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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What's wrong with the surreals?
Of all the constructions of the reals, the construction via the surreals seems the most elegant to me.
It seems to immediately capture the total ordering and precision of Dedekind cuts at a ...
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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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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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Where do surreal numbers come from and what do they mean?
I know about Conway's original discovery of the surreal numbers by way of games,
as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior
of real-valued functions, leading ...
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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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Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?
This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
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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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Surreal numbers vs. non-standard analysis
What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in ...
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Surreal exponentiation -- are the varying definitions equivalent? If not, is there agreement on which ones are better?
The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
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Integration in the surreal numbers
In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace dt=\...
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Surreal Numbers as Inductive Type?
Prompted by James Propp's recent question about surreal numbers, I was wondering whether anyone had investigated the idea of describing surreal numbers (as ordered class) in terms of a universal ...
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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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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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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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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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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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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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Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?
Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig identities) will still ...
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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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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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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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Constructing the surreal numbers as iterated Hahn series
A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with ...
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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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First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?)
It is known that locally one can ``code'' any set in the von Neumann universe $V$ by a set of ordinals. But can one do this globally? In other words, is there a first-order definable bijection $P(On)...
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Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy
Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, $\...
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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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Does this construction yield the surreal numbers?
There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals.
First given such a field one may consider rational functions over that field ...
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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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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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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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A question about J.H. Conway's SURREAL NUMBERS
My quesion is: What set theory are the mathematicians who are developing the theory of
these numbers working in-or are they, in fact, working outside any of the standard set
theories?. Each surreal ...
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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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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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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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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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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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Uniformizing the surcomplex unit circle
Is the multiplicative Group of surcomplex numbers of modulus 1 isomorphic to the additive Group of the surreal numbers modulo the sub-Group of surreal integers? And, do Norman Alling's surreal ...
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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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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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Transcendence degree of the surreals over the subfield generated by the ordinals
Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...
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Surreal Numbers and Set Theory
Hello,
I looked through MathOverflow's existing entries but couldn't find a satisfactory answer to the following question:
What is the relationship between No, Conway's class of surreal numbers, and ...
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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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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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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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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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Surreal numbers and large cardinals
This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.
Part 1 is about foundations. Much of the ...
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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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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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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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More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function
The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
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