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https://en.wikipedia.org/wiki/Ordinal_arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation.Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion.Cantor normal form provides a standardized way of writing ...
https://math.stackexchange.com/questions/278992/how-to-think-about-ordinal-exponentiation
I'm just trying to understand better how to see $\alpha^{\beta}$ for an arbitrary ordinal. I've already know that one can think about $\alpha . \beta$ as $\langle \alpha \times \beta, AntiLex\rangle$
https://mathoverflow.net/questions/319002/ordinal-exponentiation-levy-hierarchy
Added: If I didn't make any mistakes, ordinal exponentiation is $\Delta_1$ if and only if that finite support set above is $\Delta_1$ definable. Thus, in principle, there should be an easy way to get a $\Sigma_1$ definition of that set. This seems surprising to me.
https://en.wikipedia.org/wiki/Talk:Ordinal_arithmetic
Talk:Ordinal arithmetic. Jump to navigation Jump to search ... is intended to denote ordinal exponentiation, it is read to mean the ... it is not restricted to functions with finite support; and (2) the ordering on the exponent E enters into the definition in a reversed way.(Rated B-class, Mid-priority): WikiProject Mathematics
https://math.stackexchange.com/questions/222491/is-there-a-categorification-of-infinite-ordinal-arithmetic
Is there a categorification of (infinite) ordinal arithmetic? Ask Question ... one can define and construct "directed products", which categorifies ordinal exponentiation. share cite improve ... We can define $\alpha^\beta$ to be the order type of the set of functions from $\beta$ to $\alpha$ of finite support (finitely many non-$0 ...
https://www.maplesoft.com/support/help/maple/view.aspx?path=Ordinals/Power
In the two-argument case, if a , b are both nonzero, a ≠ 1 and at least one of them is an ordinal data structure, that is, an ordinal number greater or equal to ω, then the result is an ordinal data structure.Otherwise, the result is a nonnegative integer or a polynomial with positive integer coefficients. •
https://www.sciencedirect.com/science/article/pii/0168007294900760
Annals of Pure and Applied Logic 66 (1994) 1-18 North-Holland Reverse mathematics and ordinal exponentiation Jeffry L. Hirst Department of Mathematical Sciences, Appalachian State University, Boone, NC 28608, USA Communicated by A. Nerode Received 10 August 1992 Abstract Hirst, 1.L., Reverse mathematics and ordinal exponentiation, Annals of Pure and Applied Logic 66 (1994) 1-18.Cited by: 29
http://dictionary.sensagent.com/Ordinal%20arithmetic/en-en/
The ordinal ε 0 (epsilon nought) is the set of ordinal values of the finite arithmetical expressions of this form. It is the smallest ordinal that does not have a finite arithmetical expression, and the smallest ordinal such that , i.e. in Cantor normal form the exponent is not smaller than the ordinal itself. It …
https://www.arxiv-vanity.com/papers/1908.00280/
In many investigations of normal functions, ordinal exponentiation is presupposed as a starting point. Most notably, the first function in the Veblen hierarchy is usually defined as φ 0 (α) = ω α (see e. g. []).This makes a lot of sense in the context of ordinal notation systems, since a non-zero ordinal is of the form ω α if, and only if, it is closed under addition.
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