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The number $$\frac{1+\frac{1}{\omega}}{\omega}$$ for $\omega$ a transfinite number greater than all integers is a surreal number or it don't support this composition of infinitesimals?

Blue
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Iago newats
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    Keep in mind that $\omega$ is a very specific surreal number; it's not just some transfinite number greater than all integers. That said, yes, ${1+{1\over\omega}\over\omega}$ is a surreal number. – Noah Schweber May 08 '21 at 03:15
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    The surreals form an ordered field. That is, they obey all the standard algebraic properties that we are used to. So for any surreal number $\alpha$ and any rational expression in $\alpha$ that does not involve dividing by $0$, the result will be another surreal number. – Paul Sinclair May 08 '21 at 14:15

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Expanding on the comments:

There are many ordered fields: collections of "numbers" in which you can compare, add, multiply, and subtract any pair of numbers, as well as divide any pair where you're not dividing by $0$, all in ways that make sense with usual arithmetic (e.g. the sum or product of positive numbers is positive).

In any ordered field, if $a$ is a nonzero number, then $\dfrac{1+\frac{1}{a}}{a}$ is a number, and it simplifies to $\dfrac{1}{a}+\dfrac{1}{a^2}$. If $a>1$ then $0<\dfrac{1}{a}<1$, so that $\dfrac{1}{a}<\dfrac{1}{a}+\dfrac{1}{a^2}<\dfrac{2}{a}$. This still applies in a non-Archimedean ordered field where a number $a$ might be greater than $\underset{n\text{ ones}}{\underbrace{1+1+\cdots+1}}$ for any positive integer $n$.

The surreal numbers have all the properties of an ordered field (except that if you're using a standard set theory foundation, they don't fit in a set), so the above applies to the surreal numbers as well, so that $a$ above can be replaced with the surreal number $\omega$. In the surreal numbers, $\omega$ typically refers to a very specific number that is greater than any positive integer: the "simplest" such surreal number (under a technical definition of simplicitiy of a surreal). Other surreal numbers greater than any positive integer include $\omega-1$, $\omega/2$, $\sqrt{\omega}$, etc.

Mark S.
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