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7
votes
2answers
568 views

Does this construction yield the surreal numbers?

There are two simple constructions for creating arbitrarily large non-Archimedean ordered field extensions of the reals. First given such a field one may consider rational functions over that field ...
2
votes
0answers
200 views

More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
2
votes
0answers
100 views

An application of surreal numbers towards fast-growing ordinal functors?

The surreal numbers $\mathbb{SN}$ form a class of numbers introduced by J.H. Conway, which behave as an ordered field (even if technically it is not a set). In particular, Conway showed that every ...
4
votes
1answer
171 views

Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, ...
1
vote
0answers
143 views

Is a certain group, derivable from the surreal numbers, isomorphic to the surreal numbers?

Let's treat $\mathbf{No}$ as a group under addition, and forget its field structure for a little bit. I will define a "maximally Archimedean subgroup" of $\mathbf{No}$ as a subgroup which is ...
13
votes
2answers
653 views

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers?

Is it possible to generalize functions like $x^y, \ln x, \sin x, \arctan x$ to surreal numbers or surcomplex numbers? Which of their properties and relations (e.g. usual trig identities) will still ...
2
votes
1answer
460 views

Surreal numbers and large cardinals

This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject. Part 1 is about foundations. Much of the ...
14
votes
1answer
804 views

Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts. Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
12
votes
1answer
662 views

Integration in the surreal numbers

In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace ...
4
votes
1answer
317 views

Definable map from all the ordinals to the surreal numbers with a dense image?

I'm trying to understand analogies and disanalogies between ${\Bbb R}$, the reals numbers, and ${\bf No}$, the surreal numbers. ${\Bbb R}$ admits countable dense sets such as the rationals. This ...
6
votes
3answers
391 views

Uniformizing the surcomplex unit circle

Is the multiplicative Group of surcomplex numbers of modulus 1 isomorphic to the additive Group of the surreal numbers modulo the sub-Group of surreal integers? And, do Norman Alling's surreal ...
16
votes
1answer
810 views

Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
10
votes
2answers
631 views

Constructing the surreal numbers as iterated Hahn series

A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with ...
5
votes
4answers
1k views

A question about J.H. Conway's SURREAL NUMBERS

My quesion is: What set theory are the mathematicians who are developing the theory of these numbers working in-or are they, in fact, working outside any of the standard set theories?. Each surreal ...
4
votes
3answers
1k views

Surreal Numbers and Set Theory

Hello, I looked through MathOverflow's existing entries but couldn't find a satisfactory answer to the following question: What is the relationship between No, Conway's class of surreal numbers, and ...
16
votes
2answers
890 views

Surreal Numbers as Inductive Type?

Prompted by James Propp's recent question about surreal numbers, I was wondering whether anyone had investigated the idea of describing surreal numbers (as ordered class) in terms of a universal ...
18
votes
4answers
2k views

Where do surreal numbers come from and what do they mean?

I know about Conway's original discovery of the surreal numbers by way of games, as well as Kruskal's way of viewing surreal numbers in terms of asymptotic behavior of real-valued functions, leading ...