Which set theory? ZFC doesn't permit this.

Non-well-founded set theories are so non-standard that I think it's wrong, or at least misleading, to claim that unqualified "set theory" permits this.

I’m not sure how to interpret this other than Atari not wanting to compete with OpenTTD on Steam.

Presumably this is the famous (?) story of him listing his race as “mongrel” whenever asked?

* A set that is open can also be closed.

Good faith question: aside from hosting costs, what costs are there, given the reviewers are unpaid?

Comments URL: https://news.ycombinator.com/item?id=46026976

Points: 2

# Comments: 0

There are loads of 10GbE switches from Cisco/Juniper/Arista/et al.

I’m using Linux rather than AIX.

fcntl(2) locks are supported (as long as they aren't OFD), but flock(2) locks don't work across nodes.

1. Should conference "service" be something we expect of postdocs (and even PhD candidates) rather than established experts?

> Often, as a result, the PC is staffed by junior, ambitious academics intent on filling their résumés. Note that it does not matter for these résumés whether the person did a good or bad job as a referee! [...] I very much doubt that the submissions of Einstein, Curie, Planck, and such to the Solvay conferences were assessed by postdocs. Top conferences should be the responsibility of the established leaders in the field.

2. Should programme chairs strive to maintain exclusivity of their conference track, or look for important ideas that deserve to be communicated?

> As a simple example, consider a paper that introduces a new concept, but does not completely work out its implications and has a number of imperfections. In the careerist view, it is normal to reject it as not ready for full endorsement. In the scientific view, the question for the program committee (PC) becomes: is the idea important enough to warrant publication even if it still has rough edges? The answer may well be yes. [...] Since top conferences boast of their high rejection rates, typically 80% to 90%, referees must look for reasons to reject the papers in their pile rather than arguments for accepting them.

3. Is computer science suffering from a focus on orthopraxy rather than scientific method?

> What threatens to make conferences irrelevant is a specific case of the general phenomenon of bureaucratization of science. Some of the bureaucratization process is inevitable: research no longer involves a few thousand elite members in a dozen countries (as it did before the mid-1900s), but is a global academic and industry business drawing in enormous amounts of money and millions of players for whom a publication is not just an opportunity to share their latest results, but a career step.

What do you think about these?

I meant BB(748) apparently.

To elaborate on this point a bit, I specifically mean uncomputable in ZFC. There may be other foundations in which it is computable, but we can just find another n for which BB(n) is uncomputable in that framework since BB is an uncomputable function.

Direct:

Make one of the sets uncomputable, at which point the equality of the sets cannot be decided. This happens when the real defined by the Dedekind cut is itself uncomputable. BB(764) is an integer (!) that I know is uncomputable off the top of my head. The same idea (defining an object in terms of some halting property) is used in the next proof.

Via undecidability of Cauchy reals:

Equality of Cauchy reals is also undecidable. The proof is by negation: consider a procedure that decides whether a real is equal to zero; consider a sequence (a_n) with a_n = 1 if Turing machine A halts within n steps on all inputs, 0 otherwise; this is clearly Cauchy, but if we can decide whether it’s equal to 0, then we can decide HALT.

Cauchy reals and Dedekind reals are isomorphic, so equality of Dedekinds must also be undecidable.

Hopefully those two sketches show what I mean by decidable; caveat that I’m not infallible and haven’t been in academia for a while, so some/all of this may be wrong!

The equivalence classes of integers: pairs of naturals with (a, b) ~ (c, d) := (a + d) = (b + c).

The equivalence classes of rationals: pairs of integers with (a, b) ~ (c, d) := ad = bc.

It’s “easy” to tell whether two integers/rationals are equivalent, because the equivalence rule only requires you to determine whether one pair is a translation/multiple resp. of the other (proof is left to the reader).

Cauchy sequences, on the other hand, require you to consider the limit of an infinite sequence; as the GP points out, two sequences with the same limit may differ by an arbitrarily large prefix, which makes them “hard” to compare.

We can formalise this notion by pointing out that equality of integers and rationals is decidable, whereas equality of Cauchy reals is not. On the other hand, equality of Dedekind reals isn’t decidable either, so it’s not that Cauchy reals are necessarily easier than Dedekind reals, but more that they might lull one into a false sense of security because one might naively believe that it’s easy to tell if two sequences have the same limit.