Wikipedia:Reference desk/Archives/Mathematics/2013 November 16

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November 16

Solving x ^y = y ^x in Terms of the Lambert W function

I was wondering whether it is possible to transform the implicit equation above into an explicit one, expressing y in terms of x and W(x). — 79.113.237.28 (talk) 08:44, 16 November 2013 (UTC)

Try putting x's on one side and y's on the other, say both are equal to say z. Express x in terms of z using example 2 in that article, substitute what z is in terms of y, and et voilà you're done. Of course you might just end up with the original number if you take the wrong value of W ;-) Dmcq (talk) 11:20, 16 November 2013 (UTC)

Got it. Thanks ! But what if we were to have, for instance, x ^y = y ^x <=> y = F(x). Could we express the Lambert W function in terms of F ? If so, then what would this expression be ? — 79.113.237.28 (talk) 12:04, 16 November 2013 (UTC)

Nevermind, got that one too. Thanks ! — 79.113.237.28 (talk) 12:21, 16 November 2013 (UTC)

Talking about all Class Functions

Supposing we are in NBG/KM/Anything-With-Proper-Classes, is there any sensible way to define the class of all functions, including those between proper classes? Or, simpler, the class of all functions ON -> ON? My initial thought is that functions between classes should be subclasses of products, which would make them classes, so that they couldn't be a member of another class. However, category theory has categories of functors, that would seem to require that you can have a class of such functions. Finally, supposing that there is no such collection, can you quantify over all such (does that require KM?)? I'm a bit hazy when it comes to needing to reason over classes in any complicated way; I appreciate any help/insight:-)

Clarification: I only actually need F:ON -> ON -not all class functions (even supposing that is all well defined, that seems a Russel's paradox type of deal).Phoenixia1177 (talk) 09:37, 16 November 2013 (UTC)

Well, these things can be a bit philosophically problematic — if you can talk about the collection of all functions from ON to ON, then surely ON must be a completed totality, which makes it hard to understand why it's not a set. Peter Koellner told me (if I understood him correctly) that it is a completed totality, but not an individual; I'll have to see if I can find out what he meant by that.

Anyway, I generally find the easiest way to think about these things is to just look at some V_κ where κ is some strongly inaccessible cardinal (as a lower bound — you can always add stronger properties if you need them). Then when you see ON, just think κ. Unless your motivation is explicitly philosophical, that should be good enough. If it is philosophical, then I'm not sure how much more I can help than what I've already written. --Trovatore (talk) 09:48, 16 November 2013 (UTC)

While I am philosophically curious about this situation, in this instance it is a technical interest. I have a really really complicated proof that's spanning several pages (and will, probably, end up with more like it), but if I could somehow have a class of ON -> ON to work with, the trouble goes away. If you use an inaccessible cardinal, obviously, you have just such a set in V, but can you use this in your reasoning about V_κ? Or, let me try to put that better: could you prove results in general using this set, or only results of the form "there is a model in which", so that you consistency? I'd be okay with the latter, but I'd like the former. If it's not possible, what trick does category theory use to get away with this; or are they simply abusing a more formal variation that gives it legit meaning? (I apologize if any of this sounds naive, this is about the point where philosophy/mathematics/logic start blurring together for me and I end up feeling like my intuition is wrapped in a knot- this has been helping make that better, though).Phoenixia1177 (talk) 10:05, 16 November 2013 (UTC)

I think there's a general assumption that any proof in category theory could be made rigorous in ZFC via reflection. But category theorists don't care enough about foundations, and set theorists don't care enough about category theory, so no one's actually taken the time to check. For now, everything's running on faith.

An alternative approach is to use some sort of strongly-typed category theory, where the types fall into a well-founded hierarchy.--80.109.80.78 (talk) 15:21, 16 November 2013 (UTC)

That's actually a bit unsettling- but it does seem to fit most of what I've seen. I never got the seeming divide between cats and sets; they seem conceptually alike, just with a different choice in what they abstract- I'm guessing it's in what they grew out of/originated from. Do you know of any books/articles that use, or at least discuss, using strongly typed categories?Phoenixia1177 (talk) 15:53, 16 November 2013 (UTC)

I'm afraid I don't, but I'm not at all informed about the literature; I'm not a category theorist. I can explain what I meant, though: For a category of rank $\alpha$, the objects and morphisms are allowed to be categories and functors of rank $<\alpha$ and ur-elements. — Preceding unsigned comment added by 80.109.80.78 (talk) 22:15, 16 November 2013 (UTC)

Add on: I'm curious about what was meant by ON being a completed totality; for whatever reason, ON always seemed set-like compared to other classes that come up, I think it's because of the structure of the ordinals. Obviously, that's a vague gut feeling, but it's always seemed a bit "different"/grounded, it'd be interesting if someone else had a more sensible statement of this sentiment.Phoenixia1177 (talk) 10:40, 16 November 2013 (UTC)

ON can't be a set, because if it were, it would have to be an ordinal. But then it would have to be an element of itself, because it contains all ordinals. That's a contradiction even if you don't bother about the axiom of foundation, because then what about ON+1? See Burali-Forti paradox.

So what stops it from being a set? The way I think of it is, it's not a set because it's never finished, it's not a completed totality. It's like the way some of the intuitionists (Errett Bishop, maybe?) think about the natural numbers. (However, I don't see any reason to drop excluded middle from statements about the ordinals; I never quite saw the logical step from "not completed totality" to "no excluded middle".)

But I really need to track down Peter Koellner and ask him about what distinction he was making there. --Trovatore (talk) 18:49, 16 November 2013 (UTC)

That helps a lot, my understanding of classes was that we needed them because things like the set of all sets were too large, but that never jived well with "classes can't be members of classes"- except as a practical means so we don't end up just repeating set theory with classes. But going with the idea of classes never finishing makes sense, it just so happens that that makes them large. That was always were I hit a snag, I never got the intuition behind "class = large" and what you can/'t do with classes, it just seemed like sets we can't call sets because of paradoxes. Thank you, that's a wonderful insight:-) A lot of my problems when working with sets comes less from technical aspects, but more from intuitive issues- do you know of any good books/articles on the philosophical ideas that could be useful? By the way, if you do track him down, even if long after this, please leave a note on my talk page (if you don't mind), I'd be curious to hear what was meant as well.Phoenixia1177 (talk) 19:17, 16 November 2013 (UTC)