Normal function

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.
Find sources: "Normal function" – news · newspapers · books · scholar · JSTOR (March 2024)

In axiomatic set theory, a function $f : Ord \to Ord$ is called normal (or a normal function) if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:

For every limit ordinal $γ$ (i.e. $γ$ is neither zero nor a successor), it is the case that $f (γ) = sup {f (ν) : ν < γ}$ .
For all ordinals $α < β$ , it is the case that $f (α) < f (β)$ .

Examples

A simple normal function is given by $f (α) = 1 + α$ (see ordinal arithmetic). But $f (α) = α + 1$ is not normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set ${λ + 1}$ is the set ${λ}$ , which is not open when $λ$ is a limit ordinal. If $β$ is a fixed ordinal, then the functions $f (α) = β + α$ , $f (α) = β \times α$ (for $β \geq 1$ ), and $f (α) = β α$ (for $β \geq 2$ ) are all normal.

More important examples of normal functions are given by the aleph numbers $f(\alpha )=\aleph _{\alpha }$ , which connect ordinal and cardinal numbers, and by the beth numbers $f(\alpha )=\beth _{\alpha }$ .

Properties

If $f$ is normal, then for any ordinal $α$ ,

f (α) \geq α

.^[1]

Proof: If not, choose $γ$ minimal such that $f (γ) < γ$ . Since $f$ is strictly monotonically increasing, $f (f (γ)) < f (γ)$ , contradicting minimality of $γ$ .

Furthermore, for any non-empty set $S$ of ordinals, we have

f (sup S) = sup f (S)

Proof: "≥" follows from the monotonicity of $f$ and the definition of the supremum. For " $\leq$ ", set $δ = sup S$ and consider three cases:

if $δ = 0$ , then $S = {0}$ and $sup f (S) = f (0)$ ;
if $δ = ν + 1$ is a successor, then there exists $s$ in $S$ with $ν < s$ , so that $δ \leq s$ . Therefore, $f (δ) \leq f (s)$ , which implies $f (δ) \leq sup f (S)$ ;
if $δ$ is a nonzero limit, pick any $ν < δ$ , and an $s$ in $S$ such that $ν < s$ (possible since $δ = sup S$ ). Therefore, $f (ν) < f (s)$ so that $f (ν) < sup f (S)$ , yielding $f (δ) = sup {f (ν) : ν < δ} \leq sup f (S)$ , as desired.

Every normal function $f$ has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function $f ' : Ord \to Ord$ , called the derivative of $f$ , such that $f ' (α)$ is the $α$ -th fixed point of $f$ .^[2] For a hierarchy of normal functions, see Veblen functions.

Notes

^ Johnstone 1987, Exercise 6.9, p. 77
^ Johnstone 1987, Exercise 6.9, p. 77

References

Johnstone, Peter (1987), Notes on Logic and Set Theory, Cambridge University Press, ISBN 978-0-521-33692-5

Read other articles:

Lesa

Lesa comune di Italia Tempat categoria:Articles mancats de coordenades Negara berdaulatItaliaRegion di ItaliaPiemonteProvinsi di ItaliaProvinsi Novara NegaraItalia Ibu kotaLesa PendudukTotal2.164 (2023 )GeografiLuas wilayah13,58 km² [convert: unit tak dikenal]Ketinggian198 m Berbatasan denganBelgirate (en) Ispra Massino Visconti Meina Nebbiuno Stresa (en) Brovello-Carpugnino (en) Ranco SejarahSanto pelindungMartinus dari Tours Informasi tambahanKode pos28040 Zona waktuUTC+1 UTC+2 ...

Dualité des ordres de juridiction en France

Pour un article plus général, voir Dualité des ordres de juridiction. Si ce bandeau n'est plus pertinent, retirez-le. Cliquez ici pour en savoir plus. Cet article concernant le droit doit être recyclé (mai 2017). Une réorganisation et une clarification du contenu paraissent nécessaires. Améliorez-le, discutez des points à améliorer ou précisez les sections à recycler en utilisant {{section à recycler}}. Si ce bandeau n'est plus pertinent, retirez-le. Cliquez ici pour en savoir p...

Région d'Angleterre

Cet article est une ébauche concernant l’Angleterre et l’administration territoriale. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Pour les articles homonymes, voir Région. Est Londres Sud-Est Sud-Ouest Midlands de l'Est Midlands de l'Ouest Yorkshire-et-Humber Nord-Est Nord-Ouest Les régions d'Angleterre sont au nombre de neuf. Ces régions sont principalement des subdivisions administratives sans pouv...

Dewan Perwakilan Rakyat Daerah Kota Madiun

Dewan Perwakilan Rakyat Daerah Kota MadiunDewan Perwakilan RakyatKota Madiun2019-2024JenisJenisUnikameral Jangka waktu5 tahunSejarahSesi baru dimulai24 Agustus 2019PimpinanKetuaAndi Raya Bagus Miko Saputra, S.H. (PDI-P) sejak 24 September 2019 Wakil Ketua IDrs. Istono, M.Pd. (Demokrat) sejak 24 September 2019 Wakil Ketua IIDrs. H. Armaya (Perindo) sejak 24 September 2019 KomposisiAnggota30Partai & kursi PSI (2) PDI-P (6) NasDem (1) ...

العلاقات الإكوادورية البوتسوانية

العلاقات الإكوادورية البوتسوانية الإكوادور بوتسوانا الإكوادور بوتسوانا تعديل مصدري - تعديل العلاقات الإكوادورية البوتسوانية هي العلاقات الثنائية التي تجمع بين الإكوادور وبوتسوانا.[1][2][3][4][5] مقارنة بين البلدين هذه مقارنة عامة ومرجع...

Yvon Bourges

Pour les articles homonymes, voir Bourges. Yvon Bourges Yvon Bourges, octobre 1984. Fonctions Président du conseil régional de Bretagne 21 mars 1986 – 20 mars 1998 (11 ans, 11 mois et 27 jours) Prédécesseur Raymond Marcellin Successeur Josselin de Rohan Sénateur français 2 octobre 1980 – 30 septembre 1998 (17 ans, 11 mois et 28 jours) Élection 28 septembre 1980 Réélection 24 septembre 1989 Circonscription Ille-et-Vilaine Ministre de la Défense 31 j...

Thomas More

Untuk kegunaan lain, lihat Thomas More (disambiguasi). The Right HonourableSir Thomas MoreSir Thomas More,karya Hans Holbein Muda, 1527 Tuan KanselirMasa jabatanOktober 1529 – Mei 1532Penguasa monarkiHenry VIIIPendahuluThomas WolseyPenggantiThomas AudleyKanselir Kadipaten LancasterMasa jabatan31 Desember 1525 – 3 November 1529Penguasa monarkiHenry VIIIPendahuluRichard WingfieldPenggantiWilliam FitzWilliamKetua Dewan RakyatMasa jabatan16 April 1523 – 13 Agustus...

NACA airfoil

Wing shape Profile geometry – 1: Zero-lift line; 2: Leading edge; 3: Nose circle; 4: Max. thickness; 5: Camber; 6: Upper surface; 7: Trailing edge; 8: Camber mean-line; 9: Lower surface Profile lines – 1: Chord, 2: Camber, 3: Length, 4: Midline A: blue line = chord, green line = camber mean-line, B: leading-edge radius, C: xy coordinates for the profile geometry (chord = x ax...

John I Tzimiskes

Byzantine emperor from 969 to 976 John I TzimiskesEmperor and Autocrat of the RomansDetail of the Gunthertuch, a Byzantine silk tapestry depicting John Tzimiskes being greeted by the Blues and Greens at a triumph.Byzantine emperorReign11 December 969 –10 January 976Coronation19 December 969[1]PredecessorNikephoros II PhokasSuccessorBasil IIBorn925(now Çemişgezek,[2] Tunceli Province, Turkey)Died10 January 976(976-01-10) (aged 50–51)Constantinople(now ...

Бисексуальность

Флаг гордости бисексуалов Бисексуальность Сексуальные ориентации Бисексуальность Пансексуальность Полисексуальность Моносексуальность Сексуальные идентичности Би-любопытство Гетерогибкость и гомогибкость Сексуальная текучесть Исследования Шк...

李光耀之死

李光耀逝世及葬礼李光耀（1923年－2015年）日期2015年3月23日－2015年3月29日地点新加坡斯里淡马锡（私人守灵）新加坡国会大厦（民众瞻仰）新加坡国立大学文化中心（国葬）万礼火葬场（英语：Mandai Crematorium and Columbarium）（火葬）网站www.rememberingleekuanyew.sg 2015年3月23日凌晨3時18分（新加坡標準時間），新加坡建国后首任总理、前內閣资政和执政人民行动党首任秘书长李光�...

STS-105

Questa voce sull'argomento missioni spaziali è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto di riferimento. STS-105Emblema missione Dati della missioneOperatoreNASA NSSDC ID2001-035A SCN26888 ShuttleDiscovery Lancio10 agosto 2001 21:10 UTC Luogo lancioRampa 39A Atterraggio22 agosto 2001 18:23 UTC Sito atterraggioJohn F. Kennedy Space Center, Shuttle Landing Facility (pista 15) Durata11 giorni, 21 ore, 13 minuti e...

باريس تورز 1965

باريس تورز 1965 تفاصيل السباقسلسلة59. باريس تورزمنافسةسوبر برستيج بيرنود 1965التاريخ10 أكتوبر 1965البلد فرنساالمنصةالفائز غربن كارستينس (تليفايزر [الإنجليزية]‏)الثاني Gustaaf De Smet [الإنجليزية]‏ (Wiel's-Groene Leeuw [لغات أخرى]‏)الثالث Fernand Deferm [الإنجليزية]‏ ▶19641966&#...

Rhens

Rhenscittà Rhens – Veduta LocalizzazioneStato Germania Land Renania-Palatinato DistrettoNon presente CircondarioMayen-Coblenza TerritorioCoordinate50°17′N 7°37′E / 50.283333°N 7.616667°E50.283333; 7.616667 (Rhens)Coordinate: 50°17′N 7°37′E / 50.283333°N 7.616667°E50.283333; 7.616667 (Rhens) Altitudine65 m s.l.m. Superficie16,31 km² Abitanti2 939[1] (31-12-2009) Densità180,2 ab./km² Altre informazio...

Albion W. Knight Jr.

American politician Albion KnightPersonal detailsBornAlbion Williamson Knight Jr.(1924-06-01)June 1, 1924Jacksonville, Florida, U.S.DiedMay 22, 2012(2012-05-22) (aged 87)Gaithersburg, Maryland, U.S.Political partyConstitutionAlma materUnited States Military AcademyUniversity of Illinois, Urbana-ChampaignAmerican University Albion Williamson Knight Jr. (1 June 1924 – 22 May 2012) was the second Archbishop of the United Episcopal Church of North America (UECNA) from 1989 until his re...

House of York

Cadet branch of the House of Plantagenet For other uses, see York House. House of YorkAs descendants of King Edward III in the male line, the first three Dukes of York bore the arms of that King (adjusted for France modern) differenced by a label of three points argent each bearing three torteaux gules. The 4th Duke, later King Edward IV, abandoned his paternal arms in favour of new arms emphasising his descent via female lines from the royal line of Clarence/de Burgh/Mortimer, senior to that...

Canker

Common name for a tree disease This article is about the plant disease. For the human condition, see canker sore. Not to be confused with Kanker. Butternut canker is a lethal disease of butternut trees, and has no cure. A plant canker is a small area of dead tissue, which grows slowly, often over years. Some cankers are of only minor consequence, but others are ultimately lethal and therefore can have major economic implications for agriculture and horticulture. Their causes include a wide ra...

Sungai Nakdong

Sungai Nakdong낙동강Nakdong di Andong, Gyeongsang UtaraSungai NakdongPeta OpenStreetMap Koordinat: 35°3′25.384″N 128°55′28.679″E / 35.05705111°N 128.92463306°E / 35.05705111; 128.92463306LokasiNegaraKorea SelatanProvinsiGangwon, Gyeongsang Utara, Daegu, Gyeongsang Selatan, BusanCiri-ciri fisikHulu sungaiPegunungan Taebaek - lokasiPuncak Eundae, Taebaek, Gangwon[a][1] Muara sungaiLaut Jepang - lokasiEstuary Bank, Gangse...

Шёст

КоммунаШёстCheust Герб 43°03′06″ с. ш. 0°00′59″ в. д.HGЯO Страна Франция Регион Юг — Пиренеи Департамент Верхние Пиренеи Кантон Лурд-Эст Мэр Поль Лаффай(2014—2020) История и география Площадь 3,06 км² Высота центра 532–862 м Часовой пояс UTC+1:00, летом UTC+2:00 Население Население...

Tayeb Salih

Sudanese novelist and short story writer (1929–2009) Tayeb SalihBorn12 July 1929Karmakol, SudanDied18 February 2009(2009-02-18) (aged 79)London, United KingdomOccupationNovelist, columnist, civil servantLanguageModern Standard ArabicAlma materUniversity of Khartoum, University of LondonLiterary movementPostcolonialismNotable worksSeason of Migration to the North, The Wedding of Zein Tayeb Salih (Arabic: الطيب صالح, romanized: aṭ-Ṭayyib Ṣāliḥ; 12 July 1929 �...

/profillengkap.com/index.php/article/Chest (mechanical engineering)

Katedral Saint-Lizier

Demonax longevittatus

Neu Wulmstorf

The Road

Bob Vanatta

1997 Green Bay Packers season

Ram Madhvani

La Ferrassie

Cyclops (Marvel Comics)

Grzegorz Kołtan

Eleatics