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After familiarizing myself with the ordinal numbers and their arithmetic, I’ve been (perhaps sloppily) using function notation to refer to certain combinations of arithmetic operations on ordinals. For example, letting

$$f(x)=x^2+\omega x$$

Gives values like

$$f(\omega + 1) = \omega^2 + \omega$$

However, I recently realized that it might be mathematically incorrect to define a function on all ordinals. Because if asked to state the domain and codomain of this function, I would have to say “the set of all ordinals”... which isn’t a set, it’s a class. And as far as I know, functions can’t be defined on classes, only sets.

Question: Is it wrong to define functions that accept any ordinal as an argument? (By “wrong”, I mean - does it lead to contradiction or paradox?)

Franklin Pezzuti Dyer
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    Even using $+$ and $\times$ on ordinals is using a form of function notation. We use function notation as a shorthand in set theory. For example $\mathcal P(S)$ for the power set of $S.$ Here, $\mathcal P$ is not a function, but a short hand for the definition of another set. So if we say $X=\mathcal P(S)$ we mean “$X$ is the unique set that satisfies the definition for the power series of $S.$ So we allow it. I assume there are some times where you need to take care, but I can’t think of any off the top of my head. – Thomas Andrews Sep 18 '20 at 23:31
  • @ThomasAndrews The times where you'd need to take care are when doing things like talking about the "set" of all functions on the ordinals or trying to put some extra structure on this "set." Talking about individual definable functions on the ordinals should rarely get you in trouble. – Patrick Lutz Sep 19 '20 at 16:55

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