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 ...
- 236k
11
votes
Interpreting Conway's remark about using the surreals for non-standard analysis
I will provide some arguments why analysis is best developed using Nonstandard Analysis as opposed to any approach based on the surreals.
If you are interested in the constructive/computable content ...
- 4,359
9
votes
Can a game be an option of itself?
A surreal number cannot have itself as an option, because numbers are defined inductively: if $L$ and $R$ are sets of numbers, then $\{L|R\}$ is a number. Notice that $L$ and $R$ have to already ...
- 295
8
votes
Accepted
Interpreting Conway's remark about using the surreals for non-standard analysis
Conway was of course correct in saying that NSA is irrelevant to the surreals, but like Mike I found Conway’s further remarks about NSA
puzzling and I am not sure what he had in mind. What I think ...
- 6,453
7
votes
Accepted
The surreal version of $e$
$\DeclareMathOperator{\ee}{e}$If $\varepsilon$ is an infinitesimal surreal, the quantity $\log(1+\varepsilon)$ is actually equal to the formal sum à la Hahn series $\sum \limits_{n \in \mathbb{N}} \...
- 2,519
6
votes
In surreal numbers, what are the main difficulties so far in defining integration?
A preliminary answer that should really be a comment, but it's too long; everything I'm about to say is better explicated in
Integration on the Surreals, O. Costin, P. Ehrlich, 2024, arXiv:2208.14331
...
6
votes
Why is it said that all surreal numbers with birthdate $<\omega_1$ are isomorphic to a Hardy field?
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 ...
- 236k
5
votes
Accepted
Are there results unique to non-standard analysis or surreal numbers that have not been reconciled with classical analysis?
The key paper in this area is Henson and Keisler:
C. W. Henson and H. J. Keisler, On the strength of nonstandard
analysis}, J. Symbolic Logic, 51 (1986), no. 2, 377-386.
The elaborate on the point ...
- 16.6k
5
votes
What are the properties of $\operatorname{No}[i]$?
In addition to the properties mentioned in Joel's answer, $N_0[i]$ is a cogenerator in the category of all fields of characteristic $0$ by a similar argument to the one given by Keith Kearnes here, ...
- 10.1k
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, ...
- 236k
5
votes
Surreal numbers vs. non-standard analysis
Coming back to the first post. Most of modern mathematics is set-theoretic, that is, it studies sets of different kind, so that reals, real and complex functions, relations on reals, as well as a ...
- 2,281
5
votes
Interpreting Conway's remark about using the surreals for non-standard analysis
Sam Sanders' answer is very comprehensive. I'll try to add some more context.
How do you add a transfer principle to (the initial segments of) the surreal numbers? I tried to explain it with Mikhail ...
4
votes
Surreal numbers and the Axiom of Choice
This is not an answer but just a set of comments:
A cousin of 1. is a yet unanswered question of J. D. Hamkins.
Notice that if an ordered field embeds into $\mathbf{No}$, then so does, as an ordered ...
- 2,519
1
vote
An extension of the mean-value theorem for integrals?
No, a counterexample would be a function that is negative at and around $a$, but starts becoming positive as it approaches b, such that
$$\int_{a}^b f(x)g(x) dx > 0$$
Let $f(x)$ be negative ...
- 11
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