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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Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions

This question was originally asked at MSE but seems too advanced, so I'm reposting it here. In short, the idea is that many constructions for non-Archimedean fields can naturally be iterated, in some ...
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5 votes
0 answers
190 views

Surreal numbers and the ultrafilter lemma

In this question, I asked how to interpret a historical claim made by Conway regarding the potential, if not ideal, use of the surreal numbers for nonstandard analysis. In the comments and answers it ...
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8 votes
2 answers
402 views

A "surnatural numbers" as a largest model of the natural numbers

One characteristic of the surreal numbers is that they are a monster model of the first-order theory of real numbers, according to Joel David Hamkins in this post. Thus they are real-closed, and every ...
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17 votes
3 answers
1k views

Interpreting Conway's remark about using the surreals for non-standard analysis

In Conway's "On Numbers And Games," page 44, he writes: NON-STANDARD ANALYSIS We can of course use the Field of all numbers, or rather various small subfields of it, as a vehicle for the ...
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8 votes
1 answer
343 views

In surreal numbers, what is $\ln \omega$?

Can this number $\ln \omega$ be written in $\{L|R\}$ form? What's its birthday?
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2 votes
1 answer
161 views

What does it mean for the surreal numbers/partizan games to be "universally embedding"?

In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
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5 votes
1 answer
367 views

Are the Surreals a cogenerator in the category of ordered fields?

A cogenerator in a category $\mathcal{C}$ is an object $\Omega$ such that for any pair of parallel arrows $f,g:X\rightrightarrows Y$ in $\mathcal{C}$ we have $$ \forall h:Y\to\Omega\big(h\circ f=h\...
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15 votes
2 answers
934 views

Is the surreal number $\omega(\sqrt{2}+1)+1$ a prime?

In the 1986 book An Introduction to the Theory of Surreal Numbers, Gonshor, on page 117, notes that it is an open problem whether $\omega(\sqrt{2}+1)+1$ is a prime, using the standard definition of ...
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Can one represent divergent integrals or germs at infinity with surreal numbers?

I have been disliking the theory of surreal numbers for a while, but let's test it. So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\...
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2 votes
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227 views

Surreal numbers and the Collatz iteration as a game?

Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$. Each number $n$ represents a game played by left $L$ and right $R$: $$n = \{L_n | R_n \}$$ The rules ...
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6 votes
0 answers
200 views

Quantum surreal numbers

Toward Quantum Combinatorial Games presents the definition of a "quantum game", allowing a superposition of moves rather than a single classical move. This leaves me wondering: Since surreal ...
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3 votes
1 answer
277 views

Smallest ring whose field of fractions includes all the reals (subring of omnific integers?)

The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the ...
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7 votes
0 answers
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How does Conway's proposed compromise for constructing the real numbers in ONAG actually work?

I have also asked this question on Math Stack Exchange (link). In On Numbers and Games, after discussing the inclusion of the real numbers in the surreal numbers, No, Conway discusses the merits of ...
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15 votes
1 answer
794 views

In theory, how would Oneiric numbers be defined?

Background I am not a professional mathematician. I am researching Surreal numbers & games for fun (I think they are truly beautiful). If this question is not appropriate here, I beg forgiveness &...
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25 votes
1 answer
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Are Conway's combinatorial games the "monster model" of any familiar theory?

This is related to this question about a "mother of all" groups, and so seemed like it'd fit in better at MO than MSE. If I understand the answer to that question correctly, the surreal numbers have ...
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0 votes
0 answers
121 views

Is standard, affine infinity of extended reals quite small on the scale of infinities?

Some time ago I had a conversation with a guy who was into surreal numbers and he said that in surreal numbers the affine infinity is quite minor entity compared to the ordinality of natural numbers $\...
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4 votes
1 answer
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Surreal numbers and the Axiom of Choice

In ZFC and its conservative extension NBG, it can be shown that every ordered field embeds into the surreal numbers. How much choice is needed to prove this? Without choice, what is a simple example ...
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37 votes
3 answers
4k views

Who discovered the surreals?

Common folklore dictates that the Surreals were discovered by John Conway as a lark while studying game theory in the early 1970's, and popularized by Donald Knuth in his 1974 novella. Wikipedia ...
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16 votes
2 answers
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Biggest Field Of Characteristic $p$

The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every ...
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5 votes
0 answers
314 views

Algebraic Geometry Over the Surreal and Surrcomplex Numbers

I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?
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5 votes
0 answers
417 views

The surreal numbers under a change of universe

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...
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10 votes
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Genetic construction of roots of surreal polynomials

In On Numbers And Games, Conway uses the term "genetic" for definitions of operations on surreal numbers that are inductive in terms of their options. His definitions of addition and multiplication ...
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3 votes
1 answer
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'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group

What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...
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6 votes
1 answer
355 views

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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34 votes
2 answers
4k views

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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3 votes
1 answer
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Functions on a field representable by Hahn series?

It is well known (see here for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\...
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7 votes
1 answer
566 views

Is the inverse of surreal numbers actually well-defined?

J.Conway wrote in his book "On numbers and games" (1st edition, 1976) on p. 66 It seems to us, however, that mathematics has now reached the stage where formalization within some particular ...
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12 votes
1 answer
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Largest ordered "field" in NBG without axiom of global choice

I know from Wikipedia that in NBG, the surreal numbers are the largest possible ordered field (if a proper class is allowed to be a field). But then, it is written: "in theories without the axiom of ...
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4 votes
1 answer
299 views

Roots of $\omega$, larger $\gamma$-numbers

In Harry Gonshor's An Introduction to the Theory of Surreal Numbers, on page 50, Gonshor points to a method for intuitively guessing what the square root of the countable infinity is in his ...
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6 votes
2 answers
389 views

Automorphism of the transfinite rooted binary tree

I was studying combinatorical group theory recently, and I came across the infinite regular rooted binary tree and its automorphism group $Aut(T^{(2)})$with the Grigorchuk subgroup. Let me now ...
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4 votes
0 answers
158 views

Modern advances in combinatorial game theory

I'm going to take part in teaching a course in combinatorial game theory in the best of ONAG's spirit. I was wondering if there are interesting post-ONAG results that are worth mentioning in (a later ...
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10 votes
2 answers
762 views

What surreal numbers are representable by Red-Blue Hackenbush games?

Every game of Red-Blue Hackenbush represents a surreal number. Is the converse true? Assuming that it is false, what can be said about the class of surreal numbers that are representable by such ...
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2 votes
1 answer
405 views

Are Surreal Numbers the same as Trans-series?

I recently found the paper of Berarducci + Mantova [1, 2] saying that surreal numbers are equivalent to trans-series. These are very different objects: trans-series are used in physics to correct, ...
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5 votes
1 answer
416 views

Can a game be an option of itself?

My question is, can a game contain itself as an option? and can it be a surreal number? For example $A=\{A|\}$ or $B=\{C|B\}$ where $C$ is a surreal number. from the point of view of games, it is ...
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9 votes
1 answer
515 views

Surreal number: trying to construct complete ordered fields

Let $R$ be a subring of $\mathbf{No}$, the set of surreal number. We try to construct $\tilde{R}$, the Cauchy completion of $R$, just like the ordinary Cauchy completion for metric space. In the ...
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2 votes
0 answers
221 views

Factorization in the Omnific Integers

I'm wondering if there's been any work done on prime factorizations of Omnific integers as products of prime Omnific integers. I suspect that each Omnific integer has a unique prime factorization, ...
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3 votes
1 answer
93 views

Sign-expansion definition of Surreal arithmetical operations

Is there a way to define the addition and multiplication operations in Surreals numbers, defined directly on the sign-expansion notation {-,+}, i.e. without firstly convert them to the Conway notation ...
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31 votes
1 answer
1k views

Are there any interesting surreal constants?

In $\mathbf R$, we have all sorts of fascinating constant, like $e$, $\pi$, $\gamma$, ... For ordinal numbers, we have $\omega$, $\epsilon_0$, $\omega_1^{CK}$, $\omega_1$, ... Have we discovered any ...
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14 votes
1 answer
730 views

The surreal version of $e$

For a sequence $(x_{\alpha})$ of surreal numbers indexed by the set of all ordinal numbers, we say that $\lim x_{\alpha}=l$ ($l$ is a surreal number) if for each surreal $\epsilon>0$, there exists ...
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7 votes
1 answer
913 views

Going beyond the surreal numbers

Denote the class of surreal numbers No. We can create new "number", like the gap $\infty=\{\infty^L|\infty^R\}$, defined by $\infty^L=\{x:\exists n\in\mathbb N,x<n\}$ and $\infty^R=\{x:\forall n\in\...
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1 vote
0 answers
170 views

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 ...
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1 vote
0 answers
194 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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8 votes
1 answer
486 views

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 ...
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2 votes
0 answers
189 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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18 votes
0 answers
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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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5 votes
1 answer
170 views

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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9 votes
2 answers
727 views

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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9 votes
1 answer
480 views

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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18 votes
2 answers
1k views

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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3 votes
0 answers
584 views

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 ...
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