Mahlo_cardinal
Weak form of the axiom of choice
that is required to show the existence of a sequence constructed by transfinite recursion of countable length, if it is necessary to make a choice at
Axiom_of_dependent_choice
Concept in mathematics
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Axiom_of_countable_choice
American mathematician (1934–2007)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Paul_Cohen
Well-spaced set of points in a metric space
Delone sets. However, whenever the points of M have a well-ordering, transfinite induction shows that it is possible to construct an ε-net N, by including
Delone_set
Concept in axiomatic set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Axiom_of_power_set
Theorem in set theory
1895 Cantor states the theorem in his first paper on set theory and transfinite numbers. He obtains it as an easy consequence of the linear order of
Schröder–Bernstein_theorem
Elements in exactly one of two sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Symmetric_difference
Any one of the distinct objects that make up a set in set theory
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Element_of_a_set
Proof in set theory
intuitionists do not accept this relation to constitute a hierarchy of transfinite sizes. When the axiom of powerset is not adopted, in a constructive framework
Cantor's_diagonal_argument
Set of elements common to all of some sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Intersection_(set_theory)
Any collection of sets, or subsets of a set
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Family_of_sets
German mathematician (1909–1945)
done by a direct proof of the unprovability of the principle of transfinite induction, used in his 1936 proof of consistency, within Peano arithmetic
Gerhard_Gentzen
System of mathematical set theory
import of VII is that of Foundation above. Develop: Ordinal numbers, transfinite induction. Infinity: There exists a set y, such that ∅ ∈ y {\displaystyle
Morse–Kelley_set_theory
Mathematical set formed from two given sets
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Cartesian_product
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TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
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