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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
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
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)
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
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
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
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
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
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
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
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
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
Technique invented by Paul Cohen for proving consistency and independence results
within M {\displaystyle M} , defined by transfinite induction (specifically ∈ {\displaystyle \in } -induction) over the P {\displaystyle \mathbb {P} }
Forcing_(mathematics)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
Dutch mathematician and logician
the ramifications of intuitionism with respect to "transfinite judgements", e.g. transfinite induction. 1927. L. E. J. Brouwer: "On the domains of definition
L._E._J._Brouwer
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
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
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
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
Branch of mathematical logic
by Gentzen, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ε0. Ordinal analysis has been extended to many fragments
Proof_theory
Mathematical logician and philosopher
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Kurt_Gödel
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
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
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)
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
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
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
American mathematician (1926–2026)
Project Howard, W. A.; Kreisel, G. (September 1966). "Transfinite Induction and Bar Induction of Types Zero and One, and the Role of Continuity in Intuitionistic
William_Alvin_Howard
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
Proposition in mathematical logic
1090/s0273-0979-03-00981-9. S2CID 1510438. Jourdain, Philip E.B. (1905). "On transfinite cardinal numbers of the exponential form". Philosophical Magazine. Series 6
Continuum_hypothesis
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
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
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
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
System of mathematical set theory
Logic: 237. doi:10.2307/2273185. JSTOR 2273185. Kripke, S. (1964), "Transfinite recursion on admissible ordinals", Journal of Symbolic Logic, 29: 161–162
Kripke–Platek_set_theory
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
Function uniquely mapping two numbers into a single number
= α {\displaystyle \gamma (\alpha )=\alpha } can be proved by transfinite induction: If α = ω {\displaystyle \alpha =\omega } , then γ ( α ) =
Pairing_function
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
Size of a set in mathematics
of these "too large" sets "absolute infinite", separating it from the transfinite. The former he characterized by its "inconsistency", causing paradoxes
Cardinality
transfinite 1. An infinite ordinal or cardinal number (see Transfinite number) 2. Transfinite induction is induction over ordinals 3. Transfinite recursion
Glossary_of_set_theory
Infinite set that is not countable
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Uncountable_set
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)
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
Set with algorithmic membership test
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Computable_set
Theorem in order and lattice theory
of f α(0), taking α over the ordinals, where f α is defined by transfinite induction: f α+1 = f (f α) and f γ for a limit ordinal γ is the least upper
Knaster–Tarski_theorem
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)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Nested_set_collection
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
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
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
Set of elements that commute with every element of a group
be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter
Center_(group_theory)
Branch of mathematics
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Order_theory
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
Infinite ordinal number class
ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning
Limit_ordinal
Proof by Alan Turing
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Turing's_proof
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
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
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
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
induction scheme to the new language and adding the scheme ( T I ν ) : T I ( ≺ , F ) {\displaystyle (TI_{\nu }):TI(\prec ,F)} expressing transfinite induction
Theories of iterated inductive definitions
Theories_of_iterated_inductive_definitions
Extension of a mathematical field with polynomial roots
extension of K. These finitary results can be generalized using transfinite induction: The union of any chain of algebraic extensions over a base field
Algebraic_extension
German-Israeli mathematician and Zionist (1891–1965)
Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram Set types Amorphous Countable Empty Finite (hereditarily)
Abraham_Fraenkel
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
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
Set theory concept
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
Large_cardinal
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
Boy/Male
Hindu
Manifestation of the Trinity - Brahma, Vishnu and Shiva
Girl/Female
American, Australian, British, Chinese, English, German
To Cut; Tailor; Cutter of Cloth
Female
English
English name derived from the vocabulary word fawn, FAWN means "baby deer."
Boy/Male
Indian
Strong, Health
Male
French
Variant spelling of French Gervaise, GERVAIS means "spear servant."
Boy/Male
Tamil
Dikshith | தீகà¯à®·à®¿à®¤
Prepared, Initiated
Girl/Female
Muslim
Beautiful
Male
Spanish
Spanish form of Phoenician Hamilcar, AMILCAR means "friend of Melqart."Â
Surname or Lastname
German
German : habitational name from any of several places so named.German : topographic name from fields so named because they were cultivated only in the summer, from Middle High German sumer, Middle Low German somer ‘summer’ + Middle High German, Middle Low German velt ‘open country’.Jewish (Ashkenazic) : ornamental name composed of German Sommer ‘summer’ + Feld ‘field’. Compare Sommer.English : variant of Summerfield.
Boy/Male
Tamil
Vitasta | விதாஸà¯à®¤à®¾
River jhelum in Sanskrit
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
TRANSFINITE INDUCTION
n.
The property by which one body, having electrical or magnetic polarity, causes or induces it in another body without direct contact; an impress of electrical or magnetic force or condition from one body on another without actual contact.
a.
Pertaining to, or characterized by, electricity by the action of magnets; as, magneto-electric induction.
n.
That condition of a muscle in which it is in a state of continued vibratory contraction, as when stimulated by a series of induction shocks.
a.
Leading to inferences; proceeding by, derived from, or using, induction; as, inductive reasoning.
n.
The introduction of a clergyman into a benefice, or of an official into a office, with appropriate acts or ceremonies; the giving actual possession of an ecclesiastical living or its temporalities.
a.
Acting by, or in a state of, induction; relating to electrical induction.
v. t.
To derive by deduction or by induction; to conclude or surmise from facts or premises; to accept or derive, as a consequence, conclusion, or probability; to imply; as, I inferred his determination from his silence.
a.
Facilitating induction; susceptible of being acted upon by induction; as certain substances have a great inductive capacity.
a.
Of or pertaining to voltaism, or voltaic electricity; as, voltaic induction; the voltaic arc.
n.
The unit of electric induction; the induction in a circuit when the electro-motive force induced in this circuit is one volt, while the inducing current varies at the rate of one ampere a second.
n.
An induction coil.
n.
An instrument for measuring or ascertaining the degree or rate of electrical induction.
a.
Operating by induction; as, an inductive electrical machine.
n.
A canonical term for regular induction into a benefice, comprehending nomination, collation, and installation.
n.
Induction.
adv.
By induction or inference.
n.
The act or process of inferring by deduction or induction.
n.
A process of demonstration in which a general truth is gathered from an examination of particular cases, one of which is known to be true, the examination being so conducted that each case is made to depend on the preceding one; -- called also successive induction.
a.
Pertaining to, or proceeding by, induction; inductive.