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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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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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Examples of Surreal Numbers that are only Surreal Numbers?

I was just reading through the construction of the surreal numbers on wikipedia, and I read through some of the examples. I noticed that all of the examples were how certain types of already existing numbers (such as reals or hyperreals) could be…
RothX
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Is the class of ordinal numbers bounded by a fixed ordinal number really a set when defined through surreal numbers?

At his book "On Numbers and Games", Conway defines ordinal numbers as games which doesn't have right options and whose left options contain only ordinal numbers. Then, fixed an ordinal number $\alpha$, he claims that $V_\alpha=\{\beta : \beta <…
Anderson Brasil
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Surreal numbers as generalized Dedekind cuts

From the four postulates of the Dedekind cuts, namely (for (a,b) denoted as the cut, a,b being subsets of the rationals): Every rational number lies in exactly one of the sets a,b, a,b are not empty, Every element of a is smaller than every…
JtSpKg
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Inductive hypothesis in surreal multiplication proof

I'm having a hard time reading through Theorem 8 of On Numbers and Games. To prove that the multiplication of numeric games is a number, Conway sets up a simultaneous (Conway) induction with three statements: If $x$, $y$ are numbers, then $xy$ is a…
ViHdzP
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Is $\infty=\sqrt[\Omega]{\omega}$ actually a thing?

In Infinity and the Mind the author claims: Conway derives the weird equation, $\infty=\sqrt[\Omega]{\omega}$, which almost magically ties together potential infinity $\infty$, the simplest actual infinity $\omega$, and the Absolute Infinite…
user784623
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Can a set of surreal numbers be defined with arbitrary cardinality?

It is my understanding that the surreal numbers form a class rather than a set, because their collection is larger than any set. Thus it would seem to follow that for any cardinality, such as $\aleph_n$ or $\beth_n$ for a fixed $n$, a set of surreal…
user253970
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What are some construction of number system which also works when you use them "within" the surreal numbers?

I've noticed that a few construction we use to construct different number systems also work if we carry them on within the surreal numbers. A few examples of what I mean are bellow. I was wondering if there were other examples than those I have…
Mettek
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Help with Knuth's Surreal Numbers

I'm reading D. E. Knuth's book "Surreal Numbers". And I'm completely stuck in chap. 6 (The Third Day) because there is a proof I don't understand. Alice says Suppose at the end of $n$ days, the numbers are $$x_1
Antonius
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Surreal numbers - why can we continue after infinite days?

The surreals are described as a game where finitely-many numbers are generated each day. In the limit, things like the non-dyadic rationals, the reals, and even hyperreals like $\omega$ and $\epsilon$ can be defined. However, the way I've usually…
BlueRaja - Danny Pflughoeft
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How can $\pi$ be defined as a surreal number.

I want to express $\pi$ in the Surreal Number notation $\{L|R\}$. What is the most natural or intuitive way of doing so, seeing as there are many (possibly infinite) ways of expressing the same surreal number.
kam
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Why is not some numbers in nth days?

In surreal numbers : In second day we have 2 and 1/2 and ... but why in third day we don't have {1/2|2} =5/4 ?
Farshad
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Surreal numbers whose final segment is an integer.

For every ordinal $\alpha$, define $a_{\alpha} = \{0\} \ | \ \{a_{\beta} \ | \ \beta < \alpha\}$. In Harry Gonshor's approach of surreals where they are $(+,-)$ sequences of ordinal domain, $a_{\alpha}$ is a plus followed by $\alpha$ minuses. And…
nombre
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Do surreal numbers contain all other number systems?

I know that some operations are defined for surreal numbers like addition or multiplication. Are all operations for all "number systems" defined for surreal numbers in a way that they give same result as respective operations defined for those…
Kamil Szot
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Every dyadic rational is constructed as a surreal number in finitely many days

I'm looking through some notes on the surreal numbers, and have noticed that Dyadic rationals can be constructed in finitely many number of days, since they are products of integers with some power of $1/2$. I want to try prove this but I am not…
kam
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