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I was just reading through the construction of the surreal numbers on wikipedia, and I read through some of the examples. I noticed that all of the examples were how certain types of already existing numbers (such as reals or hyperreals) could be constructed. However, I'm curious about examples of numbers that are unique to the set of surreal numbers. I've heard that they are the largest possible ordered field, so I would imagine that many if not most of surreal numbers are of this type. So what are some examples?

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    $\begingroup$ I'm not sufficiently conversant with hyperreals to know whether $\sqrt{\infty}$ is one. $\endgroup$ – Gerry Myerson Apr 22 '18 at 1:06
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    $\begingroup$ @MJD $+_2$ and $*$ are games, not surreal numbers. Every number is a game but not vice versa. $\endgroup$ – Mike Earnest Apr 22 '18 at 1:15
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    $\begingroup$ God kills a kitten every time someone mentions surreal numbers and hyperreal numbers in the same sentence. They have nothing to do with each other (beyond some very superficial things). $\endgroup$ – Eric Wofsey Apr 22 '18 at 1:17
  • $\begingroup$ @MikeEarnest I don't see why $+_2$ isn't a number? As far as I can tell, $0 < $ {$0 | 2$} is true. $\endgroup$ – RothX Apr 22 '18 at 2:27
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    $\begingroup$ How about {$1,2,3,...|\omega$}? This number is smaller than $\omega$ and larger than any finite number (as far as I can tell). Not sure if any number that is not unique to the surreals acts like that. $\endgroup$ – RothX Apr 22 '18 at 15:15
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A few remarks:

$\DeclareMathOperator{\Noo}{\mathbf{No}}$ -$\Noo$ is not "the largest possible ordered field", but rather "a universal ordered field" with nice properties. Universality is to be understood in the weak sense: every ordered field embeds in it, not in a unique way in general.

-The differents fields (or frameworks whithin which one can talk) of hyperreals do not compare in a canonical way with surreals. That $\Noo$ is universal requires some choice, and for instance, no explicit embedding of $^*\mathbb{R}$ into $\Noo$ is known. Thus it makes only little sense to compare which number lies in which field, and it is more fruitful to understand global properties of the different fields.

-Although in hyperreal fields constructed by ultrafilters on $\mathbb{N}$ one can embed ordinals below $\varepsilon_0$ in a somewhat natural way preserving natural aritmetic (send $\omega$ to the class of $id_{\mathbb{N}}$ modulo the ultrafilter and embeds other ordinals using operations on ordinals and on hyperreals), it is not usual to consider that hyperreals contain distinguished infinite elements such as "$\omega$".

-However, for any infinite element $x$ in $^*\mathbb{R}$, $x-1 \in ^*\mathbb{R}$ is strictly below $x$ but infinite.

-While an hyperreal field does not canonically contain all ordinals, it is closed under extensions of real functions, most of which aren't explicitly known on $\Noo$. IN this sense many expressions such as $\sin(H)$ make sense in $^*\mathbb{R}$ but not in $\Noo$.

-To finish on a positive note, a somewhat similar question is asked here.

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  • $\begingroup$ What does it mean when you say "canonically"? $\endgroup$ – RothX Apr 22 '18 at 20:52
  • $\begingroup$ I meant it in an informal way. There is no single order embedding of $\omega_1$ into $^*\mathbb{R}$ which is "more natural", or "more simple" than another. $\endgroup$ – nombre Apr 22 '18 at 23:11

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