William Lane Craig gives a philosophical argument for the beginning of the universe using Hilbert's Paradox of the Grand Hotel.
You can watch his argument on Youtube:
If you don't know Hilbert's Hotel you can read it on Wikipedia:
Whatsoever, I was thinking about this Hilbert's Hotel Paradox and I did not found the Paradox.
Is Hilbert’s Hotel thought experiment a Paradox or an unsolved Topic?
William Lane Craig claims the following:
You can make room for every new arriving guest, even if all rooms where occupied before.
Wikipedia article of Hilberts Hotel captor Analysis:
Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true.
My argument, why I think it is not provable true:
First I want to explain another example, than we go back to Hilbert's Hotel:
Guest 1 moves out and knocks on Guest 2’s door. Guest 2 goes out. Guest 1 moves in the Room 2 and Guest 2 knocks on Guest 3’s door. Repeat this process every second. If you repeat this forever, there is always one guest n outside knocking on n+1's door. Here we have potential infinity, but never reach infinity. But after an infinite amount of time, every single guest moved. Finally, every guest n has moved in n + 1 Room. No more guest is outside anymore, because all moved. Something happens, which used to be impossible before.
Back to Hilbert's Hotel:
The mathematical or logical argument for Hilbert's Hotel Paradox is: Every guest can move to n + 1 room. So you can make room for any new guest (Peano axioms).
I would say, there is no logical or mathematical proof, that every single guest will move into the next room in this thought experiment. It's not clear what will happen in this infinity scenario. If you go with my argument, Hilbert’s Hotel is an unsolved topic and not a paradox. Saying every guest moves into the next room in Hilbert's Hotel is like saying every guest moves into the next room in my example above. I am not so sure, if every guest moves into the next room, because I don't know how this infinite sets interact with each other. If we don't know how this two infinite sets, Guests and Rooms, interacts with each other, than it is an unsolved topic.
William Lane Craig also is prepared in the Youtube video for such argument and says:
Sometimes people react to Hilbert's Hotel by saying: Because we can't understand the infinite. It's just beyond us. But this reaction is in fact mistaken and naive. Infinite set theory is highly developed and well understood branch of modern mathematics.
But here, Craigs argument is weak for me. Because infinite set theory is highly developed, it can not be beyond us? I think it can beyond us, of course. Something highly developed doesn't mean we understand it at all.
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