# Questions tagged [surreal-numbers]

For questions about the surreal numbers, which are a real-closed ordered proper-class-sized field that contains both the real numbers and the ordinal numbers. Thus they contain both infinite numbers (including the ordinals, but also infinite numbers like ω-1 and sqrt(ω)) and infinitesimal numbers (like 1/ω). They can also be identified with a subclass of two-player partisan games.

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### Surreal Numbers, Proving $x1=x$

I am trying to learn the theory of the Surreal numbers and I am therefore going over all the theorems and trying to prove them for myself. I am struggling to complete the proof of $x1 = x$. I have ...
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### Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...
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### A question about real closed fields that contain the real numbers as a proper subfield

Let F be such a real closed field and let F(C) be the algebraic closure of F. An important example of F is FSR- the field of SURREAL numbers. Suppose that one has constructed an exponential function f ...
184 views

### A (second-order) axiomatic characterization of the integers which rules out surreal/hyperreal versions

I've seen it stated, for example here, that the integers are the unique commutative ordered ring with identity whose positive elements are well-ordered. I understand why the integers are the smallest ...
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### Is $\omega^\frac{1}{\omega} > n \forall n \in \mathbb{N}$?

I was thinking about $log(\omega)$ which appears to be $\{\mathbb{N}|\omega^{\frac{1}{n}}\}_{n\in\mathbb{N}}\stackrel{?}{=}\omega^\frac{1}{\omega}$. Intuitively, there's the idea that, if the highest ...
183 views

### Extension to real number system [closed]

Suppose you have equation involving a number $s$ $s^2+ 1 = 0$, to solve it one needs to treat $s$ as complex number $s = \pm i$, and introduce $i$ as imaginary unit. Now suppose you have equation ...
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### Is the universality of the surreal number line a weak global choice principle?

I'd like to consider the principle asserting that the surreal number line is universal for all class linear orders, or in other words, that every linear order (including proper-class-sized) linear ...
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### Ordinals which embed in surreal subfields

If $k$ is an ordered field, the least ordinal $s(k)$ which doesn't embed in $(k,<)$ is regular. This is because every interval of an ordered field embeds in every infinite interval so given a ...
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### Surreal compactness

In a comment here, Joel David Hamkins said: ...I think perhaps every set-sized open cover of a bounded interval in the surreals has a finite subcover, but there are proper class open covers with no ...
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### Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...
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### Nice sign-expansions of special surreal numbers

What is the "right" surreal generalization of the fact that a real number $r$ is rational if and only if its sign-expansion is eventually periodic? I can think of more than one natural way to ...
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### Nimbers and Surreal Numbers [closed]

I have been researching Combinatorial Game Theory. One common theme is the assignment of values to games in order to classify the game as a win for a specific player. One such way is class of surreal ...