Questions tagged [surreal-numbers]
For questions about the surreal numbers, an inductively constructed ordered field that naturally contains all ordinal numbers.
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Surreal numbers by Knuth, the "bad numbers" proof.
This question is about the "Bad numbers" proof (Chapter 4, pages 24-25).
This is a proof by contradiction for sets of three numbers, so that:
$x \leq y,\ and\ y \leq z,\ then\ x \leq z$
Now,...
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Why is it said that all countable surreal numbers (with birthdate $<\omega_1$) are isomorphic to a Hardy field?
In this answer I have encountered with the following statement:
Assuming CH, every maximal Hardy field is isomorphic to
$(\bf{No}(\omega_1), \partial_{\omega_1})$, where $\bf{No}(\omega_1)$
is the ...
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Canonical embeddings in the surreal numbers
Which embeddings into surreal numbers are considered more canonical? I mean, surreal numbers are characterised by lots of automorpisms, so there is some freedom in choosing embedding of other sets and ...
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Can the surreal numbers index the class of automorphisms of the surreal numbers?
For the surreals $\mathbb{S}$ and $z$ any surreal number, can we index every possible $F:\mathbb{S} \to \mathbb{S}$ by $F_z$?
This would be unlike any other object I've encountered, the set of ...
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Why don't infinitesimals in nonstandard analysis have concrete size but infinitesimals in surreal numbers have?
I read on Wiki (https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis#Overview) that infinitesimals in NSA don't have concrete size but infinitesimals in surreal numbers have.
How is it possible?...
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Using Surreal Numbers to measure function growth rate - Tetration?
In "The Book of Numbers" by John H. Conway, pg. 299, he discusses the application of surreal numbers to quantifying the growth rate of functions.
He gives the following correspondences:
$$\...
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Nonstandard Probability Axiom Construction
I am working on extending the concept of a probability space to the surreal numbers, the main reason which being that I think that having a set whose measure is nonzero while the measure of each ...
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How can I calculate a Blue-Red Hackenbush position value using the Simplest Number Tree?
In this document (pages 5-7), the Simplest Number Tree is used to explain how to assign a value to any arbitrary Blue-Red Hackenbush position. I'm having trouble following its approach.
I think I ...
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Are $nimonics$ (nimber mnemonics) a thing?
Note: The context of the follwing is nimbers.
I found the following nimber addition mnemonic on the Wikipedia page for Fano planes. Inspired by the Fano plane mnemonic, I decided to see if I could ...
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Is $\sqrt[\omega]{\omega}$ an omnific integer? [closed]
I heard that $\omega^n$ (for any positive real, n) is an omnific integer, but dose this property also extend to numbers such as $\sqrt[\omega]{\omega}$ (aka $\omega^{1/\omega}$)?
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What Hackenbush game represents oof? [closed]
I have been reading about Hackenbush recently and have learned that the surreal numbers can be represented using RGB Hackenbush. I am having a hard time understanding On, Off, and Oof. What Hackenbush ...
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What are some applications of Surreal Numbers outside of Go endgames?
I've read through Don Knuth's book, a fair amount of "Winning Ways for your Mathematical Plays" and watched a handful of videos, and almost all the material seems to talk about definitions, ...
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Trouble with Knuth's proof that surreal numbers lie between their left and right sets
I'm reading through Donald E. Knuth's book, Surreal Numbers, and I've been struggling for days now with one little step in a single proof.
Specifically, I'm trying to work through the proof that every ...
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Defining surreal addition on signed ordinals
Consider surreal numbers as signed ordinals $\alpha\rightarrow\{-,+\}$. Suppose we already have $x<y$ defined for any two surreals, as well as $F|G$ as the simplest surreal $z$ strictly between the ...
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Transitivity of surreal numbers in their sign expansion
For any ordinal $\alpha$, we call any function $\alpha\rightarrow\{-,+\}$ a surreal number.
Denote by $D(x)$ the domain or Day of $x$, and by $\delta(x,y)$ the smallest ordinal for which $x$ and $y$ ...
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