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Set-theoretic function
an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose
Ordinal_collapsing_function
Ordinals in mathematics and set theory
uncountable ordinals. There are several ways to do this, described to some extent in the article on ordinal collapsing function. The Bachmann–Howard ordinal (sometimes
Large_countable_ordinal
Large countable ordinal
}+1}}(0)} in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories:
Takeuti–Feferman–Buchholz ordinal
Takeuti–Feferman–Buchholz_ordinal
Large countable ordinal
Howard (1972). The Bachmann–Howard ordinal is defined using an ordinal collapsing function: εα enumerates the epsilon numbers, the ordinals ε such that ωε = ε. Ω =
Bachmann–Howard_ordinal
Large countable ordinal
In mathematics, the Feferman–Schütte ordinal (Γ0) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such
Feferman–Schütte_ordinal
Topics referred to by the same term
Psi function can refer, in mathematics, to the ordinal collapsing function ψ ( α ) {\displaystyle \psi (\alpha )} the Dedekind psi function ψ ( n ) {\displaystyle
Psi_function
Type of mathematical function
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members
Ordinal_notation
Penultimate letter in the Greek alphabet
adjustment to a U-value, accounting for thermal bridge effects The ordinal collapsing function and notation In Biblical studies, as an abbreviation for the
Psi_(Greek)
Swiss mathematician (1924–2022)
observatory) in Zürich. He introduced the Bachmann–Howard ordinal and ordinal collapsing functions. "Zum Hinschied von Dr. Heinz Bachmann (1924 bis 2022)"
Heinz_Bachmann
Mathematical technique used in proof theory
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories
Ordinal_analysis
Eighth letter of the Greek alphabet
of banks in economic models The ordinal collapsing function developed by Solomon Feferman The Heaviside step function In pharmacology, the fraction of
Theta
Certain large countable ordinal
produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions. The large Veblen ordinal is sometimes denoted
Large_Veblen_ordinal
Rathjen's ψ {\displaystyle \psi } psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals M {\displaystyle
Rathjen's_psi_function
Buchholz's psi-functions are a hierarchy of single-argument ordinal functions ψ ν ( α ) {\displaystyle \psi _{\nu }(\alpha )} introduced by German mathematician
Buchholz_psi_functions
Generalization of "n-th" to infinite cases
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite
Ordinal_number
Type of transfinite numbers
enumerated by φ2 (see ordinal collapsing function.) Continuing in this vein, one can define maps φα for progressively larger ordinals α (including, by this
Epsilon_number
Certain large countable ordinal
modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The last one is
Ackermann_ordinal
Certain large countable ordinal
Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The small Veblen ordinal θ Ω
Small_Veblen_ordinal
Mathematical concept for comparing objects
\vartheta } ordinal collapsing function. (The small Veblen ordinal equals ϑ ( Ω ω ) {\displaystyle \vartheta (\Omega ^{\omega })} in this ordinal notation
Well-quasi-ordering
American philosopher and mathematician
predicative mathematics. In particular, he introduced the Feferman–Schütte ordinal as a measure of the strength of certain predicative systems. Feferman was
Solomon_Feferman
Topics referred to by the same term
Optimum Coding in the Frequency Domain Ordinal collapsing function, a process used for reaching large ordinals in the field of set theory Ordnance Clothing
OCF
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Axiomatic set theories based on the principles of mathematical constructivism
the same functions recursive. Specifically, its proof-theoretic large countable ordinal is the Bachmann–Howard ordinal. This is also the ordinal of classical
Constructive_set_theory
Topics referred to by the same term
mathematical game on a labeled tree Buchholz psi functions, a system of ordinal collapsing functions Buchholz's ID hierarchy, a hierarchy of inductively
Buchholz
3-volume treatise on mathematics, 1910–1913
own ordinals, cardinals, real numbers, and so on. This results in a lot of bookkeeping to relate the various types with each other. In ZFC functions are
Principia_Mathematica
Particular class of sets which can be described entirely in terms of simpler sets
\mathrm {Ord} \subset L\subseteq V} , properties of ordinals that depend on the absence of a function or other structure (i.e. Π 1 Z F {\displaystyle \Pi
Constructible_universe
ordinal β 1. βX is the Stone–Čech compactification of X 2. An ordinal γ A gamma number, an ordinal of the form ωα Γ The Gamma function of ordinals.
Glossary_of_set_theory
finite sequences of ordinals less than a given cardinal λ. If λ is uncountable then forcing with this poset collapses λ to ω. Collapsing a cardinal to another:
List_of_forcing_notions
Result in mathematics and set theory
In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej
Mostowski_collapse_lemma
Set-theoretic tree with uncountable branches
other two, are mapped to the same ordinal, then the three nodes are comparable. This can be done without collapsing ℵ 1 {\displaystyle \aleph _{1}} ,
Kurepa_tree
Large cardinal number
constructible universe L {\displaystyle L} satisfies "for every countable ordinal α {\displaystyle \alpha } , there is an α {\displaystyle \alpha } -Erdős
Erdős_cardinal
Type of binary relation
When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded
Well-founded_relation
System of mathematical set theory
proof-theoretic ordinal of KPω is the Bachmann–Howard ordinal. KP fails to prove some common theorems in set theory, such as the Mostowski collapse lemma. Constructible
Kripke–Platek_set_theory
Statistical model for a binary dependent variable
categories are ordered, one can use the ordinal logistic regression (for example the proportional odds ordinal logistic model). See § Extensions for further
Logistic_regression
the first such ordinal will "play the role of ω1" in M. In fact, the downward Löwenheim–Skolem theorem together with Mostowski collapse lemma can convert
Standard_model_(set_theory)
Hierarchy of formulas in set theory
restriction of the function f to a p. 23 g is the image of f on a p. 23 b is the successor ordinal of a p. 23 The Mostowski collapse of ( x , ∈ ) {\displaystyle
Lévy_hierarchy
Statement that is taken to be true
the alleged collapse of the wave function. There have been alternative approaches that either removed the need to postulate the collapse or that attempted
Axiom
Field of economics to evaluate well-being
specification of the social welfare function Further, efficiency dispenses with cardinal measures of utility, replacing it with ordinal utility, which merely ranks
Welfare_economics
Lemma in constructibility theory
{\displaystyle (X,\in )\prec (L_{\alpha },\in )} , then in fact there is some ordinal β ≤ α {\displaystyle \beta \leq \alpha } such that X = L β {\displaystyle
Condensation_lemma
Theorem that arithmetical truth cannot be defined in arithmetic
but not over sets or functions of natural numbers. The theory is strong enough to describe recursively defined integer functions such as exponentiation
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Technique invented by Paul Cohen for proving consistency and independence results
_{1})} , finite partial functions from ω {\displaystyle \omega } to ω 1 {\displaystyle \omega _{1}} , the first uncountable ordinal, one gets in V [ G ]
Forcing_(mathematics)
Axiom of set theory
( n , α ) ∣ n ∈ ω ∧ α is an ordinal } . {\textstyle \{(n,\alpha )\mid n\in \omega \land \alpha {\text{ is an ordinal }}\}\,.} Given the other axioms
Axiom_of_regularity
Infinitely many tasks in finite time
operations is uncountable. A hypertask that includes one task for each ordinal number is called an ultratask. The term "supertask" was coined by the philosopher
Supertask
Set theory concept
things get messier, as M′s interpretation of what it means to be a "function" or an "ordinal" may differ from the "external" interpretation.) Once the elements
Boolean-valued_model
School of thought in philosophy of mathematics
Russell arrived at this disheartening conclusion: that "the theory of ordinals and cardinals survives ... but irrationals, and real numbers generally
Logicism
Mathematics taught in primary and secondary school
degree 3, a cube root. Roots of higher degree are referred to by using ordinal numbers, as in fourth root, twentieth root, etc. For example: 2 is a square
Elementary_mathematics
Type of uncertainty of meaning where several interpretations are possible
type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. In mathematics and logic, ambiguity can be considered
Ambiguity
Process of assigning numbers to objects or events
sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales. Measurement is a cornerstone of trade, science
Measurement
Set of related ordination techniques used in information visualization
with interval or ratio level data), and Nonmetric MDS (which deals with ordinal data). Decide number of dimensions – The researcher must decide on the
Multidimensional_scaling
Class of green algae
the Chlorophyceae. These include: closed mitosis, the telophase spindle collapsing before cytokinesis, and a system of microtubules called a phycoplast running
Chlorophyceae
Formula for spectral line wavelengths in alkali metals
where n is the line's wavenumber, n0 is the series limit, m is the line's ordinal number in the series, m′ is a constant different for different series and
Rydberg_formula
Mathematical puzzle game
Also observe that: Disks whose ordinals have even parity move in the same sense as the smallest disk. Disks whose ordinals have odd parity move in opposite
Tower_of_Hanoi
Southwestern Iranian language
the Sasanian Empire. For some time after the Sasanian collapse, Middle Persian continued to function as a prestige language. It descended from Old Persian
Middle_Persian
Origin and evolution of the symbols used to write equations and formulas
in infinite sets. For the ordinals he employed the Greek letter ω (omega). This notation is still in use today in ordinal notation of a finite sequence
History of mathematical notation
History_of_mathematical_notation
Hungarian and American mathematician and physicist (1903–1957)
mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion
John_von_Neumann
Ordinal and numerical measures 1 group N ≥ 30 One-sample t-test N < 30 Normally distributed One-sample t-test Not normal Sign test 2 groups Independent
Location_test
Extinct ancient language of Mesopotamia
cardinal numbers from 1 to 10 as well as a few higher ones are attested. Ordinal numbers are formed with the suffix -(š)še or ši, which becomes -ze or -zi
Hurrian_language
Decentralized digital cryptocurrency
fell following the collapses of TerraUSD, a stablecoin, and the Celsius Network, a cryptocurrency loan company. In 2023, ordinals—non-fungible tokens
Bitcoin
Science fiction character
programme's 50th anniversary, and was written so as not to disturb the ordinal naming of the established Ninth, Tenth and Eleventh Doctors. In the 1986
The_Doctor
earthquake had occurred in a location indicated by a specific cardinal or ordinal direction. Although no tremors could be felt in the capital when Zhang
History_of_science
Extinct Semitic language of Mesopotamia
of the numeral. It is noted, that in the case of the numeral "one", the ordinal (masculine) and the cardinal number are the same. A metathesis occurs in
Akkadian_language
Academic field of study
created an interface between narrative and proxy data in the form of an ordinal index that allows converting disparate documentary evidence into continuous
Historical_climatology
State that has lost its ability to govern
based on the notion that while a state may function in general, polities at the substate level may collapse in terms of infrastructure, economy, and social
Failed_state
Mathematical logician and philosopher
computability and recursive functions to the point where he was able to present a lecture on general recursive functions and the concept of truth. This
Kurt_Gödel
Roman civilisation from the 8th century BC to the 5th century AD
been increasingly disinclined to childbearing. Roman hours were counted ordinally from dawn to dawn. Thus, if sunrise was at 6 am, then 6 to 7 am was called
Ancient_Rome
Management of the flow of goods and services
Zhang, Na (2021). "Sustainable Supplier Selection in Megaprojects: Grey Ordinal Priority Approach". Business Strategy and the Environment. 30 (1): 318–339
Supply_chain_management
Script used for languages in Ethiopia and Eritrea
follows; compare the colon in western text In ordered lists, after the ordinal symbol (such as a letter or number), separating it from the text of the
Geʽez_script
Austrian economist and philosopher (1899–1992)
University of Chicago", Hayek: A Collaborative Biography: Part IV, England, the Ordinal Revolution and the Road to Serfdom, 1931–50, Archival Insights into the
Friedrich_Hayek
charge number of an atomic nucleus was found to be equal to the element's ordinal position on the periodic table. The nuclear charge number thus provided
History_of_atomic_theory
Austrian psychotherapist (1870–1937)
psychological and ordinal birth order (e.g., a second child might behave like a firstborn, in which case they are considered to be an ordinal secondborn but
Alfred_Adler
years after it had first started taking contributions in bitcoin. In 2023, ordinals, non-fungible tokens (NFTs) on Bitcoin, went live. In 2024, bitcoin-holding
History_of_bitcoin
Computer bugs related to the year 2000
represented as 099 and 2001 represented as 101, etc.) and three-digit days (ordinal date in year). Only input and output instructions for the date fields had
Year_2000_problem
American mathematician and philosopher (1926–2016)
student George Boolos, he showed that the first such "non-index" is the ordinal β 0 {\displaystyle \beta _{0}} of ramified analysis (this is the smallest
Hilary_Putnam
system functions (although having meaningful internal names) were exported from user.exe as BEARNNN (where NNN is the ordinal number of the function) in
List of Easter eggs in Microsoft products
List_of_Easter_eggs_in_Microsoft_products
Language spoken in Central Asia during the time of the Mongol Empire
component may also optionally stand in the plural, e.g. 500 tabun ja'ut. Ordinal numerals are formed by the suffix -Du'ar > -Da'ar, but the shape of the
Middle_Mongol
Voter not aligned with any political party
question which is answered with a "yes" or a "no"), or by a structured ordinal question ("Generally speaking, do you consider yourself a Democrat, a Republican
Independent_voter
Types of atheism
Hungaricae Francisco-Josephinae*, 1, 199–208. Original construction of ordinals from the empty set. Enderton, H. B. (1977). *Elements of Set Theory*. Academic
Negative_and_positive_atheism
Smallest unit of a chemical element
Rutherford called the "atomic number") was found to be equal to the element's ordinal number on the periodic table and therefore provided a simple and clear-cut
Atom
Buddhist temple complex in Cambodia
the last. The two inner galleries each have four large towers at their ordinal corners (that is, North-west, North-east, South-east, and South-west) surrounding
Angkor_Wat
Unique digital identifier that is recorded on a blockchain
to add NFTs to the Bitcoin blockchain called "ordinals". By February 2023, the popularity of ordinals had led to an increase in bitcoin's payment fees
Non-fungible_token
Austroasiatic language
speak date in the format "day month year". Each month's name is just the ordinal of that month appended after the word tháng, which means "month". Traditional
Vietnamese_language
Units that are not part of a coherent system
calendar further extended this by assembling days into ten-day weeks. Ordinal dates and Julian days, the latter of which has seen use in astronomy as
List of non-coherent units of measurement
List_of_non-coherent_units_of_measurement
Russian figure skater (born 2006)
clean triple Axel, scoring near her own world record. Valieva placed four ordinals above American skater Karen Chen, moving the Russian team into first place
Kamila_Valieva
Pan-Slavic language
are inflected either as nouns of the kosť type or as soft adjectives. Ordinal numbers are formed by adding the adjective ending -y to the cardinal numbers
Interslavic
King of Georgia from 1089 to 1125
referred to as David II, III, and IV, reflecting substantial variation in the ordinals assigned to the Georgian Bagratids, especially in the early period of their
David_IV
arbitrary curved lines, especially the distances of rivers and roads on a map. ordinal directions See intercardinal directions. ordnance datum (OD) Any vertical
Glossary of geography terms (N–Z)
Glossary_of_geography_terms_(N–Z)
Military junta government of Myanmar from 2021 to 2025
report also noted that the regime's governance functions and administrative capacity were collapsing throughout the country, with much of Myanmar's townships
State_Administration_Council
Parliamentary Investigation Commission for the Coups and the Memorandums: "(Ordinal) 376, Volume 1, Page 15, Paragraph 4 (continues on page 16, the first 5
1980_Turkish_coup_d'état
Lack of economic resources
2012. Retrieved 8 August 2012. Sen, Amartya (March 1976). "Poverty: An Ordinal Approach to Measurement". Econometrica. 44 (2): 219–231. doi:10.2307/1912718
Poverty
1536–1541 disbanding of religious residences by Henry VIII
dispersed or destroyed libraries and provided for their former personnel and functions. Though the policy was originally envisaged as a way to increase the regular
Dissolution of the monasteries
Dissolution_of_the_monasteries
1707 and instead to match the ordinal usage call the monarchs kings or queens of England and Scotland. Separate ordinals were abandoned when the two states
History_of_the_United_Kingdom
Critique of planned economies
bias. They further argue that even if artificial intelligence is able to ordinally rank items like humans, they would still suffer from the same issues of
Economic_calculation_problem
Northwest Caucasian language
"тхьэ" (god) instead of the Chemguy "ӏа" and "тхьа", as well as specific ordinal numbers and verb forms. Thus, being supra-dialectal, literary Adyghe stands
Adyghe_language
First five books of the Hebrew Bible
William M.; Zurawski, Jason M.; Boccaccini, Gabriele (eds.). Torah: Functions, Meanings, and Diverse Manifestations in Early Judaism and Christianity
Torah
Declined according to case, state, gender and number
absolute ablative in Latin syntax). Adverbs can be formed from adjectives, ordinal numerals: كَثِيرًا kaṯīran "frequently, a lot, often", نَادِرًا nādiran
Arabic_nouns_and_adjectives
Trek version runs the axis through the Galactic Center. Also, rather than ordinals, quadrants are designated by the Greek letters Alpha, Beta, Gamma, and
List of Star Trek regions of space
List_of_Star_Trek_regions_of_space
as a DOS interface. Plug-ins: functions are available through an integrated development environment. Minimum function should be to list the revision
Comparison of version-control software
Comparison_of_version-control_software
Grammar of the Romanian language
Fractional: doime (half) (pronounced [do.ime]); Adverbial: de două ori (twice); Ordinal: al doilea (the second). As in all Romance languages, Romanian verbs are
Romanian_grammar
16th-century religious events during reign of English King Henry VIII
which was associated with the doctrine of purgatory. One of the primary functions of monasteries was to pray for the souls of their benefactors and for
Henrician_Reformation
Austronesian language of West Papua, Indonesia
that an action is either sudden or completed. The essive ve- functions as a verbalizer, ordinal, relativizer, or indicator of inherent properties, depending
Wamesa_language
ORDINAL COLLAPSING-FUNCTION
ORDINAL COLLAPSING-FUNCTION
Girl/Female
Hindu, Indian, Marathi
Pleased; Satisfied
Girl/Female
Tamil
Sweet girl, Variant of donald great chief
Girl/Female
Australian, Latin
Golden
Female
Italian
Feminine form of Italian Orsino, ORSINA means "bear-like."
Girl/Female
Indian
Sweet girl, Variant of donald great chief
Girl/Female
Gujarati, Hindu, Indian
Brave
Girl/Female
Latin
Ardent. Eager. Industrious.
Girl/Female
Indian
Surname or Lastname
English
English : variant of Ordway.
Female
Scottish
Scottish feminine form of English Rodney, RODINA means "Hroda's fen/island."
Girl/Female
Hindu
Surname or Lastname
English
English : variant of Cordell.
Girl/Female
Hindu, Indian
Great Chief; Variant of Donald
Girl/Female
French
Gold.
Girl/Female
Australian, Greek, Hebrew
Peace
Female
African
common, ordinay.
Girl/Female
Arabic
Pride
Girl/Female
Australian, Latin
Little Wave
Surname or Lastname
English, French, Spanish, and Dutch
English, French, Spanish, and Dutch : from Middle English, Old French cardinal ‘cardinal’, the church dignitary (Latin cardinalis, originally an adjective meaning ‘crucial’). The surname may have denoted a servant who worked in a cardinal’s household, but was probably more often bestowed as a nickname on someone who habitually dressed in red or who had played the part of a cardinal in a pageant, or on one who acted in a lordly and patronizing manner, like a prince of the Church.A bearer of the name, of unknown origin, is documented in Montreal by 1666.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh
Lotus
ORDINAL COLLAPSING-FUNCTION
ORDINAL COLLAPSING-FUNCTION
Male
Greek
(Κήυξ) Greek name, possibly KEYX means "kingfisher." In mythology, this is the name of a king of Thessaly, the son of Eosphoros.
Boy/Male
Hebrew
Ready; prepared.
Boy/Male
Hindu, Indian
With a Charming Face
Girl/Female
Arabic, Australian, Muslim
Leader; Pioneer
Girl/Female
Polish
Defends mankind.
Boy/Male
Arabic, Muslim
Great King
Girl/Female
Teutonic American English German Spanish
Oracle.
Girl/Female
Hindu, Indian, Tamil
Sweet
Boy/Male
Sikh
One who abides in lords name
Female
German
Feminine form of German Ernst, ERNSTA means "battle (to the death), serious business."
ORDINAL COLLAPSING-FUNCTION
ORDINAL COLLAPSING-FUNCTION
ORDINAL COLLAPSING-FUNCTION
ORDINAL COLLAPSING-FUNCTION
ORDINAL COLLAPSING-FUNCTION
p. pr. & vb. n.
of Collapse
a.
Pertaining to the origin or beginning; preceding all others; first in order; primitive; primary; pristine; as, the original state of man; the original laws of a country; the original inventor of a process.
n.
An officer who has original jurisdiction in his own right, and not by deputation.
a.
Having the power to suggest new thoughts or combinations of thought; inventive; as, an original genius.
a.
Indicating order or succession; as, the ordinal numbers, first, second, third, etc.
n.
Ordeal.
n.
A book containing the rubrics of the Mass.
n.
The book of forms for making, ordaining, and consecrating bishops, priests, and deacons.
n.
An original thinker or writer; an originator.
n.
Any invigorating and stimulating preparation; as, a peppermint cordial.
a.
Of common rank, quality, or ability; not distinguished by superior excellence or beauty; hence, not distinguished in any way; commonplace; inferior; of little merit; as, men of ordinary judgment; an ordinary book.
a.
Of or pertaining to trial by ordeal.
n.
Anything which is in ordinary or common use.
a.
Of or pertaining to an order.
a.
Before unused or unknown; new; as, a book full of original matter.
n.
The natural or wild species from which a domesticated or cultivated variety has been derived; as, the wolf is thought by some to be the original of the dog, the blackthorn the original of the plum.
n.
That which precedes all others of its class; archetype; first copy; hence, an original work of art, manuscript, text, and the like, as distinguished from a copy, translation, etc.
n.
A word or number denoting order or succession.
a.
Not copied, imitated, or translated; new; fresh; genuine; as, an original thought; an original process; the original text of Scripture.