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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.
17
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2
answers
2k
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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 ga …
- 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
8
votes
In hyperreal field, can ln(ε) and ln(ω) be expressed as infinite sums?
To help avoid any misunderstanding that may arise for readers of this question, let me say that when understood in the usual sense, there are no nontrivial convergent sequences or series at all in the …
- 236k
21
votes
1
answer
855
views
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 …
- 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
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
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
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
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
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
32
votes
Accepted
Are the real numbers isomorphic to a nontrivial ultraproduct of fields?
The answer is no, because such ultrapowers are always $\aleph_1$-saturated, but $\mathbb{R}$ is not.
More concretely, the ultraproduct will be an ordered field with uncountable cofinality — every cou …
- 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
10
votes
0
answers
376
views
Can one define in ZFC a directed system of embeddings on the class of all linear orders real...
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 we …
- 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
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