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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 do surreal numbers relate to real numbers?

I had the impression that surreal numbers were a subset of reals, being the smallest possible interval away from any other number you could specify. Now, after reading the book, “Surreal Numbers”, it seems that reals are a subset of surreals since…
poetasis
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What is the difference between the hyperreals and $R^3$?

I just found out about the hyperreals. Here is a visualization of the hyperreals: The problem I have with this image, is that it conveys a very clear intuition, but often times, those intuitions are very misleading, so I don't know exactly how to…
user56834
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What is a good way to represent the surreal numbers in a line?

I would like to know a good way of sketching the surreal number line. I wonder if there is a format that is widely used. In this video, Conway makes a quick drawing of a line, just like the real line (without gaps), and separates the integers to the…
MTLaurentys
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List of equivalence surreal numbers to 4 day?

I can obtain surreal numbers in n-day then I don't want only list of surreal numbers example in 2-day: -2<-1<-1/2<0<1/2<1<2 but I want list of equivalence surreal numbers exemplar 2-day: equivalence surreal numbers 2day Do one know it for 3-day and…
farshad
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Is $\frac{1+\frac{1}{\omega}}{\omega}$, for $\omega$ a transfinite number greater than all integers, a surreal number?

The number $$\frac{1+\frac{1}{\omega}}{\omega}$$ for $\omega$ a transfinite number greater than all integers is a surreal number or it don't support this composition of infinitesimals?
Iago newats
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What is the proof-theoretic strength of the theory of Surreal numbers & games?

I was reading about ordinal analysis & this made me wonder what is the proof-theoretic strength of the theory of Surreal numbers & games? Hopefully this isn't a nonsensical question.
user784623
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Is $\sqrt[\omega]{\omega}$ an omnific integer?

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 $\omega^{1/\omega}$)?
Gabriel Tellez
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Proving Proving $ \sqrt{a \cdot b} \le (a+b)/2$ with a and b in N*

We have variables $a$ and $b$ as natural numbers.. I tried using the reccurence but I got stuck proving: $\sqrt{(a+1) \cdot (b+1)} \le (a+1+b+1)/2$ May someone help me with this?
Malek Gara-Hellal
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What is new representatives for well-known values in surreal numbers?!

I know how to obtain surreal numbers in n-day and I know about <= in surreal numbers axiom 2. We will also discover a lot of new representatives for well-known values. For example : $\{-1|1\}=0$ or $\{1/2|2\}=1$ , ... But I'm confused with some new…
farshad
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