Timeline for What do we know about the computable surreal numbers?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 17 at 5:53 | comment | added | Joel David Hamkins | Another issue: since the division argument has now broken (see my answer), we don't know that we can divide computably, and this affects the $\sqrt{x}$ argument. We would no longer know that this is computable. | |
| Jun 16 at 19:25 | comment | added | Steven Stadnicki | @JoelDavidHamkins I think the biggest difference between square root and general rootfinding is the montonicity; it's very clear e.g. that $\sqrt{x^L}$ is in the left set for $\sqrt{x}$ and $\sqrt{x^R}$ is in the right set, because $x^R\gt x\implies \sqrt{x^R}\gt\sqrt{x}$. You might be able to elide this by using small enough neighborhoods to ensure that $f(x)$ is monotonic on that interval, but I suspect getting there is a little tricky in its own right. | |
| Jun 16 at 16:46 | comment | added | Joel David Hamkins | Sergei Starchenko suggests aiming at the intermediate value theorem as a way to prove real closed. By induction on degree can assume derivative has IVT. | |
| Jun 16 at 13:55 | comment | added | Joel David Hamkins | Does the square root formula generalize easily to $n$th roots? | |
| Jun 16 at 12:55 | comment | added | Joel David Hamkins | For roots of odd-degree polynomials, can we implement this recursive idea using a surreal version of Newton's method? After all, we have the formal derivative for polynomials. | |
| Jun 16 at 9:10 | comment | added | Joel David Hamkins | Thanks very much! Yes, the recursion theorem method of my other answer applies here, since there will be a program solving the recursion expressed by the formula. Conclusion: if $x$ is a computable surreal number, then so is $\sqrt{x}$, and furthermore, there is a uniform computable procedure to compute a program for $\sqrt{x}$ given a program for $x$. Great! (Now, we've just got to do the same thing for solving odd-degree polynomial equations.) | |
| Jun 16 at 1:14 | history | answered | Steven Stadnicki | CC BY-SA 4.0 |