Timeline for Are there more rational numbers than integers?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 22, 2021 at 17:17 | history | edited | RavenclawPrefect | CC BY-SA 4.0 |
corrected typos, added LaTeX, removed extraneous comments
|
| Jan 11, 2014 at 0:10 | comment | added | supercat | Is it necessary to define a bijective mapping, or would it be sufficient to show that there exists way of mapping every element in A to a unique element in B, and a (possibly different) way of mapping every element in B to a unique element in A? In some cases, it's easy to define mappings in both directions which don't use all elements of the target set, but harder to find a mapping from A to B which uses all elements in B (or vice versa). | |
| S Jan 5, 2014 at 11:50 | history | suggested | user99302 | CC BY-SA 3.0 |
felt this is such a good answer there is no need to put people through searching the pdf either
|
| Jan 5, 2014 at 11:49 | review | Suggested edits | |||
| S Jan 5, 2014 at 11:50 | |||||
| Aug 1, 2010 at 18:18 | comment | added | Jason DeVito - on hiatus | I think one of the main difficulties here is that there are many notions of "size". Off the top of my head, there are the number of elements of a set, the measure as a subset of R^n, and whether or not something has high density. The problem is that mathematically, all 3 of these concepts diverge, while our everyday experience (or at least mine) tells us the 3 should be (roughly) the same. | |
| Aug 1, 2010 at 5:22 | comment | added | Qiaochu Yuan | Another interpretation of density might be Lebesgue measure. Funnily enough, the integers and the rationals both have measure zero, as does any countable subset of R. | |
| Aug 1, 2010 at 5:21 | comment | added | Qiaochu Yuan | Sure. One can study the topology of the rational numbers and the integers as subsets of the real line, and they are very different. The rational numbers are, in the technical sense, dense (their closure is R), and the integers are discrete. That's one sense in which the rationals are more dense than the integers. | |
| Jul 31, 2010 at 21:08 | comment | added | Kurt Pfeifle | My last class in math was in the German high school equivalent, 37 years ago, so I'm a bit rusty ;-) -- However I found Jason's explanation convincing for me. -- Now I've got this idea: could one apply a concept of "density" upon both sets of numbers (integers and rationals) and somehow proof that "while both are of same size, the rationals have a higher density " ? [Of course it would all depend on a stringent definition for "density"... but maybe such a thing/concept/idea already exists in math and number theory?? -- I would be interested to know if that is the case. | |
| Jul 31, 2010 at 20:08 | vote | accept | CommunityBot | ||
| Jul 31, 2010 at 20:08 | vote | accept | CommunityBot | ||
| Jul 31, 2010 at 20:08 | |||||
| Jul 31, 2010 at 20:08 | comment | added | user510 | Excellent answer. I hadn't really considered that this was set theory (someone else added that tag). Now, I can see that this is the only way to interpret relative "size" that makes sense in this context. Thanks. | |
| Jul 31, 2010 at 20:01 | history | answered | Jason DeVito - on hiatus | CC BY-SA 2.5 |