Veteran TSZers may recall an entertaining thread in which a bunch of us tried to explain the cardinality of infinite sets to Joe G:
At UD, commenters daveS and kairosfocus are now engaged in a long discussion of the transfinite, spanning three threads:
The sticking point, which keeps arising in different forms, is that KF cannot wrap his head around this simple fact: There are infinitely many integers, but each of them is finite.
For example, KF writes:
DS, I note to you that if you wish to define “all” integers as finite -which then raises serious concerns on then claiming the cardinality of the set of integers is transfinite if such be applied…
The same confusion arises in the context of Hilbert’s Hotel:
Try, the manager inspects each room in turn, and has been doing so forever at a rate of one per second. When does he arrive at the front desk, 0?
Re: your HH explanation: If the manager was in room number -100 one hundred seconds ago, he arrives at the desk now.
Yes a manager can span the finite in finite time. But the issue is to span the proposed transfinite with an inherently finite stepwise process. KF
In the scenario I described above, the manager was in room -n n seconds ago, for each natural number n. Given any room in the hotel, I can tell you when he was there.
DS, being in room n, n seconds past does not bridge to reaching the front desk at 0 when we deal with the transfinitely remote rooms; when also the inspection process is a finite step by step process.
What KF doesn’t get is that there are no ‘transfinitely remote rooms’. Each room is only finitely remote. It’s just that there are infinitely many of them.
Any bets on when — or whether — KF will finally get it?