John

Dunno, the wording of the first line of the OP seems to indicate that, whether he means rationals or reals, he intends for both intervals to be subsets of the same set. But perhaps he'll respond and say otherwise. Certainly.

No, the cardinality of (0, 1) is the same as the cardinality of (1, infinity). The fact that there exists a bijection between the two intervals is proof. The same reasoning applies with the rationals. Edit: To add more detail, the function in question is f : (0, 1) -> (1, infinity) defined by f(x) = 1/x. The construction of the rationals and the reals ensures that in either set, every non-zero element has a multiplicative inverse. Furthermore, since every x will be less than 1 and positive, every y = 1/x will be greater than 1 and positive. Thus, our function is valid on (0, 1) in either the rationals or the reals, i.e. every x in (0, 1) will be mapped to some y = 1/x in (1, infinity). Consider two distinct elements y1 = 1/x1 and y2 = 1/x2 such that y1 = y2. This means 1/x1 = 1/x2, therefore x1 = x2. Thus the function is injective. Now take any y in (1, infinity). If y = 1/x, then we have x = 1/y. Since every non-zero element of the reals (or rationals) has a multiplicative inverse, this x certainly exists. Furthermore, since y > 1 and the positive reals (or rationals) are closed under multiplication, 1/y must be less than 1 and positive, i.e. 1/y = x must be in (0, 1). Thus the function is surjective. Since the function is injective and surjective, it is bijective by definition. And since the existence of a bijection between two sets implies the two sets are equinumerous, we have that (0, 1) and (1, infinity) are equinumerous.

The reasoning you just used is essentially the proof. By pairing each real number in [math](1, \infty)[/math] with exactly one number in [math](0,1)[/math] (taking into account, of course, the equivalence between certain numbers in the sets, e.g. 6/4 = 3/2), you've constructed a bijection (specifically, the mapping f(x) = 1/x) between the two intervals, which in turn means they have the same cardinality, i.e. the same number of elements. In fact, when dealing with the real numbers (or even just the rationals), there are as many elements in any non-empty, non-degenerate interval as there are in any other non-empty, non-degenerate interval. Edit: To be clear, pairing each real number in [math](1, \infty)[/math] with exactly one number in [math](0,1)[/math] forms an injection, but it's easy to show the mapping in question is also surjective, thus it is a bijection.

Dune is an amazing book. The last brain-related book I read was John Ratey's A User's Guide to the Brain. It's over a decade old at this point, but I think it's still worth reading.. For myself, I've been reading Andrzej Sapkowski's The Witcher series lately, since I've finally started playing the video games and wanted to know the backstory. Some of the translation is a bit weird, but it's been mostly enjoyable so far anyway. Also, in terms of nonfiction, I've just started William Cook's In Pursuit of the Traveling Salesman, which (as the name implies) is about computational complexity. Reviews on Amazon say it's a pretty decent book, recreational in tone but with some real substance. I'm only two chapters in so far, though, so I'll have to wait and see.

My point was that Python is a fairly easy language to get into. Java isn't bad either, but it may be a bit more complicated initially. I personally started with C++, so perhaps my perspective is skewed. However, I stand by my second paragraph. Most languages aren't that difficult to learn. Programming concepts are the bigger hurdle.

Neither Python nor Java is particularly difficult to learn and understand. Both are commonly used as beginner languages in computer science curricula. Python in particular is very clear. Java is perhaps slightly more complex at the beginning, but it has the advantages of being fairly widely used and sharing a lot of syntax with handy languages like C/C++ and Perl. In any case, the main challenge in learning to program is understanding programming concepts themselves. Outside of a few deliberately designed esoteric languages, learning syntax is a small part of the process and not generally difficult. BASIC strikes me as an odd recommendation in 2013, heh. I suppose Visual Basic is handy if you're into that sort of thing, and TI-BASIC might be fun if you happen to have a TI graphing calculator and want to write new programs for it, but generally speaking, there are much more useful and still simple to learn languages available these days.

The IS Theory wiki is now hosted by BYU at http://istheory.byu.edu/wiki/Main_Page. There is a link on the original page to the new Wiki, but I figured a direct link from this thread would be handy.

Oh, really?

I'm really not much of a forum person.

Hi, I'm also John, and I'm not new at all, though I haven't really been active very much in the course of my ~10-year membership. I could have sworn I'd posted in this thread already, but the search results indicate I haven't. I'm a mathematics student just starting in my studies, and I enjoy learning about pretty much everything, though my levels of interest in various subjects rise and fall, with only a few maintaining a consistently high level (those being indicated in my profile---along with everything else I've posted here so far, but, no matter). In my spare time, I like to read, look around, breathe, sleep, wake up, and make conversations awkward by interrupting discussions of things like professional sports with mention of things like probability theory.

The Language of Mathematics: Making the Invisible Visible by Keith Devlin. Pretty good overview of some major areas of mathematics. It's a little outdated now (for instance, the Poincare conjecture has since been proved), but it's still a good read. The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory by Brian Greene. Also a decent book, a bit more technical and oriented towards String Theory than Fabric of the Cosmos. Man's Search for Meaning by Viktor Frankl. Starts with Frankl's description of his time in a concentration camp, then introduces the reader to logotherapy, probably my favorite of the three major Viennese schools of psychology. Death by Black Hole: And Other Cosmic Quandaries by Neil deGrasse Tyson. Easy-to-read discussion of various topics in astrophysics. It's a fun read, not very technical. I may add some more later, but those are some of my favorites from my reading in the last few years.