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Results tagged with surreal-numbers
Search options questions only
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user 10059
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.
10
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1
answer
681
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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?
- 10.1k
8
votes
2
answers
906
views
In surreal numbers, what exactly is $\omega_1$?
This answer refers to $\omega_1$ in context of surreal numbers, and calls it "first uncountable ordinal".
But what exactly does it mean? How can it be represented in the $\{L|R\}$ form?
How do we know …
- 10.1k
6
votes
2
answers
964
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In surreal numbers, what is the successor of all the germs in the Hardy field?
I have my own totally ordered hierarchy of quantities, including infinite ones. Can I embeed them in surreal numbers somehow?
For instance, I have the quantity $\omega$, which I identify with the simi …
- 10.1k
6
votes
1
answer
488
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In surreal numbers, what are the main difficulties so far in defining integration?
I know, there were several (including unsuccessful) attempts at defining integration on surreal numbers, so I am asking for a good summary of what have been the main difficulties so far.
Particularly, …
- 10.1k
6
votes
1
answer
752
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Are periodic functions such as sine and cosine defined on surreal numbers?
Surely, one can compose a power series for them, and any partial sum of those series would be defined, But are they defined in the limit?
I mean, what is $\cos \omega$, for instance?
Does the trigonom …
- 10.1k
1
vote
1
answer
122
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Question on derivation of $\omega$ in surreal numbers
This paper gives a derivation definition on log-atomic surreal numbers:
where the logarithm with lower index means iterated logarithm.
I think — I may be wrong — that $\omega$ is a log-atomic number …
- 10.1k
1
vote
1
answer
249
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Confusion regarding $\ln \omega$
This answer says that in surreal numbers $\ln \omega=\omega^{1/\omega}$.
At the same time, this Wikipedia article says that transseries $\mathbb{T}^{LE}$ are isomorphic to a subfield of $No$ with its …
- 10.1k
1
vote
1
answer
197
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Representing the set of rationals $\mathbb{Q}$ as a germ or surreal number
Let us define natural equivalence between elements of Hardy fields and integrals of Dirac comb-like functions.
Let us assume a natural embedding of Hardy field into surreal numbers ($[x]=\omega$). De …
- 10.1k
0
votes
0
answers
117
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Can one represent divergent integrals or germs at infinity with surreal numbers?
I have been disliking the theory of surreal numbers for a while, but let's test it.
So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\inft …
- 10.1k
0
votes
0
answers
138
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Is standard, affine infinity of extended reals quite small on the scale of infinities?
Some time ago I had a conversation with a guy who was into surreal numbers and he said that in surreal numbers the affine infinity is quite minor entity compared to the ordinality of natural numbers $ …
- 10.1k
0
votes
0
answers
142
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Why is the property of linearity against an infinitely-large factor considered essential for...
Why the property $(b)$ in Proposition 14 in this paper on surreal integration is considered essential?
The Proposition lists the desired properties of the surreal integration, and among others lists t …
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-1
votes
1
answer
268
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Why is it said that all surreal numbers with birthdate $<\omega_1$ are isomorphic to a Hardy...
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 s …
- 10.1k
-1
votes
1
answer
109
views
In surreal numbers, do the automorphisms allow us to define $\omega_2=\partial(\omega_1)$?
Consider surreal numbers as an H-field with operation of derivation.
In such setting for any surreal number $\alpha$ such that $0<\alpha<e^\omega$, $\partial(\alpha)<\alpha$ and for $\alpha>e^\omega$, …
- 10.1k
-4
votes
0
answers
30
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In surreal numbers, why log-atomic numbers are not EL-numbers?
In surreal numbers, the log-atomic numbers are those numbers that can be obtained from $\omega$ and its powers via iterated logarithm or exponential function.
At the same time, Timothy Chow's EL-numbe …
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