Existence and Infinity (Warning: major rambling)
September 3, 2012 at 4:02 pm
(This post was last modified: September 3, 2012 at 4:06 pm by Categories+Sheaves.)
(September 3, 2012 at 3:02 pm)idunno Wrote: That's absolutely an answer. But at this point it seems best to agree to disagree. Perhaps if C+S chimes in on this it'll become clearer, apparently he understands I'm making a distinction between an actual and potential infinite.I mean, we can talk intelligently about points at infinity (cf. extended reals, projective spaces) or else it wouldn't be on the table. We've all seen functions with vertical asymptotes (e.g. 1/x) so we can also talk intelligently about traveling infinite distances in finite amounts of time--but given that universal speed limit (3*108 yadda yadda) it's not going to have a ton of physical relevance.
The 'actual'/'potential' thing... I mean, there's a difference between being able to bet any amount of money on an auction (it can be arbitrarily large, so go ahead and break out the Conway chained arrow notation... But you're still offering finite amounts) and being able to bet infinity dollars. What would the latter even mean?
That being said, it's possible for the auction to keep going of you're clear about how the infinities work ("aleph-null dollars!", "aleph-null plus fifty dollars!", "aleph-one plus aleph-null plus a hundred dollars!" etc.). That being said, I have no idea how you would make physical sense of the result when the auction ended. Is there some time-point after the infinite amount of time we spend traveling an infinite distance?
