141–150 of 359 posts
As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…
As a constructivist I'll be over in the corner that says that there are only a countable number of real numbers, and the unimaginable number of unimaginable infinities that classical mathematics insiste exists is all made up nonsense. That, in fact, things that can't ever be named, even in principle, don't actually exist. What is interesting is that as shocking as constructivism may be, there is no logical flaw in it…
Mathematicians using sets to argue about the size of uncountably infinite Reals feels like the old joke of philosophers arguing over how many angels can fit on the head of a pin. I often wonder if we've used set theory (and formalisms of math based on logic) to substitute one metaphysical theory for another equivalent one.
I can understand the frustration of mathematicians of the past being told they have to ensure their theories align with some spiritual mythology. But it sort of feels like I'm being forced to accept some kind of metaphysical truth in the name of ensuring the calculus of infinitesimals is grounded in logic.
Earlier quoted context omitted.
The non-sandwich analogy is called Hilbert’s hotel. Saying that two sets have the same cardinality is equivalent to them having a bijection between them. So the claim is that the natural numbers and the natural numbers plus a sandwich have the same cardinality. This can be proved by the bijection: 0 -> sandwich 1 -> 0 2 -> 1 3 -> 2 . . . n -> n-1 . . . There is actually more though! If you had an infinite but countab…
Because everyone gave me such good and well meaning answers perhaps you’ll permit me a follow up. As I understand it we can say the cardinality of the reals is 2^aleph_0. Why is it cheating to create a bijection thusly: 0 -> 0 1 -> 1/(2^aleph_0) 2 -> 2/(2^aleph_0) etc?
Earlier quoted context omitted.
Why does “the set containing the natural numbers and a sandwich” not have cardinality between the two?
For the same reason that integers have the same cardinality as only odd numbers. You can create a 1:1 mapping between them.
I'm no mathematician, but I have always found it strange infinities are talked about as physical states (towers of tall towers, etc), and not functions. Integer number counting is essentially a successor function; take N, add 1, output N+1. One input, one output. Real number counting is a bit more loose. To find the numbers between .1 and .2, you find all fractionals of a given size, normally 1/10. So we get .10, .11…
Unfortunately the fractionals method does not work, since there are numbers (in fact, uncountably infinitely many!) irrational numbers which cannot be expressed as fractions, which are nonetheless real numbers (such as the square root of two and pi).
If people are interested in this stuff, there’s a great course called Paradox and Infinity going on right now on edX. You’ve missed the first two homework assignments, but there’s still time to get going in the course. The course is based on or supported by the book On the Brink of Paradox .
The existence of addition implies that the answer is infinity. Too simple an explanation for mathematicians obviously.
Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…
Maddy’s work is great and worth reading (even as an antirealist!), but it’s worth keeping in mind that debates about foundations are basically irrelevant to the working lives of the large majority of mathematicians. If you’re studying, say, extremal graph theory or the Langlands program or low-dimensional topology, the cardinality of the continuum is simply not relevant.
https://pavpanchekha.com/blog/proof-system-os.html
I tried to restrict my post to just set theory because this philosophical point is often hard for people to grasp.
Earlier quoted context omitted.
The non-sandwich analogy is called Hilbert’s hotel. Saying that two sets have the same cardinality is equivalent to them having a bijection between them. So the claim is that the natural numbers and the natural numbers plus a sandwich have the same cardinality. This can be proved by the bijection: 0 -> sandwich 1 -> 0 2 -> 1 3 -> 2 . . . n -> n-1 . . . There is actually more though! If you had an infinite but countab…
Because everyone gave me such good and well meaning answers perhaps you’ll permit me a follow up. As I understand it we can say the cardinality of the reals is 2^aleph_0. Why is it cheating to create a bijection thusly: 0 -> 0 1 -> 1/(2^aleph_0) 2 -> 2/(2^aleph_0) etc?
Great article. Since it looks like a lot of folks are interested in this article, some extra background. First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow's "A beginner's guide to forcing" [1] which does a good job not only laying out the mathematical details at a high level, but also really cle…
> but at the same time there is no "right answer"—all of these axioms, after all, are independent of ZFC, and it's "ok" to add any of them. Minor quible: Just because a potential axiom is independent of ZF(C) doesn't make it necessarily "okay" to add. Potential axioms can be unsound, for example if they prove new / untrue Sigma_1 statements. As an example, even in the likely circumstance that ¬Con(ZFC) is independent…