Wikipedia:Reference desk/Archives/Mathematics/2016 October 29

= October 29 =

Is there possibly a relationship between Russell's Paradox, Godel's Incompleteness Theorems, and the Vacuous Truth?
Hello, let me state what my views are on Russell's Paradox:

I think Russell was correct to redefine some of the language. If one understands the true meaning of a "set" as a collection of elements that has no specific existence in and of itself, that is, it's not a predicate, then one can easily dissolve the dilemma in my opinion.

This would make R not a member of itself because the definition itself is arbitrary (i.e. not "real"), just like the definition of a set itself. However, R would also be a member of itself because of the definition of what goes under R. This is a contradiction only because I think one fails to differentiate between what actually exists and what is being defined. For example if I have a blue car, then I can say accurately that "I drive a blue car." However, if I had to break my car down into individual atoms, then I'd be driving something that's neither blue nor a car (something similar to Sorites' Paradox or Theseus' Ship).

So the "barber who shaves all men who don't shave themselves" example can best be understood by acknowledging that one hasn't defined the relationship between customer and barber properly. This might seems contradictory, but just like our "blue car" example, it's ultimately an example of the fallacy of language: such a barber does not "exist," just like sets don't "exist" in and of themselves.

So if any of these ideas of mine have a possible grain of understanding, I was wondering how they might relate to the vacuous truth (R has a property that it doesn't have because it doesn't "exist"), Godel's Incompleteness Theorems (which if I understand correctly, imply that one cannot fully prove anything to be what it is without making some assumptions), and Kant's idea that "existence," isn't a predicate (which I think relates to the 'possible worlds' of philosophers). --Cornelius (talk) 03:05, 29 October 2016 (UTC)


 * see here and here, and here Count Iblis (talk) 07:08, 29 October 2016 (UTC)

Coastlines
I've been reading List of countries by length of coastline, and understand how the coastline of Britain is fractal and varies as the measuring stick used changes. I am puzzled by the coastline of GB being stated as 28000km (what is the superscript 1.43?) yet the table list the bigger UK as only 12429km.

Anyway that apart, how does the Hausdorff measure work? I've read the article and don't understand a word of it. Could someone summarize how it works in English rather than in Maths please. One (semi-random) idea inspired by the superscript number is that in some fashion the coastline length remains stable with changing measure stick at some particular fractional dimension. (Or not ...) -- SGBailey (talk) 08:17, 29 October 2016 (UTC)
 * The 1.43 is the Hausdorff dimension and you should probably read that article before the one on Hausdorff measure. The Hausdorff measure is first of all a measure which is an abstraction of concepts like length and area, but as an abstraction it's difficult to summarize what it means. For a non-specialist, the main thing is that 1.43 is greater than 1, so the coastline does not have a a length in the usual sense, and is less than 2, so it has no area. The Coastline paradox article has more on this. The fact that 1.43 is not an integer means that the coastline is, according to one definition, a fractal. --RDBury (talk) 16:57, 29 October 2016 (UTC)
 * Well, it does have an area. It just happens to be zero.  Arguably, it also has a length, which just happens to be infinity. --Trovatore (talk) 22:18, 29 October 2016 (UTC)

Can an ultrafilter, on an infinitely countable set, be countable?
HOTmag (talk) 21:26, 29 October 2016 (UTC)
 * You mean, literally? Can the set of all measure-one sets be countable?  No, of course not.  There are $$2^{2^{\aleph_0}}$$ sets that have to be measured, and for every measure-zero set, there's a unique corresponding measure-one set (its complement), so the cardinality of the ultrafilter has to be $$2^{2^{\aleph_0}}$$.  Did you mean something else? --Trovatore (talk) 22:13, 29 October 2016 (UTC)
 * What?
 * Our article ultrafilter, defines an ultrafilter on a given S - "as a filter on S that cannot be enlarged", whereas our article filter "Define[s] a filter F on S as a nonempty subset of P(S)" - with some additional properties indicated ibid. So, an ultrafilter on S is a subset of P(S) (with some additional properties). Consequently, the cardinality of an ultrafilter of an infinitely countable set, cannot be larger than the cardinality of the powerset of S. HOTmag (talk) 23:13, 29 October 2016 (UTC)
 * Well, this is somewhat context-dependent, but generally what one means by an ultrafilter "on" the natural numbers is a subset of P(P(N)).
 * It is a little bit confusing, because you can also speak of an ultrafilter "on" a Boolean algebra, and that is indeed a subset of the powerset of the Boolean algebra. This is a generalization of the first usage (the relevant Boolean algebra in the first case would already be P(P(N)) . --Trovatore (talk) 23:19, 29 October 2016 (UTC)
 * Hold on, sorry, I'm getting mixed up here. You're right; you can certainly speak of an ultrafilter on the naturals as being a subset of P(N).  I reached into a part of my brain I haven't looked in in a while, and some dust came out.
 * In any case, no, it can't be countable. It partitions P(N) into two pieces of equal cardinality (every set in the ultrafilter is uniquely matched with its complement, which is not in the ultrafilter), so its cardinality has to be $$2^{\aleph_0}$$. --Trovatore (talk) 23:28, 29 October 2016 (UTC)
 * Thankxs. HOTmag (talk) 23:33, 29 October 2016 (UTC)
 * (So more generally any ultrafilter on an infinite set X has the same cardinality of  P(X)  ) --pm a  18:02, 30 October 2016 (UTC)