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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.
74
votes
Accepted
What's wrong with the surreals?
At a recent conference in Paris on Philosophy and Model Theory (at which I also spoke), Philip Ehrlich gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying ma …
- 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
42
votes
Accepted
Who wins two player sudoku?
Update. I made a blog post about Infinite Sudoku and the Sudoku game, following up on ideas in this post and the comments below.
I claim that the second player wins the even-sized empty Sudoku boa …
- 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
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
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
18
votes
0
answers
885
views
Is the universality of the surreal number line a weak global choice principle?
I'd like to consider the principle asserting that the surreal
number line is universal for all class linear orders, or in other
words, that every linear order (including proper-class-sized)
linear ord …
- 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
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
17
votes
2
answers
2k
views
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
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
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
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
10
votes
Accepted
Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hier...
This kind of analysis is very well understood in ultrapowers, and one often sees this kind of thinking with ultrapowers, where one performs calculations with the representing function for an object. W …
- 236k