Skip to main content

Questions tagged [surreal-numbers]

For questions about the surreal numbers, an inductively constructed ordered field that naturally contains all ordinal numbers.

Filter by
Sorted by
Tagged with
24 votes
2 answers
8k views

What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain: real numbers a hierarchy of infinitesimal ...
Nathan Reed's user avatar
21 votes
4 answers
2k views

Are surreal numbers actually well-defined in ZFC?

Thinking about surreal numbers, I've now got doubts that they are actually well-defined in ZFC. Here's my reasoning: The first thing to notice is that the surreal numbers (assuming they are well ...
celtschk's user avatar
  • 43.5k
14 votes
2 answers
2k views

Surreal and ordinal numbers

Is there a surjective map between the (class of) ordinal numbers On and the set No (Conway's surreal numbers) and is it constructable, In Conway's system we have for example: $\omega_0 = < 0,1,2,3,...
user avatar
15 votes
1 answer
1k views

Surreal numbers without the axiom of infinity

Let $ZF^\times$ denote the set of axioms of Zermelo-Fraenkel set theory without the axiom of infinity. The set $V_\omega$ of all hereditarily finite sets is a model of $ZF^\times$, and $Ord^{V_\omega}=...
Asaf Karagila's user avatar
  • 396k
6 votes
1 answer
1k views

Curiosity with surreal numbers

I'm a high school student who is interested in surreal numbers and studying it by myself. Suddenly a weird idea popped in my mind. For example, can we define an event $A$ that has $P(A)=\frac1{\omega}$...
user avatar
4 votes
1 answer
172 views

Prove that a surreal number is born in a finite stage if and only if it is of the form $\frac m{2^n}$.

We define surreal numbers here. My attempt is to first prove this lemma: Lemma 1. Suppose in the $n$th stage, we have already constructed 2 surreals $a<b$, with no other surreals constructed ...
Trebor's user avatar
  • 4,776
3 votes
2 answers
376 views

A "complete" ordered field of ordinals

First, a note regarding proper classes: One can construct a tuple of proper classes, for example with the Morse definition. Same goes for proper-class-sized algebraic structures, as in the field of ...
user76284's user avatar
  • 6,017
3 votes
1 answer
1k views

$\omega$th iteration of Cayley-Dickson construction

The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D). The ...
user820789's user avatar
-1 votes
1 answer
106 views

Cauchy completion of transfinite "rationals"

Let the Hessenberg power $\alpha^\beta$ be the supremum of ordinals that are order-isomorphic to some well-order on the set of finite-support functions $\beta \rightarrow \alpha$ that extends the ...
user76284's user avatar
  • 6,017
14 votes
1 answer
673 views

Non-standard measure

Just a bit of a strange question. Modern formulations of probability theory rest upon measure theory. This poses an issue for non-measurable sets. Typically, one simply excludes these sets from the ...
E8xE8's user avatar
  • 404
7 votes
1 answer
824 views

A question about something in Conway's "On Numbers and Games"

In the book mentioned in the title, which deals with (among other things), Conway's "surreal numbers", there is a small section (pp. 37-38) where the "gaps" in the surreal number line are discussed. ...
The_Sympathizer's user avatar
7 votes
3 answers
1k views

Why is epsilon not a rational number?

I was wondering why epsilon, the smallest positive number, isn't a rational number. I was watching a video a few days ago about surreal numbers, and I've learned that, in the field of surreal numbers, ...
Asix's user avatar
  • 555
6 votes
2 answers
164 views

What is the “maximal hyperreal field”?

In many SE posts and the Wikipedia article on the surreal numbers I’ve seen references to a “maximal” hyperreal field that’s isomorphic to the surreals. If they’re isomorphic, then why is it that ...
Lave Cave's user avatar
  • 1,159
5 votes
1 answer
313 views

Can the surreal numbers be completed to form an ordered field?

The surreal number line isn’t Cauchy complete as it’s filled with “gaps”. When constructing the real numbers from the rationals, one could take the ring of all Cauchy sequences and take the quotient ...
Lave Cave's user avatar
  • 1,159
2 votes
1 answer
168 views

Clarify about Knuth's Surreal Numbers

I'm reading Surreal Numbers, by Knuth, and I don't understand well the formalism at the very beginning. Here is what it is unclear: We define that a number $x $ corresponds to a couple of set (of ...
Dr. Scotti's user avatar
  • 2,503
2 votes
1 answer
341 views

What exactly is surreal star? What does it mean that it is incomparable to zero?

I have been thinking about this for a while & I am perplexed. What exactly is surreal star? I am aware that it is a fuzzy game. What I don't understand is what exactly that means. Wikipedia says ...
user773237's user avatar
2 votes
1 answer
386 views

Conway's 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 ...
mathoverflowUser's user avatar
1 vote
4 answers
175 views

List of equivalence surreal numbers to 4 day?

I can obtain surreal numbers in n-day then I don't want only list of surreal numbers example in 2-day: -2<-1<-1/2<0<1/2<1<2 but I want list of equivalence surreal numbers exemplar 2-...
farshad's user avatar
  • 11
1 vote
3 answers
660 views

Real equivalent of the surreal number {0.5|}

I've been reading up on Surreal numbers, but have some questions. Some equivalent real and surreal numbers. ...
alan2here's user avatar
  • 1,017
1 vote
1 answer
495 views

Conway Notation for Large Countable Ordinals

I have not previously seen anything online that dives deeply into On: In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of ...
user820789's user avatar