Nondisjunction is the failure of homologous chromosomes or sister chromatids to separate properly during cell division (mitosis/meiosis). There are three forms of nondisjunction: failure of a pair of homologous chromosomes to separate in meiosis I, failure of sister chromatids to separate during meiosis II, and failure of sister chromatids to separate during mitosis. Nondisjunction results in daughter cells with abnormal chromosome numbers (aneuploidy).

Calvin Bridges and Thomas Hunt Morgan are credited with discovering nondisjunction in Drosophila melanogaster sex chromosomes in the spring of 1910, while working in the Zoological Laboratory of Columbia University.

Types
In general, nondisjunction can occur in any form of cell division that involves ordered distribution of chromosomal material. Higher animals have three distinct forms of such cell divisions: Meiosis I and  meiosis II are specialized forms of cell division occurring during generation of gametes (eggs and sperm) for sexual reproduction,  mitosis is the form of cell division used by all other cells of the body.

Meiosis II
Ovulated eggs become arrested in metaphase II until fertilization triggers the second meiotic division. Similar to the segregation events of mitosis, the pairs of sister chromatids resulting from the separation of bivalents in meiosis I are further separated in anaphase of meiosis II. In oocytes, one sister chromatid is segregated into the second polar body, while the other stays inside the egg. During spermatogenesis, each meiotic division is symmetric such that each primary spermatocyte gives rise to 2 secondary spermatocytes after meiosis I, and eventually 4 spermatids after meiosis II. Meiosis II-nondisjunction may also result in aneuploidy syndromes, but only to a much smaller extent than do segregation failures in meiosis I.

Mitosis
Division of somatic cells through mitosis is preceded by replication of the genetic material in S phase. As a result, each chromosome consists of two sister chromatids held together at the centromere. In the anaphase of mitosis, sister chromatids separate and migrate to opposite cell poles before the cell divides. Nondisjunction during mitosis leads to one daughter receiving both sister chromatids of the affected chromosome while the other gets none. This is known as a chromatin bridge or an anaphase bridge. Mitotic nondisjunction results in somatic mosaicism, since only daughter cells originating from the cell where the nondisjunction event has occurred will have an abnormal number of chromosomes. Nondisjunction during mitosis can contribute to the development of some forms of cancer, e.g., retinoblastoma (see below). Chromosome nondisjunction in mitosis can be attributed to the inactivation of topoisomerase II, condensin, or separase. Meiotic nondisjunction has been well studied in Saccharomyces cerevisiae. This yeast undergoes mitosis similarly to other eukaryotes. Chromosome bridges occur when sister chromatids are held together post replication by DNA-DNA topological entanglement and the cohesion complex. During anaphase, cohesin is cleaved by separase. Topoisomerase II and condensin are responsible for removing catenations.

Molecular mechanisms

Central role of the spindle assembly checkpoint
The spindle assembly checkpoint (SAC) is a molecular safe-guarding mechanism that governs proper chromosome segregation in eukaryotic cells. SAC inhibits progression into anaphase until all homologous chromosomes (bivalents, or tetrads) are properly aligned to the spindle apparatus. Only then, SAC releases its inhibition of the anaphase promoting complex (APC), which in turn irreversibly triggers progression through anaphase.

Sex-specific differences in meiosis
Surveys of cases of human aneuploidy syndromes have shown that most of them are maternally derived. This raises the question: Why is female meiosis more error prone? The most obvious difference between female oogenesis and male spermatogenesis is the prolonged arrest of oocytes in late stages of prophase I for many years up to several decades. Male gametes on the other hand quickly go through all stages of meiosis I and II. Another important difference between male and female meiosis concerns the frequency of recombination between homologous chromosomes: In the male, almost all chromosome pairs are joined by at least one crossover, while more than 10% of human oocytes contain at least one bivalent without any crossover event. Failures of recombination or inappropriately located crossovers have been well documented as contributors to the occurrence of nondisjunction in humans.

Age-related loss of cohesin ties
Due to the prolonged arrest of human oocytes, weakening of cohesive ties holding together chromosomes and reduced activity of the SAC may contribute to maternal age-related errors in segregation control. The cohesin complex is responsible for keeping together sister chromatids and provides binding sites for spindle attachment. Cohesin is loaded onto newly replicated chromosomes in oogonia during fetal development. Mature oocytes have only limited capacity for reloading cohesin after completion of S phase. The prolonged arrest of human oocytes prior to completion of meiosis I may therefore result in considerable loss of cohesin over time. Loss of cohesin is assumed to contribute to incorrect microtubule-kinetochore attachment and chromosome segregation errors during meiotic divisions.

Consequences
The result of this error is a cell with an imbalance of chromosomes. Such a cell is said to be aneuploid. Loss of a single chromosome (2n-1), in which the daughter cell(s) with the defect will have one chromosome missing from one of its pairs, is referred to as a monosomy. Gaining a single chromosome, in which the daughter cell(s) with the defect will have one chromosome in addition to its pairs is referred to as a trisomy. In the event that an aneuploidic gamete is fertilized, a number of syndromes might result.

Monosomy
The only known survivable monosomy in humans is Turner syndrome, where the affected individual is monosomic for the X chromosome (see below). Other monosomies are usually lethal during early fetal development, and survival is only possible if not all the cells of the body are affected in case of a mosaicism (see below), or if the normal number  of chromosomes is restored via duplication of the single monosomic chromosome ("chromosome rescue").

Turner syndrome (X monosomy) (45, X0)

Complete loss of an entire X chromosome accounts for about half the cases of Turner syndrome. The importance of both X chromosomes during embryonic development is underscored by the observation that the overwhelming majority (>99%) of fetuses with only one X chromosome (karyotype 45, X0) are spontaneously aborted.

Autosomal trisomy
The term autosomal trisomy means that a chromosome other than the sex chromosomes X and Y is present in 3 copies instead of the normal number of 2 in diploid cells.

Down syndrome (trisomy 21)

Down syndrome, a trisomy of chromosome 21, is the most common anomaly of chromosome number in humans. The majority of cases result from nondisjunction during maternal meiosis I. Trisomy occurs in at least 0.3% of newborns and in nearly 25% of spontaneous abortions. It is the leading cause of pregnancy wastage and is the most common known cause of intellectual disability. It is well documented that advanced maternal age is associated with greater risk of meiotic nondisjunction leading to Down syndrome. This may be associated with the prolonged meiotic arrest of human oocytes potentially lasting for more than four decades.

Edwards syndrome (trisomy 18) and Patau syndrome (trisomy 13)
Human autosomal trisomies compatible with live birth, other than Down syndrome (trisomy 21), are Edwards syndrome (trisomy 18) and Patau syndrome (trisomy 13). Complete trisomies of other chromosomes are usually not viable and represent a relatively frequent cause of miscarriage. Only in rare cases of a mosaicism, the presence of a normal cell line, in addition to the trisomic cell line, may support the development of a viable trisomy of the other chromosomes.

Sex chromosome aneuploidy
The term sex chromosome aneuploidy summarizes conditions with an abnormal number of sex chromosomes, i.e., other than XX (female) or XY (male). Formally, X chromosome monosomy (Turner syndrome, see above) can also be classified as a form of sex chromosome aneuploidy.

Klinefelter syndrome (47, XXY)
Klinefelter syndrome is the most common sex chromosome aneuploidy in humans. It represents the most frequent cause of hypogonadism and infertility in men. Most cases are caused by nondisjunction errors in paternal meiosis I. About eighty percent of individuals with this syndrome have one extra X chromosome resulting in the karyotype XXY. The remaining cases have either multiple additional sex chromosomes (48,XXXY; 48,XXYY; 49,XXXXY), mosaicism (46,XY/47,XXY), or structural chromosome abnormalities.

XYY Male (47, XYY)
The incidence of XYY syndrome is approximately 1 in 800–1000 male births. Many cases remain undiagnosed because of their normal appearance and fertility, and the absence of severe symptoms. The extra Y chromosome is usually a result of nondisjunction during paternal meiosis II.

Trisomy X (47,XXX)
Trisomy X is a form of sex chromosome aneuploidy where females have three instead of two X chromosomes. Most patients are only mildly affected by neuropsychological and physical symptoms. Studies examining the origin of the extra X chromosome observed that about 58–63% of cases were caused by nondisjunction in maternal meiosis I, 16–18% by nondisjunction in maternal meiosis II, and the remaining cases by post-zygotic, i.e., mitotic, nondisjunction.

Uniparental disomy
Uniparental disomy denotes the situation where both chromosomes of a chromosome pair are inherited from the same parent and are therefore identical. This phenomenon most likely is the result of a pregnancy that started as a trisomy due to nondisjunction. Since most trisomies are lethal, the fetus only survives because it loses one of the three chromosomes and becomes disomic. Uniparental disomy of chromosome 15 is, for example, seen in some cases of Prader-Willi syndrome and Angelman syndrome.

Mosaicism syndromes
Mosaicism syndromes can be caused by mitotic nondisjunction in early fetal development. As a consequence, the organism evolves as a mixture of cell lines with differing ploidy (number of chromosomes). Mosaicism may be present in some tissues, but not in others. Affected individuals may have a patchy or asymmetric appearance. Examples of mosaicism syndromes include Pallister-Killian syndrome and Hypomelanosis of Ito.

Mosaicism in malignant transformation

Development of cancer often involves multiple alterations of the cellular genome (Knudson hypothesis). Human retinoblastoma is a well studied example of a cancer type where mitotic nondisjunction can contribute to malignant transformation: Mutations of the RB1 gene, which is located on chromosome 13 and encodes the tumor suppressor retinoblastoma protein, can be detected by cytogenetic analysis in many cases of retinoblastoma. Mutations of the RB1 locus in one copy of chromosome 13 are sometimes accompanied by loss of the other wild-type chromosome 13 through mitotic nondisjunction. By this combination of lesions, affected cells completely lose expression of functioning tumor suppressor protein.

Diagnosis

Preimplantation genetic diagnosis
Pre-implantation genetic diagnosis (PGD or PIGD) is a technique used to identify genetically normal embryos and is useful for couples who have a family history of genetic disorders. This is an option for people choosing to procreate through IVF. PGD is considered difficult due to it being both time consuming and having success rates only comparable to routine IVF.

Karyotyping
Karyotyping involves performing an amniocentesis in order to study the cells of an unborn fetus during metaphase 1. Light microscopy can be used to visually determine if aneuploidy is an issue.

Polar body diagnosis
Polar body diagnosis (PBD) can be used to detect maternally derived chromosomal aneuploidies as well as translocations in oocytes. The advantage of PBD over PGD is that it can be accomplished in a short amount of time. This is accomplished through zona drilling or laser drilling.

Blastomere biopsy
Blastomere biopsy is a technique in which blastomeres are removed from the zona pellucida.  It is commonly used to detect aneuploidy. Genetic analysis is conducted once the procedure is complete. Additional studies are needed to assess the risk associated with the procedure.

Lifestyle/environmental hazards
Exposure of spermatozoa to lifestyle, environmental and/or occupational hazards may increase the risk of aneuploidy. Cigarette smoke is a known aneugen (aneuploidy inducing agent). It is associated with increases in aneuploidy ranging from 1.5 to 3.0-fold. Other studies indicate factors such as alcohol consumption, occupational exposure to benzene, and exposure to the insecticides fenvalerate and carbaryl also increase aneuploidy.

References

Genetics

Nondisjunction


Maskelyne este un crater lunar solitar de impact situat în partea de sud–est a Mării Liniștii, având un diametru de 22 km. A fost numit după astronomul britanic Nevil Maskelyne. Rama exterioară are o formă oarecum poligonală, deși este în general circulară. Pereții interiori sunt terasați iar fundul craterului prezintă o excrescență situată central.

Locul de aselenizare al expediției Apollo 11 se află situat la aproximativ 250 de kilometri spre vest–sud–vest. În nord–est sunt craterele Wallach și Aryabhata, în sud–est se află craterul Censorinus, iar în partea de sud sunt situați munții lunari cunoscuți sub denumirile informale de Duke Island și Boot Hill. Depresiunile sinuoase aflate la sud–vest și la vest de Maskelyne au fost denumite informal de către echipajul Apollo 10 Sidewinder, respectiv Diamondback. Acestea au fost menționate ca atare de către misiunile ulterioare, în special Apollo 11.

Craterele satelite

Prin convenție, craterele vecine sunt identificate pe hărțile lunare prin plasarea de litere alături de numele craterului central, în acest caz Maskelyne.

Următoarele cratere au fost redenumite de către  Uniunea Astronomică Internațională:

 Maskelyne E — Aryabhata
 Maskelyne H — Wallach

Note

Bibliografie

Legături externe

 LTO-61D4 Maskelyne Orientalis, Lunar Topographic Orthophotomap (LTO) Series
 Location of Maskelyne Crater in Google Moon
Cratere pe Lună

Maskelyne (crater lunar)


Bonillos es una localidad del municipio de Brazuelo, en la provincia de León, Comunidad Autónoma de Castilla y León (España).

Situación
Se encuentra al O de Brimeda; al SO de Magaz de Cepeda, al SE de Requejo de Pradorrey y al NO de Astorga.

Evolución demográfica

Localidades de La Maragatería
Localidades de Brazuelo
Poblaciones del Camino de Santiago por Manzanal

Bonillos


est un shōnen manga écrit et dessiné par Kentarō Yabuki. Il est prépublié entre le  et le  dans le Weekly Shōnen Jump. Sa sérialisation a ensuite été transférée sur la plateforme numérique Shōnen Jump+ entre le 25 avril 2022 et le 25 septembre 2023. Le manga est publié en volumes reliés par l'éditeur japonais Shūeisha. La version française est éditée par Delcourt/Tonkam depuis le . La diffusion d'une adaptation en série télévisée animée produite par le studio d'animation Connect est diffusée depuis .

Synopsis
Le Japon regorge de mystérieux monstres appelés Ayakashi. Pour les exorciser et contrer la menace une organisation ninja s'est formée. Matsuri est un jeune ninja exorciste qui affronte régulièrement ces créatures et protège son amie d'enfance Suzu, qui possède elle aussi la capacité de les voir. Mais lorsqu'elle s'attache à un chat Ayakashi nommé Shirogane, les choses basculent.

Personnages

C'est un jeune ninja exorciste appartenant au clan Kazamaki. Il s'entraîne pour prendre la place de son grand-père à la tête du clan. Il est insouciant mais responsable. Dans sa jeunesse, il avait peur des ayakashi.

Suzu est une jeune fille capable de voir et d'attirer les ayakashi. Elle fait confiance aux ayakashi mais se retrouve souvent dans des situations dangereuses.

Il s'agit du roi des Ayakashi. Sa véritable force est un gigantesque félin mais il apparaît souvent sous la forme d'un chat obèse vêtu d'un bavoir. Il accompagne Matsuri, il est jaloux des sentiments de Matsuri envers Suzu.

Manga
Ayakashi Triangle est dessiné par Kentarō Yabuki. La série débute sa publication dans le numéro 28 du magazine Weekly Shōnen Jump de l'éditeur Shūeisha paru le  et se termine dans le  numéro  du Weekly Shōnen Jump avant d'être transféré sur la plateforme Shōnen Jump+ à partir du . L'éditeur Shūeisha publie les chapitres en  tankōbon avec un premier volume sorti le . Quinze volumes sont sortis au . La version française est publiée par Delcourt/Tonkam avec un premier volume sorti le .

Liste des volumes

|}

Anime
Une adaptation en anime est annoncée le  durant la Jump Festa 22. La série est produite par Connect et réalisée par Noriaki Akitaya, avec l'assistant Kei Umabiki, les scénarios sont écrits par Shogo Yasukawa, le design des personnages est assuré par Hideki Furukawa, et la musique est composée par Rei Ishizuka. La diffusion commence le  mais est interrompue sine die après le 4 épisode en raison des effets de l'épidémie de Covid-19 sur le calendrier de production. Les épisodes 5 et 6 sont diffusés en mars, avant une nouvelle pause, et un redémarrage de la diffusion en juillet.

Liste des épisodes

Produits dérivés

Vomic
Une adaptation vomic (voice comic) dAyakashi Triangle débute le 20 novembre 2020 avec des épisodes publiés sur la chaîne YouTube de Jump Comics. Le vomic montre les planches du manga qui apparaissent à l'écran tandis que les acteurs vocaux, la musique et les effets sonores sont joués. Les voix de Matsuri Kazamaki sont réalisés par Hiromu Mineta pour la voix masculine et Yūki Kyōka pour la voix féminine. Suzu Kanade est doublée par Saya Aizawa. Shirogane est doublé par Mitsuteru Nagato (forme de chat) et par Hikaru Fujikura (forme Ayakashi), ainsi que d'autres voix.

Accueil

Ventes
Au Japon, le premier volume dAyakashi Triangle a débuté à la 10e place du classement Shoseki avec plus de 23 350 exemplaires vendus le 2 octobre 2020. Il est ensuite tombé au 20e rang, puis au 35e rang le 4 octobre. Au 13 octobre 2020, Ayakashi Triangle s'est vendu à plus de , dépassant les ventes des volumes de Mashle et de Undead Unluck. Au 24 décembre 2021, le manga s'est vendu à plus d'un million d'exemplaires à travers le monde.

Critiques
Après avoir lu les deux premiers chapitres dAyakashi Triangle, Shawn Hacaga, de The Fandom Post, a qualifié l'histoire de « décente » mais de « rien de spécial ». En plus de faire l'éloge de l'art de Yabuki, qu'il qualifie de « magnifique » et de « fantastique », Hacaga a également déclaré que l'histoire était néanmoins amusant et a admis avoir été surpris par l'inversion des genres à la fin du premier chapitre. Dans sa critique pour UT Daily Beacon, Connor Holt a donné au manga une note de 3/5, en disant quAyakashi Triangle est moyen quand il est considéré comme un manga ecchi, mais échoue quand il est considéré comme une comédie romantique.

Jacob Parker-Dalton dOTAQUEST a fait l'éloge du style artistique de Yabuki, en plus de comparer Ayakashi Triangle à Rosario + Vampire d'Akihisa Ikeda : « bête, excité, mais définitivement conçu avec soin et attention ». Dalton a également exprimé son opinion sur la façon dont Ayakashi Triangle ressemble au premier volet d'un manga du milieu des années 2000, puisque la série, selon lui, « réutilise plusieurs éléments typiques de cette période : les démons et les yōkai, pour commencer, mais aussi l'objectivation éhontée, le gender bending, et même une mascotte animale mignonne ». Dans une critique ultérieure, Dalton a souligné la « qualité » de l'écriture dAyakashi Triangle, la comparant à JoJo's Bizarre Adventure de Hirohiko Araki en déclarant que « l'un des derniers chapitres de la série, en particulier, était suffisamment bien écrit pour justifier une comparaison ». Dans une troisième critique dAyakashi Triangle, Dalton a fait une remarque positive sur le développement de l'intrigue de la série en disant qu'il s'agit d'une « histoire bien construite avec de nombreux moments d'émotion », et que le manga « s'avère rapidement beaucoup plus compétent qu'il ne semblait l'être au départ ».

Notes et références

Notes

Références

Œuvres

Édition japonaise

Édition française

Liens externes
  sur Weekly Shōnen Jump

Série manga
Manga paru dans le Weekly Shōnen Jump
Manga paru dans le Shōnen Jump+
Manga des années 2020
Animation réalisée par Connect
Série diffusée sur Crunchyroll
Anime reporté en raison de la pandémie de Covid-19

Ayakashi Triangle


Policijska akademija je filmska franšiza koja je počela 1984. s filmom Policijska akademija (1984.) (Police Academy). Malo je poznato da je zamisao za film došla kad je scenarist na snimanju jednog drugog filma vidio stvarne policijske novake karikaturne pojave, unovačenih po obrascu koji je spomenut u prvom nastavku, da se primi širi krug ljdui. Prvi film iz serijala postigao je veliki uspjeh, što je stvorilo zahtjeve za nastavkom koji je brzo proizveden. Njegovi nastavci također su bili uspješni, pa su gotovo svaku godinu izlazili novi nastavci. 1985. godine u kina je došla Policijska akademija 2 (Police Academy 2: Their First Assignment), koji je također postigao financijski uspjeh, premda ne kao prvi. 1986. je godine izašla Policijska akademija 3 (Police Academy 3: Back in Training). Četvrti nastavak bio je Policijska akademija 4 (Police Academy 4: Citizens on Patrol) objavljen 1987. godine. Peti nastavak bio je 1988. godine, Policijska akademija 5 (Police Academy 5: Assignment Miami Beach), u kojem su se "maknuli" iz svog grada u floridsko okružje. Šesti nastavak bio je 1989. godine, Policijska akademija 6 (Police Academy 6: City Under Siege). "Međunarodno" su pošli 1994., u Rusiju, koja je izabrana u duhu sveopće liberalizacije i otopljavanja odnosa dviju supersila: Policijska akademija: Misija Moskva (Police Academy: Mission to Moscow). Kroz cijeli serijal provlači se desetak stalnih protagonista i antagonista, a u nastavcima su se pojavili likovi koji su postali dijelom stalne postave junaka serije.

Iznjedrila je i dvije serije, jednu igranu, Policijska akademija (televizijska serija) (Police Academy: The Series, 1987. — 1988.), koja je doživjela samo jednu sezonu i u kojoj je samo jedan glumac bio izvorni, dok su ostali likovi bili u nekakvom srodstvu ili prijateljstvu s izvornim likovima iz filma. Animirana serija Policijska akademija (animirana serija) (Police Academy: The Animated Series, 1989. — 1991.) imala je više epizoda.

Osmi nastavak iz serijala je u razvoju od 2003. godine. Izrada predugo traje, zastala je u samim prvim dionicama 2003., pa 2012. godine. Glumac Steve Guttenberg izjavio je da se sprema osmi nastavak, ali nije iznosio detalje.

Izvori

Američki filmovi
Multimedijske franšize

Policijska akademija (franšiza)


Mezoterapia (de la Mesos greacă, "mijloc", și terapie de la therapeia greacă, "pentru a trata medical") este un tratament cosmetic non chirurgical. Mezoterapia utilizează injectări multiple de medicamente farmaceutice și homeopate, extracte de plante, vitamine și alte ingrediente în grăsimea subcutanată. Mezoterapia prin injectări se presupune că vizează celulele adipoase grase, aparent prin inducerea lipolizei, a rupturii și a morții celulelor în rândul adipocitelor .

Utilizare
Există studii publicate privind tratamentele clinice și efectele acestor medicații și numeroase cocktailuri de compuși chimici combinați pe corp au fost raportate în Europa și America de Sud de mai mulți ani. Nu există o dovadă concludentă a cercetării că acești compuși chimici au în mod specific ca țintă țesutul adipos (celulele grase). Liza celulelor, care rezultă din acțiunea detergentă a deoxicolicului, poate determina orice efect clinic.

În anul 2012, un laborator francez a inventat o modalitate de a introduce un tratament  mezoterapic într-un săculeț lichid. Acest săculeț este apoi conectat la un abur facial care aplică tratamentul porilor faciali ai utilizatorului prin abur. Aceasta a fost prima invenție de acest gen care permitea tratamentele mezoterapic direct acasă.

Substanțele utilizate includ:
Phosphatidylcholine
T3-T4 thyroid,
Isoproterenol
Aminophylline
Pentoxifylline
L-carnitine
L-arginine
Hyaluronidase
Collagenase
Yohimbine
Co-enzyme cofactors
Dimethylethanolamine
Gerovital
Glutathione
Tretinoin
Alpha lipoic acid
Vitamin C
Procaine
Lidocaine
Ginkgo biloba
Melilotus
C-adenosine monophosphate
Multiple vitamins
Trace mineral elements
Carbon dioxide
Mesoglycan

Istoric
Michel Pistor (1924-2003) a efectuat cercetări clinice și a fondat domeniul mezoterapiei.  Cercetarea multinațională în terapia intradermică a culminat cu munca lui Pistor din 1948 până în 1952 în tratamentele mezoterapeutice umane. Presa franceză a inventat termenul de mezoterapie în 1958.  Academia Națională de Medicină din Franța a recunoscut mezoterapia ca specialitate de medicină în 1987. Societatea franceză de mezoterapie recunoaște folosirea acesteia ca tratament pentru diferite condiții, dar nu menționează utilizarea acesteia în chirurgia plastică.
Populară în țările europene și America de Sud, mezoterapia este practicată de aproximativ 18.000 de medici din întreaga lume.

Critică
Tratamentele mezoterapice au fost efectuate pe întreg teritoriul Europei, Americii de Sud și mai recent în Statele Unite, de peste cincizeci de ani. Cu toate acestea, medicii și-au exprimat îngrijorarea cu privire la eficacitatea mezoterapiei, argumentând că tratamentul nu a fost studiat suficient pentru a face o determinare. Problema primară este că mezoterapia pentru tratamentul stărilor cosmetice nu a făcut obiectul studiilor clinice standard aur; totuși, procedura a fost studiată pentru ameliorarea durerii în alte afecțiuni, cum ar fi tendinita, calcifierea tendonului, procedurile dentare, cancerul, cervicobrahialgia, artrita, limfedemul și staza venoasă.
Mai mult, au existat serii de cazuri și numeroase lucrări medicale privind mezoterapia ca tratament cosmetic, precum și studii care utilizează ingredientele utilizate în mezoterapie.

Cealaltă parte a dezbaterii este exprimată de Rod Rohrich, președintele Departamentului de Chirurgie Plastică al Southwestern Medical Center al Universității din Dallas, Texas: "Nu există pur și simplu date, din câte știu, că mezoterapia ar avea vreun efect."  Societatea Americana de Chirurgie Plastică a emis o declarație de poziție care nu aprobă mezoterapia, dar această delimitare este obiectul unor controverse. Deoarece mezoterapia nu este un tratament chirurgical, ci mai degrabă o alternativă non invazivă la chirurgia plastică, tratamentul concurează cu chirurgia plastică pentru aceiași pacienți.

În urma reacțiilor adverse observate la câțiva pacienți ai unui medic francez, a fost publicată o ratificare oficială în Franța în aprilie 2011 pentru a interzice mezoterapia ca metodă de eliminare a depozitelor de grăsime. Această interdicție a fost anulată în iunie 2011 de către Consiliul de Stat al Franței, deoarece ancheta a demonstrat că aceste efecte nedorite nu se datorează mezoterapiei însăși, ci faptului că a fost practicată în condiții proaste și fără respectarea principiilor de igienă.

Studii clinice
Într-un studiu prospectiv, 10 pacienți au suferit patru sesiuni de mezoterapie facială folosind multivitamine la intervale lunare.  Acest studiu a constatat că nu a existat nici un beneficiu relevant din punct de vedere clinic.

Acidul deoxiolic a primit aprobarea FDA ca injectabil pentru a dizolva grăsimea submentală în iunie 2015. Aceasta s-a bazat pe un studiu randomizat, de fază III, pe 2600 de pacienți. 68,2% au prezentat un răspuns prin măsurarea depozitului de grăsime. 81% au prezentat reacții adverse ușoare de vânătăi, umflături, durere, amorțeală, eritem și rigidizare în jurul zonei tratate.

Legături externe
 Consiliul American de mezoterapie estetică
 Societatea franceză de mezoterapie
 Consiliul profesional al mesoterapiei Atlanta

Referințe

Pseudoștiință
Cosmetică

Mezoterapie


Karl Emanuel 2. af Savoyen (italiensk: Carlo Emanuele II di Savoia (født 20. juni 1634 i Torino, død 12. juni 1675 samme sted) var hertug af Savoyen fra 1638 til 1675.

Liv og gerning
Han blev født i Torino som søn af Viktor Amadeus 1. af Savoyen og Christine af Frankrig. Hans morforældre var Henrik IV af Frankrig og hans anden kone Marie de Medici. I 1638, efter hans broders Francis Hyacinths død, lykkedes det Karl Emanuel at blive udpeget til Savoyens hertugdommes regent i en alder af 4 år. Hans moder regerede i hans sted, og selv efter at han var kommet i voksenalderen i 1648, inviterede han hende til at fortsætte at regere. Karl Emanuel fortsatte et liv med fornøjelser langt væk fra statens anliggender.

Han blev berygtet forsin forfølgelse af vaudoiserne (Waldenserne), der kulminerede i massakren i 1655. Massakren var så brutal, at den fik den engelske digter John Milton til at skrive sonet Om den sene massakre i Piemonte. Oliver Cromwell, Lord Protector, opfordrede til at man handlede hurtig i England og foreslog at sende den britiske flåde, hvis massakren ikke blev stoppet, mens der blev indsamlet midler til at hjælpe Waldenserne. Sir Samuel Morland blev bestilt af denne opgave. Han skrev senere Historien om de evangeliske kirker i Piedont dalene (1658).

Kun efter sin moders død i 1663 fik han virkelig magten. Han havde ikke held med at få en passage til havet på bekostning af Genova (anden krig mellem Geneve og Savoyen, 1672-1673), og havde vanskeligheder med at skærme mod indflydelsen fra sin magtfulde nabo Frankrig.

Men han forbedrede i høj grad handel og rigdom i hertugdømmet, der udviklede Nice havn og byggede en vej gennem Alperne mod Frankrig. Han reformerede også hæren, som indtil da hovedsagelig var sammensat af lejesoldater: han dannede i stedet fem piemontanske regimenter og genskabte kavaleriet, samt introducerede uniformer. Han restaurerede også befæstninger. Han konstruerede mange smukke bygninger i Torino, for eksempel Palazzo Reale.

Han døde 12. juni 1675 og efterlod sin anden kone som regent for sin søn. Han er begravet i katedralen i Torino.

Noter

Litteratur
(alleanza monarchica, Italian)

Kongelige fra Italien
Huset Savoyen
Hertuger af Savoyen

Karl Emanuel 2. af Savoyen


Die Liste der denkmalgeschützten Objekte in Veľké Chyndice enthält die zwei nach slowakischen Denkmalschutzvorschriften geschützten Objekte in der Gemeinde Veľké Chyndice im Okres Nitra.

Denkmäler

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Velke Chyndice


In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice.

John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined class and set. Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis.

Classes in set theory

The uses of classes
Classes have several uses in NBG:
 They produce a finite axiomatization of set theory.
 They are used to state a "very strong form of the axiom of choice"—namely, the axiom of global choice: There exists a global choice function  defined on the class of all nonempty sets such that  for every nonempty set  This is stronger than ZFC's axiom of choice: For every set  of nonempty sets, there exists a choice function  defined on  such that  for all
 The set-theoretic paradoxes are handled by recognizing that some classes cannot be sets. For example, assume that the class  of all ordinals is a set. Then  is a transitive set well-ordered by . So, by definition,  is an ordinal. Hence, , which contradicts  being a well-ordering of  Therefore,  is not a set. A class that is not a set is called a proper class,  is a proper class.
 Proper classes are useful in constructions. In his proof of the relative consistency of the axiom of global choice and the generalized continuum hypothesis, Gödel used proper classes to build the constructible universe. He constructed a function on the class of all ordinals that, for each ordinal, builds a constructible set by applying a set-building operation to previously constructed sets. The constructible universe is the image of this function.

Axiom schema versus class existence theorem
Once classes are added to the language of ZFC, it is easy to transform ZFC into a set theory with classes. First, the axiom schema of class comprehension is added. This axiom schema states: For every formula  that quantifies only over sets, there exists a class  consisting of the  satisfying the formula—that is,  Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified to handle classes: If two classes have the same elements, then they are identical. The other axioms of ZFC are not modified.

This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.

To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms. Then these axioms are used to prove the class existence theorem, which implies every instance of the axiom schema. The proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula.

Axiomatization of NBG

Classes and sets
NBG has two types of objects: classes and sets. Intuitively, every set is also a class. There are two ways to axiomatize this. Bernays used many-sorted logic with two sorts: classes and sets. Gödel avoided sorts by introducing primitive predicates:  for " is a class" and  for " is a set" (in German, "set" is Menge). He also introduced axioms stating that every set is a class and that if class  is a member of a class, then  is a set. Using predicates is the standard way to eliminate sorts. Elliott Mendelson modified Gödel's approach by having everything be a class and defining the set predicate  as  This modification eliminates Gödel's class predicate and his two axioms.

Bernays' two-sorted approach may appear more natural at first, but it creates a more complex theory. In Bernays' theory, every set has two representations: one as a set and the other as a class. Also, there are two membership relations: the first, denoted by "∈", is between two sets; the second, denoted by "η", is between a set and a class. This redundancy is required by many-sorted logic because variables of different sorts range over disjoint subdomains of the domain of discourse.

The differences between these two approaches do not affect what can be proved, but they do affect how statements are written. In Gödel's approach,  where  and  are classes is a valid statement. In Bernays' approach this statement has no meaning. However, if  is a set, there is an equivalent statement: Define "set  represents class " if they have the same sets as members—that is,  The statement  where set  represents class  is equivalent to Gödel's

The approach adopted in this article is that of Gödel with Mendelson's modification. This means that NBG is an axiomatic system in first-order predicate logic with equality, and its only primitive notions are class and the membership relation.

Definitions and axioms of extensionality and pairing
A set is a class that belongs to at least one class:  is a set if and only if .
A class that is not a set is called a proper class:  is a proper class if and only if .
Therefore, every class is either a set or a proper class, and no class is both.

Gödel introduced the convention that uppercase variables range over classes, while lowercase variables range over sets. Gödel also used names that begin with an uppercase letter to denote particular classes, including functions and relations defined on the class of all sets. Gödel's convention is used in this article. It allows us to write:

  instead of
  instead of

The following axioms and definitions are needed for the proof of the class existence theorem.

Axiom of extensionality. If two classes have the same elements, then they are identical.

This axiom generalizes ZFC's axiom of extensionality to classes.

Axiom of pairing. If  and  are sets, then there exists a set  whose only members are  and .

As in ZFC, the axiom of extensionality implies the uniqueness of the set , which allows us to introduce the notation

Ordered pairs are defined by:

Tuples are defined inductively using ordered pairs:

Class existence axioms and axiom of regularity
Class existence axioms will be used to prove the class existence theorem: For every formula in  free set variables that quantifies only over sets, there exists a class of  that satisfy it. The following example starts with two classes that are functions and builds a composite function. This example illustrates the techniques that are needed to prove the class existence theorem, which lead to the class existence axioms that are needed.

{| class="wikitable"
|- style="text-align: left; vertical-align: top; style="background: white"
|Example 1: If the classes  and  are functions, then the composite function  is defined by the formula:  Since this formula has two free set variables,  and  the class existence theorem constructs the class of ordered pairs:

Because this formula is built from simpler formulas using conjunction  and existential quantification , class operations are needed that take classes representing the simpler formulas and produce classes representing the formulas with  and . To produce a class representing a formula with , intersection used since  To produce a class representing a formula with , the domain is used since

Before taking the intersection, the tuples in  and  need an extra component so they have the same variables. The component  is added to the tuples of  and  is added to the tuples of :
 and
In the definition of  the variable  is not restricted by the statement  so  ranges over the class  of all sets. Similarly, in the definition of  the variable  ranges over  So an axiom is needed that adds an extra component (whose values range over ) to the tuples of a given class.

Next, the variables are put in the same order to prepare for the intersection:
 and
To go from  to  and from  to  requires two different permutations, so axioms that support permutations of tuple components are needed.

The intersection of  and  handles :

Since  is defined as , taking the domain of  handles  and produces the composite function:

So axioms of intersection and domain are needed.
|}

The class existence axioms are divided into two groups: axioms handling language primitives and axioms handling tuples. There are four axioms in the first group and three axioms in the second group.

Axioms for handling language primitives:

Membership. There exists a class  containing all the ordered pairs whose first component is a member of the second component.

Intersection (conjunction). For any two classes  and , there is a class  consisting precisely of the sets that belong to both  and .

Complement (negation). For any class , there is a class  consisting precisely of the sets not belonging to .

Domain (existential quantifier). For any class , there is a class  consisting precisely of the first components of the ordered pairs of .

By the axiom of extensionality, class  in the intersection axiom and class  in the complement and domain axioms are unique. They will be denoted by:   and  respectively. On the other hand, extensionality is not applicable to  in the membership axiom since it specifies only those sets in  that are ordered pairs.

The first three axioms imply the existence of the empty class and the class of all sets: The membership axiom implies the existence of a class  The intersection and complement axioms imply the existence of , which is empty. By the axiom of extensionality, this class is unique; it is denoted by  The complement of  is the class  of all sets, which is also unique by extensionality. The set predicate , which was defined as , is now redefined as  to avoid quantifying over classes.

Axioms for handling tuples:

Product by . For any class , there is a class  consisting of the ordered pairs whose first component belongs to .

Circular permutation. For any class , there is a class  whose 3tuples are obtained by applying the circular permutation  to the 3tuples of .

Transposition. For any class , there is a class  whose 3tuples are obtained by transposing the last two components of the 3tuples of .

By extensionality, the product by  axiom implies the existence of a unique class, which is denoted by  This axiom is used to define the class  of all :  and  If  is a class, extensionality implies that  is the unique class consisting of the  of  For example, the membership axiom produces a class  that may contain elements that are not ordered pairs, while the intersection  contains only the ordered pairs of .

The circular permutation and transposition axioms do not imply the existence of unique classes because they specify only the 3tuples of class  By specifying the 3tuples, these axioms also specify the  for  since:  The axioms for handling tuples and the domain axiom imply the following lemma, which is used in the proof of the class existence theorem.

One more axiom is needed to prove the class existence theorem: the axiom of regularity. Since the existence of the empty class has been proved, the usual statement of this axiom is given.

Axiom of regularity. Every nonempty set has at least one element with which it has no element in common.

This axiom implies that a set cannot belong to itself: Assume that  and let  Then  since  This contradicts the axiom of regularity because  is the only element in  Therefore,  The axiom of regularity also prohibits infinite descending membership sequences of sets:

Gödel stated regularity for classes rather than for sets in his 1940 monograph, which was based on lectures given in 1938. In 1939, he proved that regularity for sets implies regularity for classes.

Class existence theorem

The theorem's proof will be done in two steps:
 Transformation rules are used to transform the given formula  into an equivalent formula that simplifies the inductive part of the proof. For example, the only logical symbols in the transformed formula are , , and , so the induction handles logical symbols with just three cases.
 The class existence theorem is proved inductively for transformed formulas. Guided by the structure of the transformed formula, the class existence axioms are used to produce the unique class of  satisfying the formula.

Transformation rules. In rules 1 and 2 below,  and  denote set or class variables. These two rules eliminate all occurrences of class variables before an  and all occurrences of equality. Each time rule 1 or 2 is applied to a subformula,  is chosen so that  differs from the other variables in the current formula. The three rules are repeated until there are no subformulas to which they can be applied. This produces a formula that is built only with , , , , set variables, and class variables  where  does not appear before an .

  is transformed into
 Extensionality is used to transform  into
 Logical identities are used to transform subformulas containing  and  to subformulas that only use  and

Transformation rules: bound variables. Consider the composite function formula of example 1 with its free set variables replaced by  and :  The inductive proof will remove , which produces the formula  However, since the class existence theorem is stated for subscripted variables, this formula does not have the form expected by the induction hypothesis. This problem is solved by replacing the variable  with  Bound variables within nested quantifiers are handled by increasing the subscript by one for each successive quantifier. This leads to rule 4, which must be applied after the other rules since rules 1 and 2 produce quantified variables.

 If a formula contains no free set variables other than  then bound variables that are nested within  quantifiers are replaced with  These variables have (quantifier) nesting depth .

{| class="wikitable"
|- style="text-align: left; vertical-align: top; style="background: white"
|Example 2: Rule 4 is applied to the formula  that defines the class consisting of all sets of the form  That is, sets that contain at least  and a set containing  — for example,  where  and  are sets.

Since  is the only free variable,  The quantified variable  appears twice in  at nesting depth 2. Its subscript is 3 because  If two quantifier scopes are at the same nesting depth, they are either identical or disjoint. The two occurrences of  are in disjoint quantifier scopes, so they do not interact with each other.
|}

Proof of the class existence theorem. The proof starts by applying the transformation rules to the given formula to produce a transformed formula. Since this formula is equivalent to the given formula, the proof is completed by proving the class existence theorem for transformed formulas.

Gödel pointed out that the class existence theorem "is a metatheorem, that is, a theorem about the system [NBG], not in the system …" It is a theorem about NBG because it is proved in the metatheory by induction on NBG formulas. Also, its proof—instead of invoking finitely many NBG axioms—inductively describes how to use NBG axioms to construct a class satisfying a given formula. For every formula, this description can be turned into a constructive existence proof that is in NBG. Therefore, this metatheorem can generate the NBG proofs that replace
uses of NBG's class existence theorem.

A recursive computer program succinctly captures the construction of a class from a given formula. The definition of this program does not depend on the proof of the class existence theorem. However, the proof is needed to prove that the class constructed by the program satisfies the given formula and is built using the axioms. This program is written in pseudocode that uses a Pascal-style case statement.

Let  be the formula of example 2. The function call  generates the class  which is compared below with  This shows that the construction of the class  mirrors the construction of its defining formula

Extending the class existence theorem
Gödel extended the class existence theorem to formulas  containing relations over classes (such as  and the unary relation ), special classes (such as ), and operations (such as  and ). To extend the class existence theorem, the formulas defining relations, special classes, and operations must quantify only over sets. Then  can be transformed into an equivalent formula satisfying the hypothesis of the class existence theorem.

The following definitions specify how formulas define relations, special classes, and operations:
 A relation  is defined by:
 A special class  is defined by:
 An operation  is defined by:

A  is defined by:
 Variables and special classes are terms.
 If  is an operation with  arguments and  are terms, then  is a term.

The following transformation rules eliminate relations, special classes, and operations. Each time rule 2b, 3b, or 4 is applied to a subformula,  is chosen so that  differs from the other variables in the current formula. The rules are repeated until there are no subformulas to which they can be applied.  and  denote terms.

 A relation  is replaced by its defining formula
 Let  be the defining formula for the special class
 Let  be the defining formula for the operation
 Extensionality is used to transform  into

{| class="wikitable"
|- style="text-align: left; vertical-align: top; style="background: white"
|Example 3: Transforming

|}

{| class="wikitable"
|- style="text-align: left; vertical-align: top; style="background: white"
|Example 4: Transforming

This example illustrates how the transformation rules work together to eliminate an operation.
|}

Set axioms
The axioms of pairing and regularity, which were needed for the proof of the class existence theorem, have been given above. NBG contains four other set axioms. Three of these axioms deal with class operations being applied to sets.

Definition.  is a function if

In set theory, the definition of a function does not require specifying the domain or codomain of the function (see Function (set theory)). NBG's definition of function generalizes ZFC's definition from a set of ordered pairs to a class of ordered pairs.

ZFC's definitions of the set operations of image, union, and power set are also generalized to class operations. The image of class  under the function  is  This definition does not require that  The union of class  is  The power class of  is  The extended version of the class existence theorem implies the existence of these classes. The axioms of replacement, union, and power set imply that when these operations are applied to sets, they produce sets.

Axiom of replacement. If  is a function and  is a set, then , the image of  under , is a set.

Not having the requirement  in the definition of  produces a stronger axiom of replacement, which is used in the following proof.

Axiom of union. If  is a set, then there is a set containing

Axiom of power set. If  is a set, then there is a set containing

Axiom of infinity. There exists a nonempty set  such that for all  in , there exists a  in  such that  is a proper subset of .

The axioms of infinity and replacement prove the existence of the empty set. In the discussion of the class existence axioms, the existence of the empty class  was proved. We now prove that  is a set. Let function  and let  be the set given by the axiom of infinity. By replacement, the image of  under , which equals , is a set.

NBG's axiom of infinity is implied by ZFC's axiom of infinity:  The first conjunct of ZFC's axiom, , implies the first conjunct of NBG's axiom. The second conjunct of ZFC's axiom, , implies the second conjunct of NBG's axiom since  To prove ZFC's axiom of infinity from NBG's axiom of infinity requires some of the other NBG axioms (see Weak axiom of infinity).

Axiom of global choice
The class concept allows NBG to have a stronger axiom of choice than ZFC. A choice function is a function  defined on a set  of nonempty sets such that  for all  ZFC's axiom of choice states that there exists a choice function for every set of nonempty sets. A global choice function is a function  defined on the class of all nonempty sets such that  for every nonempty set  The axiom of global choice states that there exists a global choice function. This axiom implies ZFC's axiom of choice since for every set  of nonempty sets,  (the restriction of  to ) is a choice function for  In 1964, William B. Easton proved that global choice is stronger than the axiom of choice by using forcing to construct a model that satisfies the axiom of choice and all the axioms of NBG except the axiom of global choice. The axiom of global choice is equivalent to every class having a well-ordering, while ZFC's axiom of choice is equivalent to every set having a well-ordering.

Axiom of global choice. There exists a function that chooses an element from every nonempty set.

History

Von Neumann's 1925 axiom system
Von Neumann published an introductory article on his axiom system in 1925. In 1928, he provided a detailed treatment of his system. Von Neumann based his axiom system on two domains of primitive objects: functions and arguments. These domains overlap—objects that are in both domains are called argument-functions. Functions correspond to classes in NBG, and argument-functions correspond to sets. Von Neumann's primitive operation is function application, denoted by [a, x] rather than a(x) where a is a function and x is an argument. This operation produces an argument. Von Neumann defined classes and sets using functions and argument-functions that take only two values, A and B. He defined x ∈ a if [a, x] ≠ A.

Von Neumann's work in set theory was influenced by Georg Cantor's articles, Ernst Zermelo's 1908 axioms for set theory, and the 1922 critiques of Zermelo's set theory that were given independently by Abraham Fraenkel and Thoralf Skolem. Both Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of natural numbers and Zn+1 is the power set of Zn. They then introduced the axiom of replacement, which would guarantee the existence of such sets. However, they were reluctant to adopt this axiom: Fraenkel stated "that Replacement was too strong an axiom for 'general set theory'", while "Skolem only wrote that 'we could introduce' Replacement".

Von Neumann worked on the problems of Zermelo set theory and provided solutions for some of them:
 A theory of ordinals
 Problem: Cantor's theory of ordinal numbers cannot be developed in Zermelo set theory because it lacks the axiom of replacement.
 Solution: Von Neumann recovered Cantor's theory by defining the ordinals using sets that are well-ordered by the ∈-relation, and by using the axiom of replacement to prove key theorems about the ordinals, such as every well-ordered set is order-isomorphic with an ordinal. In contrast to Fraenkel and Skolem, von Neumann emphasized how important the replacement axiom is for set theory: "In fact, I believe that no theory of ordinals is possible at all without this axiom."
 A criterion identifying classes that are too large to be sets
 Problem: Zermelo did not provide such a criterion. His set theory avoids the large classes that lead to the paradoxes, but it leaves out many sets, such as the one mentioned by Fraenkel and Skolem.
 Solution: Von Neumann introduced the criterion: A class is too large to be a set if and only if it can be mapped onto the class V of all sets. Von Neumann realized that the set-theoretic paradoxes could be avoided by not allowing such large classes to be members of any class. Combining this restriction with his criterion, he obtained his axiom of limitation of size: A class C is not a member of any class if and only if C can be mapped onto V.
 Finite axiomatization
 Problem: Zermelo had used the imprecise concept of "definite propositional function" in his axiom of separation.
 Solutions: Skolem introduced the axiom schema of separation that was later used in ZFC, and Fraenkel introduced an equivalent solution. However, Zermelo rejected both approaches "particularly because they implicitly involve the concept of natural number which, in Zermelo's view, should be based upon set theory." Von Neumann avoided axiom schemas by formalizing the concept of "definite propositional function" with his functions, whose construction requires only finitely many axioms. This led to his set theory having finitely many axioms. In 1961, Richard Montague proved that ZFC cannot be finitely axiomatized.
 The axiom of regularity
 Problem: Zermelo set theory starts with the empty set and an infinite set, and iterates the axioms of pairing, union, power set, separation, and choice to generate new sets. However, it does not restrict sets to these. For example, it allows sets that are not well-founded, such as a set x satisfying x ∈ x.
 Solutions: Fraenkel introduced an axiom to exclude these sets. Von Neumann analyzed Fraenkel's axiom and stated that it was not "precisely formulated", but it would approximately say: "Besides the sets ... whose existence is absolutely required by the axioms, there are no further sets." Von Neumann proposed the axiom of regularity as a way to exclude non-well-founded sets, but did not include it in his axiom system. In 1930, Zermelo became the first to publish an axiom system that included regularity.

Von Neumann's 1929 axiom system

In 1929, von Neumann published an article containing the axioms that would lead to NBG. This article was motivated by his concern about the consistency of the axiom of limitation of size. He stated that this axiom "does a lot, actually too much." Besides implying the axioms of separation and replacement, and the well-ordering theorem, it also implies that any class whose cardinality is less than that of V is a set. Von Neumann thought that this last implication went beyond Cantorian set theory and concluded: "We must therefore discuss whether its [the axiom's] consistency is not even more problematic than an axiomatization of set theory that does not go beyond the necessary Cantorian framework."

Von Neumann started his consistency investigation by introducing his 1929 axiom system, which contains all the axioms of his 1925 axiom system except the axiom of limitation of size. He replaced this axiom with two of its consequences, the axiom of replacement and a choice axiom. Von Neumann's choice axiom states: "Every relation R has a subclass that is a function with the same domain as R."

Let S be von Neumann's 1929 axiom system. Von Neumann introduced the axiom system S + Regularity (which consists of S and the axiom of regularity) to demonstrate that his 1925 system is consistent relative to S. He proved:
 If S is consistent, then S + Regularity is consistent.
 S + Regularity implies the axiom of limitation of size. Since this is the only axiom of his 1925 axiom system that S + Regularity does not have, S + Regularity implies all the axioms of his 1925 system.
These results imply: If S is consistent, then von Neumann's 1925 axiom system is consistent. Proof: If S is consistent, then S + Regularity is consistent (result 1). Using proof by contradiction, assume that the 1925 axiom system is inconsistent, or equivalently: the 1925 axiom system implies a contradiction. Since S + Regularity implies the axioms of the 1925 system (result 2), S + Regularity also implies a contradiction. However, this contradicts the consistency of S + Regularity. Therefore, if S is consistent, then von Neumann's 1925 axiom system is consistent.

Since S is his 1929 axiom system, von Neumann's 1925 axiom system is consistent relative to his 1929 axiom system, which is closer to Cantorian set theory. The major differences between Cantorian set theory and the 1929 axiom system are classes and von Neumann's choice axiom. The axiom system S + Regularity was modified by Bernays and Gödel to produce the equivalent NBG axiom system.

Bernays' axiom system

In 1929, Paul Bernays started modifying von Neumann's new axiom system by taking classes and sets as primitives. He published his work in a series of articles appearing from 1937 to 1954. Bernays stated that:

Bernays handled sets and classes in a two-sorted logic and introduced two membership primitives: one for membership in sets and one for membership in classes. With these primitives, he rewrote and simplified von Neumann's 1929 axioms. Bernays also included the axiom of regularity in his axiom system.

Gödel's axiom system (NBG)

In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global choice. Gödel used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.

Several reasons have been given for Gödel choosing NBG for his monograph:
 Gödel gave a mathematical reason—NBG's global choice produces a stronger consistency theorem: "This stronger form of the axiom [of choice], if consistent with the other axioms, implies, of course, that a weaker form is also consistent."
 Robert Solovay conjectured: "My guess is that he [Gödel] wished to avoid a discussion of the technicalities involved in developing the rudiments of model theory within axiomatic set theory."
 Kenneth Kunen gave a reason for Gödel avoiding this discussion: "There is also a much more combinatorial approach to L [the constructible universe], developed by ... [Gödel in his 1940 monograph] in an attempt to explain his work to non-logicians. ... This approach has the merit of removing all vestiges of logic from the treatment of L."
 Charles Parsons provided a philosophical reason for Gödel's choice: "This view [that 'property of set' is a primitive of set theory] may be reflected in Gödel's choice of a theory with class variables as the framework for ... [his monograph]."

Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades. In 1963, Paul Cohen proved his independence proofs for ZF with the help of some tools that Gödel had developed for his relative consistency proofs for NBG. Later, ZFC became more popular than NBG.  This was caused by several factors, including the extra work required to handle forcing in NBG, Cohen's 1966 presentation of forcing, which used ZF, and the proof that NBG is a conservative extension of ZFC.

NBG, ZFC, and MK
NBG is not logically equivalent to ZFC because its language is more expressive: it can make statements about classes, which cannot be made in ZFC. However, NBG and ZFC imply the same statements about sets. Therefore, NBG is a conservative extension of ZFC. NBG implies theorems that ZFC does not imply, but since NBG is a conservative extension, these theorems must involve proper classes. For example, it is a theorem of NBG that the global axiom of choice implies that the proper class V can be well-ordered and that every proper class can be put into one-to-one correspondence with V.

One consequence of conservative extension is that ZFC and NBG are equiconsistent. Proving this uses the principle of explosion: from a contradiction, everything is provable. Assume that either ZFC or NBG is inconsistent. Then the inconsistent theory implies the contradictory statements ∅ = ∅ and ∅ ≠ ∅, which are statements about sets. By the conservative extension property, the other theory also implies these statements. Therefore, it is also inconsistent. So although NBG is more expressive, it is equiconsistent with ZFC. This result together with von Neumann's 1929 relative consistency proof implies that his 1925 axiom system with the axiom of limitation of size is equiconsistent with ZFC. This completely resolves von Neumann's concern about the relative consistency of this powerful axiom since ZFC is within the Cantorian framework.

Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see .

Morse–Kelley set theory has an axiom schema of class comprehension that includes formulas whose quantifiers range over classes. MK is a stronger theory than NBG because MK proves the consistency of NBG, while Gödel's second incompleteness theorem implies that NBG cannot prove the consistency of NBG.

For a discussion of some ontological and other philosophical issues posed by NBG, especially when contrasted with ZFC and MK, see Appendix C of .

Models
ZFC, NBG, and MK have models describable in terms of the cumulative hierarchy Vα and the constructible hierarchy Lα. Let  V include an inaccessible cardinal κ, let X ⊆ Vκ, and let Def(X) denote the class of first-order definable subsets of X with parameters. In symbols where "" denotes the model with domain  and relation , and "" denotes the satisfaction relation:

Then:
 (Vκ, ∈) and (Lκ, ∈) are models of ZFC.
 (Vκ, Vκ+1, ∈) is a model of MK where Vκ consists of the sets of the model and Vκ+1 consists of the classes of the model. Since a model of MK is a model of NBG, this model is also a model of NBG.
 (Vκ, Def(Vκ), ∈) is a model of Mendelson's version of NBG, which replaces NBG's axiom of global choice with ZFC's axiom of choice. The axioms of ZFC are true in this model because (Vκ, ∈) is a model of ZFC. In particular, ZFC's axiom of choice holds, but NBG's global choice may fail. NBG's class existence axioms are true in this model because the classes whose existence they assert can be defined by first-order definitions. For example, the membership axiom holds since the class  is defined by:
 (Lκ, Lκ+, ∈), where κ+ is the successor cardinal of κ, is a model of NBG. NBG's class existence axioms are true in (Lκ, Lκ+, ∈). For example, the membership axiom holds since the class  is defined by:  So E ∈ 𝒫(Lκ). In his proof that GCH is true in L, Gödel proved that 𝒫(Lκ) ⊆ Lκ+. Therefore, E ∈ Lκ+, so the membership axiom is true in (Lκ, Lκ+, ∈). Likewise, the other class existence axioms are true. The axiom of global choice is true because Lκ is well-ordered by the restriction of Gödel's function (which maps the class of ordinals to the constructible sets) to the ordinals less than κ. Therefore, (Lκ, Lκ+, ∈) is a model of NBG.

Category theory
The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. For instance, in some developments of category theory, a "large category" is defined as one whose objects and morphisms make up a proper class. On the other hand, a "small category" is one whose objects and morphisms are members of a set. Thus, we can speak of the "category of all sets" or "category of all small categories" without risking paradox since NBG supports large categories.

However, NBG does not support a "category of all categories" since large categories would be members of it and NBG does not allow proper classes to be members of anything. An ontological extension that enables us to talk formally about such a "category" is the conglomerate, which is a collection of classes. Then the "category of all categories" is defined by its objects: the conglomerate of all categories; and its morphisms: the conglomerate of all morphisms from A to B where A and B are objects. On whether an ontology including classes as well as sets is adequate for category theory, see .

Notes

References

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External links

Foundations of mathematics
John von Neumann
Systems of set theory
Works by Kurt Gödel

Von Neumann–Bernays–Gödel set theory