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Jan
29
awarded  Yearling
Jan
17
comment Nice sign-expansions of special surreal numbers
And thanks yet again, Mark!
Jan
5
awarded  Favorite Question
Dec
30
awarded  Nice Answer
Dec
23
comment Do mathematical objects disappear?
Following up on Tim's example of infinitesimals, I would opine that "horn angles" as such have ceased to play a role in modern mathematics. But if I'm wrong I hope someone will correct me!
Dec
22
asked The orbit-structure of an automorphism of the full shift
Dec
15
awarded  Popular Question
Dec
5
comment Propositions equivalent to the completeness of the real numbers
If I'm interpreting that thread correctly, then one can't use ZF to prove the existence of an ordinal with uncountable cofinality. (If this is wrong, please let me know!)
Dec
2
comment Nice sign-expansions of special surreal numbers
Thanks again, Mark.
Dec
1
awarded  Nice Question
Nov
30
comment Nice sign-expansions of special surreal numbers
For instance, must the sign expansion of a ratio of ordinals have the finite-orbit property I mentioned in the original post? And must a surreal number whose sign-sequence satisfies the finite-orbit property belong to the Field generated by the ordinals?
Nov
30
comment Nice sign-expansions of special surreal numbers
Since Mark S. has answered my "two very concrete questions" (with the answers "yes" and "no", respectively), I'll approve his solution. Thanks, Mark! However, it seems to me that the original broader question is still worth exploration. It would be satisfying to have a quasi-combinatorial understanding of the sign-sequences associated with ratios of ordinals and with elements of the Field generated by ordinals.
Nov
30
accepted Nice sign-expansions of special surreal numbers
Nov
29
comment Nice sign-expansions of special surreal numbers
Is this a ratio of ordinals? I don't think so, but I don't see how to prove it. Note that the presence of a minus sign is not decisive here, since for instance $\omega^2 -\omega+1$ is expressible as the ordinal $\omega^3+1$ divided by the ordinal $\omega+1$.
Nov
24
comment Propositions equivalent to the completeness of the real numbers
I don't understand the role of the axiom of choice in the theory of ordinals deeply enough to see what the problem is, but let me sidestep my ignorance and ask, what's an example of an ordinal that can be shown in ZF to have uncountable cofinality?
Nov
19
awarded  Nice Question
Nov
17
awarded  Popular Question
Nov
14
comment Comparing really big numbers
I agree, but how does this relate to the question? I was asking about a routine that accepts as input a pair of definitions of natural numbers (like a googolplex and Graham's number) and outputs a single bit that says which one is bigger. So the notion of a "bigger output" does not immediately apply.
Nov
14
asked Comparing really big numbers
Oct
17
awarded  Good Question