What does hartogs number mean?
Definitions for hartogs number
har·togs num·ber
This dictionary definitions page includes all the possible meanings, example usage and translations of the word hartogs number.
Wiktionary
Hartogs numbernoun
For some set X, the least ordinal u03B1 such that there is no injection from u03B1 into X.
Wikipedia
Hartogs number
In mathematics, specifically in axiomatic set theory, a Hartogs number is an ordinal number associated with a set. In particular, if X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X can be well-ordered then the cardinal number of α is a minimal cardinal greater than that of X. If X cannot be well-ordered then there cannot be an injection from X to α. However, the cardinal number of α is still a minimal cardinal not less than or equal to the cardinality of X. (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.) The map taking X to α is sometimes called Hartogs's function. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets. The existence of the Hartogs number was proved by Friedrich Hartogs in 1915, using Zermelo–Fraenkel set theory alone (that is, without using the axiom of choice).
Wikidata
Hartogs number
In mathematics, specifically in axiomatic set theory, a Hartogs number is a particular kind of cardinal number. It was shown by Friedrich Hartogs in 1915, from ZF alone, that there is a least well-ordered cardinal greater than a given well-ordered cardinal. To define the Hartogs number of a set it is not in fact necessary that the set be well-orderable: If X is any set, then the Hartogs number of X is the least ordinal α such that there is no injection from α into X. If X cannot be well-ordered, then we can no longer say that this α is the least well-ordered cardinal greater than the cardinality of X, but it remains the least well-ordered cardinal not less than or equal to the cardinality of X. The map taking X to α is sometimes called Hartogs' function.
Numerology
Chaldean Numerology
The numerical value of hartogs number in Chaldean Numerology is: 4
Pythagorean Numerology
The numerical value of hartogs number in Pythagorean Numerology is: 8
Translation
Find a translation for the hartogs number definition in other languages:
Select another language:
- - Select -
- 简体中文 (Chinese - Simplified)
- 繁體中文 (Chinese - Traditional)
- Español (Spanish)
- Esperanto (Esperanto)
- 日本語 (Japanese)
- Português (Portuguese)
- Deutsch (German)
- العربية (Arabic)
- Français (French)
- Русский (Russian)
- ಕನ್ನಡ (Kannada)
- 한국어 (Korean)
- עברית (Hebrew)
- Gaeilge (Irish)
- Українська (Ukrainian)
- اردو (Urdu)
- Magyar (Hungarian)
- मानक हिन्दी (Hindi)
- Indonesia (Indonesian)
- Italiano (Italian)
- தமிழ் (Tamil)
- Türkçe (Turkish)
- తెలుగు (Telugu)
- ภาษาไทย (Thai)
- Tiếng Việt (Vietnamese)
- Čeština (Czech)
- Polski (Polish)
- Bahasa Indonesia (Indonesian)
- Românește (Romanian)
- Nederlands (Dutch)
- Ελληνικά (Greek)
- Latinum (Latin)
- Svenska (Swedish)
- Dansk (Danish)
- Suomi (Finnish)
- فارسی (Persian)
- ייִדיש (Yiddish)
- հայերեն (Armenian)
- Norsk (Norwegian)
- English (English)
Word of the Day
Would you like us to send you a FREE new word definition delivered to your inbox daily?
Citation
Use the citation below to add this definition to your bibliography:
Style:MLAChicagoAPA
"hartogs number." Definitions.net. STANDS4 LLC, 2024. Web. 26 Apr. 2024. <https://www.definitions.net/definition/hartogs+number>.
Discuss these hartogs number definitions with the community:
Report Comment
We're doing our best to make sure our content is useful, accurate and safe.
If by any chance you spot an inappropriate comment while navigating through our website please use this form to let us know, and we'll take care of it shortly.
Attachment
You need to be logged in to favorite.
Log In