Poll #13: "0,1,2,3... keep counting, you're heading towards infinity. Take a number, divide it in half, divide the result in half, keep dividing... you're heading towards infinity Are each of those infinities the same, or is there more than one version of infinity?"
(Poll ended May 3 2008)
In the end, 53% said there is more than one version of infinity, and the remainder said all infinities are the same. This poll was nip and tuck right to the end, with each answer taking the lead over the other throughout the time the poll was running.
There has been a spirited discussion about questions related to single/multiple infinities at the tenth dimension forum: "Infinity, Eternity, Endlessness and Number System" has generated over 130 comments there so far. I believe part of the controversy here is merely language-related: when someone says "infinity", what exactly do they mean? Are they talking about a number, or a path to a number, or something else entirely?
Interestingly, doing a google search for the phrase "there can be only one infinity" (in quotes) returns mostly theological and philosophical sites. On the other hand, doing a google search for the phrase "multiple infinities" (in quotes) returns mostly math and physics sites.
Plus.maths.org and WorseleySchool.net say there's more than one infinity. PhysicsForums.com says there's different degrees of infinity. Maria Henderson's TCMTechnology blog (a teacher's blog about teaching college math) says there are countable and uncountable infinities, and this blog entry includes a great set of parody lyrics for the song "Hotel California" which explains some of the seemingly paradoxical ideas behind there being multiple infinities. (Edit: Maria has now made her blog accessible by invitation only, too bad. Here's a different link to the lyrics to that parody.)
Robert Kaplan's "The Art of the Infinite: the Pleasures of Mathematics" says there can be infinite infinities. Rudy Rucker's Infinity and the Mind: the Science and Philosophy of the Infinite" is described by Martin Gardner as "a dizzying glimpse into that boundless region of blinding light where the mysteries of transcendence shatter the clarity of logic, set theory, proof theory, and contemporary physics". Click here for a site that talks about Cantor and his proposals that some infinities can be subsets of other infinities, while clicking on the following words takes you to a review of of the book "A Brief History of Infinity", which also mentions Cantor again and his idea that there are multiple infinities, and some infinities are larger than others. This blog says there can be small infinities and big infinities. And finally here is a mathematical dissertation that says there are many roads to infinity.
Infinity is not a number, it's a concept, and even though one infinite set can be a subset of another infinite set, ultimately all infinite sets are the same because they are all of indeterminate size... (just like the point we start from in the tenth dimension animation!). That's why I think "there are many roads to infinity" is a great phrase, as it shows us how different infinities can appear to be of different sizes as we travel towards them, and this is why saying that there are multiple infinities in the tenth dimension animation has strong roots in mainstream theory. In the animation, I say:
"but how can there be anything more than infinity? The answer is, there can be other completely different infinities created through initial conditions which are different from our own big bang."So, those different infinities we're talking about all come from the same background of unobserved quantum indeterminacy, which is the place where all infinities are the same.
There are many roads to infinity.
Click here for the blog discussing polls 1 through 10.
Click here for the blog discussing polls 11 through 15.
Next: Poll 14 - Agree or disagree? Time is a direction, and our physical bodies can't travel in the opposite direction.