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Results tagged with surreal-numbers
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user 1946
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.
9
votes
In surreal numbers, what is the successor of all the germs in the Hardy field?
The central construction feature of the surreal numbers is that it is Ord-saturated, which means that for any sets of surreal numbers $A$ and $B$, with $A<B$ in the sense that every element of $A$ is …
- 236k
15
votes
Accepted
Surreal Numbers, Proving $x1=x$
If you look at the Wikipedia entry for surreal multiplication, you find
The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right set …
- 236k
9
votes
Are periodic functions such as sine and cosine defined on surreal numbers?
Since the surreal numbers form a saturated real-closed field, it follows that they serve as a (proper class) version of the hyperreal numbers, including the transfer principle. This means that absolut …
- 236k
7
votes
Accepted
Definable map from all the ordinals to the surreal numbers with a dense image?
The answer is that the existence of a definable class embedding
like that is independent of ZFC. In fact, it is equivalent to the
axiom V=HOD.
Theorem. The following are equivalent.
There is a def …
- 236k
13
votes
Accepted
Largest ordered "field" in NBG without axiom of global choice
There is no problem defining the surreal field without global choice.
One can define it in ZFC and considerably weaker theories, for
example with the hereditary birthday construction of left-sets and
…
- 236k
19
votes
Accepted
Going beyond the surreal numbers
Relentlessly filling cuts is of course the main construction idea of the surreal numbers---at every ordinal birthday, one fills all the cuts that exist in the previously-born surreals. Your proposal i …
- 236k
17
votes
What do we know about the computable surreal numbers?
Here is some partial progress. I claim that the computable surreal numbers include some noncomputable real numbers, confirming my guess in connection with question 2.
For each TM program $e$ we can wr …
- 236k
10
votes
What do we know about the computable surreal numbers?
Let me explicate fuller details about the computable surreal number operations. Let's start by showing that they form a ring.
Theorem. The computable surreal numbers form a ring.
Proof. We have to sho …
- 236k
18
votes
Accepted
In surreal numbers, what exactly is $\omega_1$?
There is nothing special about $\omega_1$ or indeed any infinite number in the surreals, and they cannot be defined purely from the field structure of the surreals. What I claim is that all infinite n …
- 236k
43
votes
4
answers
3k
views
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 …
- 236k
5
votes
Accepted
What are the properties of $\operatorname{No}[i]$?
The surreal complex field $\text{No}[i]$, known as the surcomplex field, is a proper-class-sized set-saturated algebraically closed field. It is universal for all fields of characteristic 0. Indeed, u …
- 236k
6
votes
First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V...
$\newcommand\Ord{\text{Ord}}\newcommand\HOD{\text{HOD}}$Let me
point out that the existence of a definable bijection of $V$ with
$P(\Ord)$ is equivalent to the assertion that the universe is
Leibnizia …
- 236k
5
votes
Surreal compactness
$\newcommand\No{\text{No}}$
Here is a proof of the second claim, that there is a proper class-sized open
cover of the surreal init interval with no set-sized subcover.
In particular, there is no finit …
- 236k
15
votes
Accepted
Can you build the surreal numbers as a simple direct limit of ordered fields?
Here is one way to get a positive answer to the title question.
Theorem. There is a definable class $\mathcal{F}$ of ordered fields, containing isomorphic copies of any given field, and a directed ord …
- 236k
6
votes
Why is it said that all surreal numbers with birthdate $<\omega_1$ are isomorphic to a Hardy...
As Philip Ehrlich had mentioned in the other post, the initial claim of your question is Corollary B of the following paper
Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven, Filling ma …
- 236k