Linked Questions
10 questions linked to/from Is the universality of the surreal number line a weak global choice principle?
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Surreal numbers vs. non-standard analysis
What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in ...
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17
votes
2
answers
3k
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Does ZFC prove the universe is linearly orderable?
It is consistent with ZFC that the universe is well-ordered, e.g. in $V=L$ where global choice holds. I also know that it is consistent that global choice fails (although I have no immediate example ...
- 39.7k
17
votes
1
answer
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Is there a stronger form of recursion?
I'm wondering if there are any recursion principles more general than the following, first given by Montague, Tarski and Scott (1956):
Let $\mathbb{V}$ be the universe, and $\mathcal{R}$ be a well-...
- 10.1k
14
votes
1
answer
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Largest ordered "field" in NBG without axiom of global choice
I know from Wikipedia that in NBG, the surreal numbers are the largest possible ordered field (if a proper class is allowed to be a field). But then, it is written: "in theories without the axiom of ...
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9
votes
2
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A "surnatural numbers" as a largest model of the natural numbers
One characteristic of the surreal numbers is that they are a monster model of the first-order theory of real numbers, according to Joel David Hamkins in this post. Thus they are real-closed, and every ...
- 4,897
12
votes
1
answer
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For which theories does ZFC without global choice prove the existence of a proper class monster model?
Proper class sized monster models are typically formulated in a class theory like $NBG$ and they can reasonably be formalized in $ZFC$ with some kind of global choice, but for some theories you don't ...
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6
votes
1
answer
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Are the Surreals a cogenerator in the category of ordered fields?
A cogenerator in a category $\mathcal{C}$ is an object $\Omega$ such that for any pair of parallel arrows $f,g:X\rightrightarrows Y$ in $\mathcal{C}$ we have
$$
\forall h:Y\to\Omega\big(h\circ f=h\...
- 10.1k
4
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1
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Surreal numbers and the Axiom of Choice
In ZFC and its conservative extension NBG, it can be shown that every ordered field embeds into the surreal numbers.
How much choice is needed to prove this?
Without choice, what is a simple example ...
- 4,897
10
votes
0
answers
381
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Can one define in ZFC a directed system of embeddings on the class of all linear orders realizing the surreal line as the direct limit?
Consider the surreal line $\langle\newcommand\No{\text{No}}\No,\leq\rangle$, in its order structure only. This is a proper class linear order, which is universal for all set-sized linear orders, as ...
- 236k
3
votes
1
answer
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'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group
What is the smallest subfield $F\subset N_0$ such that $$(F,+,\times,\leq)\ncong(N_0,+,\times,\leq)$$ but $$(F,+,\leq)\cong(N_0,+,\leq)?$$ Since these are all going to be proper classes cardinality is ...
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