Timeline for Measuring big stuff
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 26, 2023 at 19:59 | history | edited | Joel David Hamkins |
edited tags
|
|
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
|
|
| Nov 26, 2012 at 10:31 | vote | accept | Vidit Nanda | ||
| Nov 23, 2012 at 8:18 | comment | added | Gerhard Paseman | Motivated by a rereading of Qwfwq's comment above, here is a suggestion: consider instead the lattice of subvarieties of groups, which is dual to the lattice of (some modifier which eludes me, perhaps invariant?) congruences on a free countably generated group. One can try a measure on either lattice to make sense of the statement "most groups are Abelian", since many classes of groups of interest are equational classes or closely related (e.g. quasivarieties). Gerhard "Seeking Measures Which Measure Up" Paseman, 2012.11.23 | |
| Nov 22, 2012 at 23:51 | history | edited | Vidit Nanda |
changed tags in light of joel's answer
|
|
| Nov 22, 2012 at 18:42 | answer | added | Gerhard Paseman | timeline score: 1 | |
| Nov 22, 2012 at 15:24 | comment | added | Qfwfq | (...) Of course there are as many Abelian groups as there are groups (a proper class thereof, or a well determined cardinal if you only count groups below a certain fixed cardinality of elements), but we would nevertheless like to be able to say rigorously that "most groups are not Abelian". | |
| Nov 22, 2012 at 15:23 | comment | added | Qfwfq | If I interpret correctly the question, there's a difference between "most rationals are proper fractions" and "most sets are infinite", in that in the former case we've got a crystal clear framework (in fact, more than one: at least measure theory and topology) to compare the part and the whole, whereas in the latter case we lack a sensible and useful framework. (...) | |
| Nov 22, 2012 at 14:44 | answer | added | Joel David Hamkins | timeline score: 8 | |
| Nov 22, 2012 at 11:36 | history | edited | Vidit Nanda | CC BY-SA 3.0 |
fixed typo "examples --> example"
|
| Nov 22, 2012 at 11:34 | comment | added | Vidit Nanda | Asaf, "most sets are infinite" is ambiguous precisely because we do not yet agree on a way to measure sub-classes of the class of all sets (independently of whether or not we agree to treat two sets as equal in this class if there exists a set-bijection between them). The question asks whether there is a measure theory on classes which provides some framework for making such statements non-ambiguous. | |
| Nov 22, 2012 at 11:24 | comment | added | Asaf Karagila♦ | Well, of course that I meant the Lebesgue measure (and the standard topology for the other one). I was remarking that to say that "most sets are infinite" is an extremely ambiguous thing to say, and it makes it slightly difficult to guess what you are aiming for (at least for me). You wrote that the question is fuzzy, and I pointed out something which for me makes it harder to focus on your intention, that is all. | |
| Nov 22, 2012 at 10:14 | comment | added | Vidit Nanda | Asaf, either you don't get my question or I don't get your comments. Saying that measure theoretically there are as many integers as Cantor points assumes the existence of a common measure space (maybe Reals with Lebesgue measure?) into which both parties can inject, and then you use that implicitly assumed measure to compare. The fact that another measure (or another common space) might yield a different answer hardly undermines the usefulness of measure theory itself. | |
| Nov 22, 2012 at 6:27 | comment | added | Asaf Karagila♦ | (in addition to my previous comment, note that measure theoretically there are as many integers as there are points in the Cantor set; and topologically speaking there are more rational numbers than there are points in the Cantor set... so all these notions are very very different from one another as measurements of size) | |
| Nov 22, 2012 at 6:25 | comment | added | Asaf Karagila♦ | I think that saying that most sets are infinite is much like saying that most rational numbers are proper fractions. It's true in some sense, but very false in another. And the question is whether one seeks a measure of cardinality (even for classes), in which there are as many singletons as there are sets; or does one seek a measure theoretic-like tool or a topological density-equivalent, in which it would make sense to say "there are more of this proper class than this proper class, despite the existence of a bijection between them". | |
| Nov 22, 2012 at 4:22 | history | asked | Vidit Nanda | CC BY-SA 3.0 |