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Questions tagged [surreal-numbers]

For questions about the surreal numbers, an inductively constructed ordered field that naturally contains all ordinal numbers.

The surreal numbers are an inductively constructed proper class which has the structure of an ordered field. Surreal numbers were originally discovered in the context of combinatorial game theory, as they form a very special class of "games" in the sense of combinatorial game theory. For more information, see https://en.wikipedia.org/wiki/Surreal_number.

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How does Conway's proposed compromise for constructing the real numbers actually work?

My question is about understanding a remark John Conway made in On Numbers and Games (ONAG), where he proposes a method for constructing the real numbers from the rationals. I will have to assume familiarity with Conway's construction of the surreal…
Mike Earnest
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What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain: real numbers a hierarchy of infinitesimal numbers like $\epsilon, \epsilon^2, \epsilon^3,…
Nathan Reed
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Are surreal numbers actually well-defined in ZFC?

Thinking about surreal numbers, I've now got doubts that they are actually well-defined in ZFC. Here's my reasoning: The first thing to notice is that the surreal numbers (assuming they are well defined, of course) form a proper class. Now, quoting…
celtschk
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Modeling numbers with vectors of vectors?

I stumbled upon the strange representation of integers where $$8=\langle\langle0,\langle0^{\infty}\rangle,0^{\infty}\rangle,0^{\infty}\rangle$$ I'll try explain the representation in a natural way. What I'm wondering -- as is common -- does this…
user377597
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Surreal and ordinal numbers

Is there a surjective map between the (class of) ordinal numbers On and the set No (Conway's surreal numbers) and is it constructable, In Conway's system we have for example: $\omega_0 = < 0,1,2,3,... | > $ and: $\epsilon = < 0 | 1, 1/2, 1/4, 1/8,…
Willem Noorduin
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Surreal numbers without the axiom of infinity

Let $ZF^\times$ denote the set of axioms of Zermelo-Fraenkel set theory without the axiom of infinity. The set $V_\omega$ of all hereditarily finite sets is a model of $ZF^\times$, and $Ord^{V_\omega}=\omega$. The surreal numbers, in $ZFC$ form a…
Asaf Karagila
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Why does the inverse of surreal numbers exist?

Problem I'm working with the book "On numbers and games" from John Conway, first edition from 1976. On page 20 he writes Summary. Numbers form a totally ordered Ring. Note that in view of Theorem 8 and the distributive law, we can assert, for…
SK19
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Surcomplex numbers and the largest algebraically closed field

It's well known that the surreal numbers $\mathbf{No}$ are the largest ordered "field" (more accurately, they form a proper class with field structure, which is sometimes called a Field with capital F), in the sense that every other ordered field…
pregunton
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In the surreal numbers, is it fair to say $0.9$ repeating is not equal to $1$?

I find the surreal numbers very interesting. I have tried my best to work through John Conway's On Numbers and Games and teach myself from some excellent online resources. I have prepared a short video to introduce surreal numbers, but I want to…
Presh
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Non-standard measure

Just a bit of a strange question. Modern formulations of probability theory rest upon measure theory. This poses an issue for non-measurable sets. Typically, one simply excludes these sets from the analysis and considers only measurable subsets…
E8xE8
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More than the real numbers: hyperreals, superreals, surreals ...?

I've read something about extensions of the real numbers, as hyperreals, superreals, surreals and, as I can understand, all these extensions contain some new kinds of infinitesimal and infinite ''numbers''. And, if I well understand, we have a chain…
Emilio Novati
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Why are the surreals considered "recreational" mathematics?

One of my college math professors once remarked to me that it was interesting that John Conway's two "biggest" contributions to math were both recreational: the Game of Life and the Surreals. No one, he said, was really working in the Surreals. I…
kuzzooroo
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Is multiplication of games that are equivalent to numbers well-defined?

It's well-known that if you take the definition of surreal multiplication and attempts to generalize it to all games, the result is not well-defined, in that it does not respect equivalence of games. What I'm wondering is this: What if we consider…
Harry Altman
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Are all nimbers included in the surreals?

I guess the question says it all. The **nimber* (https://en.wikipedia.org/wiki/Nimber) concept, sometimes called "Sprague-Grundy numbers" embodies the "values" of positions in impartial games which can be added together to form larger games. The…
Mark Fischler
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Proof of Conway's "Simplicity Rule" for Surreal Numbers

A "number" in the sense of Combinatorial Game Theory is a game $G = \{ a,b,c,\dots | \; d,e,f,\dots \}$ such that $a,b,c < d,e,f$. Then our game is between the left and right options: $$ a,b,c < G < d,e,f $$ However, we still don't know which…
cactus314
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