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12 votes
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Surcomplex numbers and the largest algebraically closed field

Let $F$ be an algebraically closed field of characteristic zero. Let $\kappa$ be the cardinal of a transcendance basis of $F$ over its prime subfield $\mathbb{Q}$. Since the ${\omega_0}^{{\omega_0}^{...
nombre's user avatar
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11 votes
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Why does the inverse of surreal numbers exist?

[I have the 2nd edition, not the 1st, but there don't seem to be any differences important for this question.] First, an important correction. You write "the existence of the multiplicative inverse ...
Eric M. Schmidt's user avatar
10 votes
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How are the cardinalities of infinite sets described in systems where the continuum hypothesis doesn't hold?

Cardinality of sets are indeed something of interest in set theory. And set theory is generally considered as a list of rules for objects which we call "sets", and they come to model our intuition as ...
Asaf Karagila's user avatar
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9 votes
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Are there countably infinte surreal number?

Note that that tree representation of the surreals is actually more complicated than it may first appear - it is infinitely tall! (Indeed, the surreal numbers constructed at a finite height are merely ...
Noah Schweber's user avatar
9 votes
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Question about the $* = \{0\mid 0\}$ Game in "On Numbers and Games"

Once Left plays $0$, the position is now $\{\ \mid\ \}$, from which neither player has any valid moves. Since it is now Right's turn, Right loses. It seems like you might be misinterpreting $\{0\mid0\}...
Théophile's user avatar
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8 votes
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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 ...
Joe's user avatar
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7 votes

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, ...
JDH's user avatar
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7 votes
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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 ...
Noah Schweber's user avatar
7 votes

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 ...
nombre's user avatar
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7 votes
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Does every total order embed into the surreal numbers?

In this answer, I consider a first order language $\mathcal{L}_i$, a theory $T_i$ in $\mathcal{L}_i$ and its model companion $T_i'$ which is complete. Moreover the natural interpretation of $\mathcal{...
nombre's user avatar
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7 votes
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How much arithmetic can we find definably in the surreals?

The ordinals (with their ordering) can be interpreted inside $\mathfrak{S}$ as the equivalence classes with respect to the simplicity order. Inside the ordinals we can define the finite ordinals as ...
Eric Wofsey's user avatar
6 votes

In the surreal numbers, is it fair to say $0.9$ repeating is not equal to $1$?

The answer to 3. is affirmative. We can prove it with a straight calculation. Fun with surreals Now let $(a_n)$ be a real sequence with $0< a_n$ for all $n\in\mathbb{N}$ and $\displaystyle\lim_{n\...
SK19's user avatar
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6 votes
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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 ...
Ted's user avatar
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6 votes
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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.
hmakholm left over Monica's user avatar
6 votes
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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 ...
Paolo Lipparini's user avatar
6 votes
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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 ...
Angel_IG's user avatar
  • 127
5 votes

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 ...
Arthur's user avatar
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5 votes
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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 ...
nombre's user avatar
  • 5,117
5 votes

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 ...
PMar's user avatar
  • 51
5 votes
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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 ...
Mark S.'s user avatar
  • 24.2k
5 votes

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 ...
Eric M. Schmidt's user avatar
5 votes

Examples of Surreal Numbers that are only Surreal Numbers?

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 ...
mjqxxxx's user avatar
  • 41.9k
5 votes
Accepted

Proof that $x+(-x)=0$ for surreal numbers

Where does $x+(-x)^L\le0$ come from? It is a typo that your question has that my copy of the book does not, possibly related to confusion about notational conventions. My copy says $x+-x^L$, not $x+(-...
Mark S.'s user avatar
  • 24.2k
5 votes

Could a coordinate space be defined over the surreal numbers?

When working in ZFC or similar, a "class" basically amounts to a condition. For example, there are various methods, but one nice bunch of ways to set up the surreals is to build them up inductively. ...
Mark S.'s user avatar
  • 24.2k
5 votes

Is $\infty=\sqrt[\Omega]{\omega}$ actually a thing?

So there's a formalization of this relation. Given strictly monotonous function $f:X \rightarrow Y$ between dense linear orders $(X,<)$ and $(Y,<)$, there is a natural extension of $f$ to a ...
nombre's user avatar
  • 5,117
5 votes
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Does surreal numbers with bounded birthdays form a field?

The specific result you are asking about is true and was proved for any uncountable ordinal (irrespective of it being regular or not) by Harry Gonshor in his book An introduction to the theory of ...
nombre's user avatar
  • 5,117
5 votes
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Hyperreals and Suslin problem

No non-Archimedean (let alone hyperreal) field is complete: if $F$ is non-Archimedean, the set $I_F$ of infinitesimal elements of $F$ is bounded but has no supremum in $F$. Two quick endnotes: First, ...
Noah Schweber's user avatar
4 votes

What's the difference between hyperreal and surreal numbers?

According to the recent work by Ehrlich, the surreals are also a subset of the hyperreals. More precisely, maximal class-size fields of the surreals and the hyperreals are isomorphic. When one ...
Mikhail Katz's user avatar
  • 43.9k
4 votes
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Surcomplex Transfinite Maths

Surcomplex numbers are defined to be numbers of the form $a + bi$ for $a,b$ surreals. In particular, $b$ is allowed to be $0$, so the surcomplex numbers contain the surreals inside them. That means, ...
Reese Johnston's user avatar

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