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See all on WikipediaLarge countable ordinal - Wikipedia
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations … See more
Ordinal notations
Computable ordinals (or recursive ordinals) are certain countable ordinals: loosely speaking those represented by a computable function. … See moreWithin the scheme of notations of Kleene some represent ordinals and some do not. One can define a recursive total ordering that is a subset of the Kleene notations and has an initial … See more
Predicative definitions and the Veblen hierarchy
We have already mentioned (see Cantor normal form) … See moreThe Church–Kleene ordinal
The supremum of the set of recursive ordinals is the smallest ordinal that cannot be described in a recursive way. (It is not the order type of any … See moreWikipedia text under CC-BY-SA license Ordinal number - Wikipedia
As mentioned above (see Cantor normal form), the ordinal ε0 is the smallest satisfying the equation , so it is the limit of the sequence 0, 1, , , , etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the -th ordinal such that is called , then one could go on trying to find the -th ordinal such that , "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is used to define an…
Wikipedia · Text under CC-BY-SA license- Estimated Reading Time: 8 mins
Ordinal arithmetic - Wikipedia
WEBSuch ordinals are known as large countable ordinals. The operations of addition, multiplication and exponentiation are all examples of primitive recursive ordinal functions …
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Large Countable Ordinals (Part 1) | Azimuth
WEBJun 29, 2016 · • countable ordinals index the strength of axioms for arithmetic: the proof-theoretic ordinal of a theory is the smallest countable ordinal that the theory is unable …
Ordinal number - Encyclopedia of Mathematics
WEBDec 4, 2017 · An ordinal number is called a limit ordinal number if and only if it does not have a predecessor. Thus, $ 0 $ is a limit ordinal number. Any ordinal number can be …
Ordinal Number -- from Wolfram MathWorld
WEBAug 22, 2024 · In common usage, an ordinal number is an adjective which describes the numerical position of an object, e.g., first, second, third, etc. In formal set theory, an …
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Biggest countable ordinal number - Mathematics Stack Exchange
WEBThere isn't one. If $\alpha$ is a countable ordinal number, then $\alpha+1$ is also a countable ordinal number which is strictly bigger, so there cannot be a largest one.
Ordinal number - Wikipedia
WEBIn set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one …
Large Veblen ordinal - Wikipedia
WEBIn mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. There is no standard notation for ordinals beyond the Feferman–Schütte …
How is $\epsilon_0$ countable? - Mathematics Stack Exchange
WEBIn Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why $\epsilon_0$ is …
How much can " (recursively) large ordinal axioms" prove?
WEBJan 10, 2017 · There are references to some of this in Wikipedia's article on ordinal analysis. There is also some relevant information in its article on large countable …
The structure of countable ordinals - Mathematics Stack Exchange
WEBTo see that it is countable (assuming the natural formalization of your picture) notice that it is the limit of the ω-sequence of countable ordinals appearing directly above it in the …
Computable ordinal - Wikipedia
WEBComputable ordinal - Wikipedia. Contents. hide. (Top) See also. References. Computable ordinal. In mathematics, specifically computability and set theory, an ordinal is said to be …
logic - How large is the supremum of all countable ordinals that …
WEBJan 26, 2023 · An almost-$\mathsf{ZFC}$-absolute ordinal is an ordinal $\theta$ such that for some formula $\varphi(x)$ and some set $c$, every transitive model …
WEBL[j] satis es an analogous covering lemma; i.e., either L[j] covers V or there is an embedding from L[j] to L[j]. The latter yields yet another canonical model. Proceeding this way, one …
Ordinal collapsing function - Wikipedia
WEBOrdinal collapsing function. In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain …
logic - What is the cardinality of the countable ordinals ...
WEBEvery countable ordinal α α can be written uniquely in Cantor canonical form as a finite arithmetical expression, say C(α) C (α). We thus have the 1-1 correspondence between …
Aleph number - Wikipedia
WEBAleph-one. ℵ 1 is, by definition, the cardinality of the set of all countable ordinal numbers. This set is denoted by ω 1 (or sometimes Ω). The set ω 1 is itself an ordinal number …
Ordinal notation - Wikipedia
WEBIn mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a …
Talk:Large countable ordinal - Wikipedia
WEBAny definable uncountable cardinal of the given model will be represented in the submodel by a countable ordinal which is larger than all of those. So your "large countable …
Question on the hierarchy of named Ordinals
WEBDec 18, 2021 · A good overview of the issues involved: en.wikipedia.org/wiki/Large_countable_ordinal. – BrianO. Dec 18, 2021 at 23:52. 2ω …
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