Questions tagged [surreal-numbers]
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.
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Intermediate Value Theorem for Hahn series
Let $R[\![\Gamma]\!]$ be the set of Hahn series over $R$ with a linearly ordered value group $\Gamma$, i.e. the set of functions $\Gamma \to R$ with well-founded support. Recall that $R[\![\Gamma]\!]$ ...
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Monotonicity Proof for Simple Genetic Functions
I define a simple genetic function to be a function on the surreals that has is defined as left and right sets of other simple genetic functions, and itself applied to simpler inputs. All constant ...
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Birthday of a surreal number
There are two ways that one could reasonably define the birthday of a surreal number $x$:
The smallest birthday among all forms $\{L|R\}$ that are equal to $x$.
The smallest birthday among all ...
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Is $\log\omega$ an omnific integer?
Question: Is $\log\omega$ an omnific integer?
Is $\log\omega\in\mathbf{Oz}$?
Conway [ONAG, pg.46] defines an omnific integer to be divisible
iff $x$ is divisible by every finite nonzero integer.
...
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Addition as the "simplest" operation having certain properties
I once went to a talk by John Conway in which presented his theory of surreal numbers in a different way than the approaches taken in "Surreal Numbers", "On Numbers and Games", or &...
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Proper class sized hyperreals
In his paper [1], pp. 36-37, Ehrlich quotes Keisler on a possible construction of proper class sized hyperreals. Keisler indicates to construct four objects $\mathbb{R}$, $\mathbb{R}^*$, $<^*$, $^*$...
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Making multiplication closure work on surreal numbers as sign sequences
I've been trying to understand the form of induction used to prove multiplication closure on surreal numbers and I'm a bit stumped.
In Gonshor's Introduction to the theory of surreal numbers he proves ...
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Homomorphism from field of hyperreals to field of reals? [closed]
I am curious if it is possible to construct a homomorphism from a field of hyperreal numbers to the field of real numbers? (Similarly, a homomorphism from the surreals to the reals?)
Assuming that ...
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Do the surreal numbers enjoy the transfer principle in ZFC?
The surreal field $\newcommand\No{№}\No$ is definable in ZFC, and it is easy to see that the surreal order is $\kappa$-saturated for every cardinal $\kappa$, precisely because we fill any specified ...
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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 ...
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In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
In the hyperreal field, we can use Taylor series to express e^(ε) and e^(ω) as:
e^(ε) = 1 + ε + (ε^2)/2! + ...
e^(ω) = 1 + ω + (ω^2)/2! + ...
Is it similarly possible to express ln(ε) and ln(ω) as ...
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Growth of the hyperexponential
Am I correct that the hyperexponential $\exp_{\omega}$ is a bijection on positive infinite surreals?
An exponential level is an equivalence class for the relation $a \asymp_L b \Leftrightarrow \exists ...
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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&...
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What makes the surreals special among other surreal-like fields?
Pre-setup: Let $\kappa = \aleph_\xi$ be an uncountable regular cardinal. Its role in this question is merely to sidestep the technical difficulties surrounding the (imho quite uninteresting) notion ...
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Birthday of combinatorial game product
Recall that the birthday $B(G)$ of a combinatorial game $G$ is recursively defined as the least ordinal $\alpha$ such that $G = \{L | R\}$ for some sets of games $L$, $R$ with birthdays less than $\...
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Computing the truncations (“ancestors”) of a surreal number from its Hahn series representation (“normal form”)
If $x$ is a surreal number and $\alpha$ an ordinal, let us denote $T_\alpha(x)$ and call $\alpha$-truncation of $x$ the surreal number whose sign sequence is obtained by truncating the sign sequence ...
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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,
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Is there a minimal (least?) countably saturated real-closed field?
I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.
Is there a soft model-theoretic construction ...
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Labelling non-Archimedean sets
I was reading papers 1 and 2 on numerosities. These present a way of comparing the size of sets as equivalences of the size of the set intersected with finite subsets. I am trying to extend the work ...
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What do we know about the computable surreal numbers?
The surreal numbers are built up in a natural iterative process, by which at any ordinal stage, if one has two sets of surreal numbers $L$ and $R$, with every number $x_L$ in $L$ strictly below every ...
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Why is it said that all 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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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 ($[x]=\omega$). ...
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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. ...
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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 ...
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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 ...
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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 ...
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Are there results unique to non-standard analysis or surreal numbers that have not been reconciled with classical analysis?
I am exploring areas where non-standard analysis or the theory of surreal numbers has yielded results that remain exclusive to these frameworks without analogs or proofs in classical analysis. For ...
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Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
Let $K_1, K_2, \dots$ be a countable sequence of fields, and let $\prod_{\mathcal F} K_i$ be the ultraproduct with respect to some nonprincipal ultrafilter $\mathcal F$.
Question: Can there be a field ...
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Are gaps and loopy games interchangeable in the Surreal Numbers?
The class of surreal numbers (commonly called $No$) is not complete: it contains gaps. Some people have studied the "Dedekind completion" of the surreal numbers in order to do limits and ...
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What are the properties of $\operatorname{No}[i]$?
I was looking for a complex extension of the surreals and then I found $\operatorname{No}[i]$, what are its properties and how do I express $x+yi$ in the $a | b$ notation?
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Proof of Theorem Concerning Conway's "Nim Field"
I have a question about the proof of theorem 44 in "On Numbers and Games" on page 58, concerning the "Nim field" ${ON}_2$. As background, ${ON}_2$ turns the ordinals into a field ...
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Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?
Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...
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Can you build the surreal numbers as a simple direct limit of ordered fields?
The surreal numbers are sometimes called the "universally embedding" ordered field, in that every ordered field embeds into them. What "universally embedding" means seems to be ...
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Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions
This question was originally asked at MSE but seems too advanced, so I'm reposting it here.
In short, the idea is that many constructions for non-Archimedean fields can naturally be iterated, in some ...
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Surreal numbers and the ultrafilter lemma
In this question, I asked how to interpret a historical claim made by Conway regarding the potential, if not ideal, use of the surreal numbers for nonstandard analysis. In the comments and answers it ...
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A "surnatural numbers" as a largest model of the natural numbers
One characteristic of the surreal numbers is that they are a monster model of the first-order theory of real numbers, according to Joel David Hamkins in this post. Thus they are real-closed, and every ...
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Interpreting Conway's remark about using the surreals for non-standard analysis
In Conway's "On Numbers And Games," page 44, he writes:
NON-STANDARD ANALYSIS
We can of course use the Field of all numbers, or rather various small
subfields of it, as a vehicle for the ...
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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?
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What does it mean for the surreal numbers/partizan games to be "universally embedding"?
In "On numbers and games", Conway writes that the surreal Numbers form a universally embedding totally ordered Field. Later Jacob Lurie proved that (the equivalence classes of) the partizan ...
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Are the Surreals a cogenerator in the category of ordered fields?
A cogenerator in a category $\mathcal{C}$ is an object $\Omega$ such that for any pair of parallel arrows $f,g:X\rightrightarrows Y$ in $\mathcal{C}$ we have
$$
\forall h:Y\to\Omega\big(h\circ f=h\...
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Is the surreal number $\omega(\sqrt{2}+1)+1$ a prime?
In the 1986 book An Introduction to the Theory of Surreal Numbers, Gonshor, on page 117, notes that it is an open problem whether $\omega(\sqrt{2}+1)+1$ is a prime, using the standard definition of ...
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Can one represent divergent integrals or germs at infinity with surreal numbers?
I have been disliking the theory of surreal numbers for a while, but let's test it.
So, we have a set of divergent improper integrals of continuous functions with the following ordering: $\int_0^\...
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Surreal numbers and the Collatz iteration as a game?
Let us define a game based on the Collatz function $C(n) = n/2$ if $n$ is even, otherwise $=3n+1$.
Each number $n$ represents a game played by left $L$ and right $R$:
$$n = \{L_n | R_n \}$$
The rules ...
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Quantum surreal numbers
Toward Quantum Combinatorial Games presents the definition of a "quantum game", allowing a superposition of moves rather than a single classical move. This leaves me wondering: Since surreal ...
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Smallest ring whose field of fractions includes all the reals (subring of omnific integers?)
The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the ...
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How does Conway's proposed compromise for constructing the real numbers in ONAG actually work?
I have also asked this question on Math Stack Exchange (link).
In On Numbers and Games, after discussing the inclusion of the real numbers in the surreal numbers, No, Conway discusses the merits of ...
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In theory, how would Oneiric numbers be defined?
Background I am not a professional mathematician. I am researching Surreal numbers & games for fun (I think they are truly beautiful). If this question is not appropriate here, I beg forgiveness &...
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Are Conway's combinatorial games the "monster model" of any familiar theory?
This is related to this question about a "mother of all" groups, and so seemed like it'd fit in better at MO than MSE.
If I understand the answer to that question correctly, the surreal numbers have ...
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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 $\...
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Surreal numbers and the Axiom of Choice
In ZFC and its conservative extension NBG, it can be shown that every ordered field embeds into the surreal numbers.
How much choice is needed to prove this?
Without choice, what is a simple example ...
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