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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Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...
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Are gaps and loopy games interchangeable in the Surreal Numbers?
The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and ...
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What are the properties of $\operatorname{No}[i]$?
I was looking for a complex extension of the surreals and then I found $\operatorname{No}[i]$, what are its properties and how do I express $x+yi$ in the $a | b$ notation?
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Proof of Theorem Concerning Conway's "Nim Field"
I have a question about the proof of theorem 44 in "On Numbers and Games" on page 58, concerning the "Nim field" ${ON}_2$. As background, ${ON}_2$ turns the ordinals into a field ...
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Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?
Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...
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Can you build the surreal numbers as a simple direct limit of ordered fields?
The surreal numbers are sometimes called the "universally embedding" ordered field, in that every ordered field embeds into them. What "universally embedding" means seems to be ...
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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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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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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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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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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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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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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
$$
\forall h:Y\to\Omega\big(h\circ f=h\...
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Is the surreal number $\omega(\sqrt{2}+1)+1$ a prime?
In the 1986 book An Introduction to the Theory of Surreal Numbers, Gonshor, on page 117, notes that it is an open problem whether $\omega(\sqrt{2}+1)+1$ is a prime, using the standard definition of ...
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Can one represent divergent integrals or germs at infinity with surreal numbers?
I have been disliking the theory of surreal numbers for a while, but let's test it.
So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\...
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Surreal numbers and the Collatz iteration as a game?
Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...
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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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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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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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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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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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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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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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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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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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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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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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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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'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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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