How to Count Past Infinity

https://www.youtube.com/watch?v=SrU9YDoXE88

This is so cool. I kind of know how to count this now. Thanks Akkarin for a wonderful post.

Hmm, I'd be happy to say that I do, too, but I really don't...

For example, one of the earlier questions/arguments in the video is that if you have this countably infinite set of bars you still get to insist that another bar be counted after you are "done" counting the aforementioned set. But countability does not imply that one is ever finished counting, and the presenter says this much that the union of the set of bars with a set containing the new bar would of course still have the same cardinality as the bar set. So you never actually get to the end of it and counting the added bar on the right. In order to ever actually count it (and by count I mean have a counting function map it to an integer) it has to correspond to a finite integer.
It gets even more bizarre though. If we are free to isolate a bar and put it "after" all the others and if our initial set of bars is countable that means that the set of such translations is countable, too, and it then stands to reason that they are reversible. I.e. if we have an additional bar "after" our infinite bars, we are just as free to move it back to the beginning of our set without at all disturbing our counting process.
In essence, countability means the existence of such an injective counting function. Countability does not imply this, rather this is what we mean when we say a set is countable. To go beyond Aleph-null is equivalent to leaving the realms of what is countable. To "count past infinity" is literally a contradiction in terms - unless we first redefine our terms, of course.

Another nitpick I have with something the presenter says towards the end of the video. To paraphrase, he says that even if the physical sciences would never find an application for the fringes of set theory as these topics are, it would still be great for it would mean that with but the power of our minds we had discovered something true that is beyond reality itself. Beautiful though this may sound it betrays a fundamental flaw in rationalism. How is something we just declare to be true, just to keep pondering for pondering's sake, a discovery, if whenever else we speak of discoveries we mean recognitions of aspects of or things within nature? How can something that may not have any use be anything other than use-less? How is something real if it is no part of reality? And how is something true if it has no way of being false?
At the end of the day, modern mathematics proposes assertions to feed into formal logic and sees what it can find on the other side. This is a most important endeavour indeed for it helps us predict what to expect whenever what we actually see matches up with some of the assertions hitherto proposed. While every second spent pondering the implications of something that will not ever be found is a second wasted, until the end of times comes we do not know what those things are and are thus stuck spending time with them for as long as they can keep us entertained.

Thanks for the post

Gisteron wrote: For example, one of the earlier questions/arguments in the video is that if you have this countably infinite set of bars you still get to insist that another bar be counted after you are "done" counting the aforementioned set. But countability does not imply that one is ever finished counting, and the presenter says this much that the union of the set of bars with a set containing the new bar would of course still have the same cardinality as the bar set. So you never actually get to the end of it and counting the added bar on the right. In order to ever actually count it (and by count I mean have a counting function map it to an integer) it has to correspond to a finite integer.

There is more than one kind of infinity because there is more than one way to count. He presumes that most people are unaware of these intricacies, because most are. You probably don't watch this to learn about maths if you have a highly informed knowledge of the subject.