# 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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### 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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### Why is the property of linearity against an infinitely-large factor considered essential for surreal integration?

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 ...

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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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### Who discovered the surreals?

Common folklore dictates that the Surreals were discovered by John Conway as a lark while studying game theory in the early 1970's, and popularized by Donald Knuth in his 1974 novella.
Wikipedia ...

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### Biggest Field Of Characteristic $p$

The Surreal nummbers, $\boldsymbol{No}$, are according to Wikipidia the biggest ordered field, and the Surrcomplex numbers are the biggest field of characteristic 0. Biggest in the sense that every ...

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### Algebraic Geometry Over the Surreal and Surrcomplex Numbers

I was wondering whether or not there is some kind of theory of algebraic geometry over the field of Surreal and Surrcomplex numbers?

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### The surreal numbers under a change of universe

Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...

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### Genetic construction of roots of surreal polynomials

In On Numbers And Games, Conway uses the term "genetic" for definitions of operations on surreal numbers that are inductive in terms of their options. His definitions of addition and multiplication ...

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### 'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group

What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...

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### Surreal Numbers, Proving $x1=x$

I am trying to learn the theory of the Surreal numbers and I am therefore going over all the theorems and trying to prove them for myself.
I am struggling to complete the proof of $x1 = x$.
I have ...

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### Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...

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### Functions on a field representable by Hahn series?

It is well known (see here for example) that a function over $\mathbb{R}$ is representable by a power series iff its analytic continuation to $\mathbb{C}$ is holomorphic on some open subset of $\...

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### Is the inverse of surreal numbers actually well-defined?

J.Conway wrote in his book "On numbers and games" (1st edition, 1976) on p. 66
It seems to us, however, that mathematics has now reached the stage where formalization within some particular ...

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### Largest ordered "field" in NBG without axiom of global choice

I know from Wikipedia that in NBG, the surreal numbers are the largest possible ordered field (if a proper class is allowed to be a field). But then, it is written: "in theories without the axiom of ...