# Tag Info

Accepted

• 24.6k
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### Can the surreal numbers be completed to form an ordered field?

Great question. let me say a few things that might be interesting. First, “Cauchy Completeness” isn’t going to be the best way to think about this. We usually think of Cauchy completions for metric ...
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### Why is not some numbers in nth days?

The surreal number $\{A\mid B\}$ is the first-born number $x$ for which $a<x$ for all $a\in A$ and $x<b$ for all $b\in B$. That is, while there will be many surreal numbers filling that cut, ...
• 44.3k
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### Do surreals prove reals are countable?

The mistake is your claim there are a countable number of these infinitesimals in any interval (since ${1\over \omega}\cdot\omega=1$). You seem to be conflating the surreal number $\omega$ and the ...
• 248k

### Examples of Surreal Numbers that are only Surreal Numbers?

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 ...
• 5,117
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• 3,131
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### Why is $\uparrow+\ast=\{\ast,0|0\}$?

You can't remove a position just because it leaves a win for the opponent. The main idea behind combinatorial game theory (not emphasized in most expositions on the subject, but it is implicit in how ...
• 34.2k
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### Can a set of surreal numbers be defined with arbitrary cardinality?

Yes: Since every ordinal is a surreal number, the sets you're looking for can be taken to be the initial ordinals that represent $\aleph_n$, $\beth_n$, and so forth.
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### Is multiplication of games that are equivalent to numbers well-defined?

(the counterexample has been worked out in collaboration with Harry Altman. All errors are mine) The counterexample provides a game $K$ such that $K=0$ but $K^2\neq 0$. Here $\cong$ is identity of ...
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### What Hackenbush game represents oof?

$\textbf{on}$, $\textbf{off}$ and $\textbf{oof}$ are loopy game values in combinatorial game theory, meaning they occur in games in which a position can be repeated. $\textbf{on}$ is the value of a ...
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### List of equivalence surreal numbers to 4 day?

Ok, calculation time, just to make it clear what I'm talking about in my comment above: on day $3$, the number $3$ appears. It appears in many different forms, namely all the forms that has empty ...
• 201k
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### Are the hyperreals emerging at some stage of the surreal construction?

$\DeclareMathOperator{\Noo}{No}$ First note that in ZFC, there can be several non-isomorphic fields of hyperreal numbers built as ultrapowers of $\mathbb{R}$ to the power $\mathbb{N}$. If CH is ...
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### Surreal arithmetic with $\frac{1}{2}\omega$

Your calculation looks right. What you are overlooking is a comment in the earlier chapter which discusses various tricks that help one calculate values. Conway notes that one can often remove ...
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### How does the empty set work in arithmetic of surreal numbers?

$1$ is not really defined as $(0,\emptyset)$, but as $(\{0\},\emptyset)$. Then $$1+1=\left(\{0+1\}\cup\{0+1\},(\emptyset+1)\cup(\emptyset+1)\right)$$ However, $\emptyset+1$ is empty since there are no ...
• 24.2k

### How does the empty set work in arithmetic of surreal numbers?

A surreal number is, in particular, a ordered pair of sets of surreal numbers, so $\emptyset$ isn't a surreal number at all. In the formula for addition, the expressions like $X_R + y$ do not ...
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For a simple answer to the original question, per Gerry Myerson's comment, $$\sqrt{\omega} \equiv \left\{1,2,3,\ldots \;\vert\; \omega, \omega/2, \omega/3, \ldots \right\}$$ is a surreal number that ...