Hilbert's Infinite Hotel - 60-Second Adventures in Thought (4/6)

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A never-ending hotel, always full of guests, helps to explain the nature of infinity.

(Part 4 of 6)
Playlist link - http://www.youtube.com/playlist?list=PL73A886F2DD959FF1

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http://www3.open.ac.uk/study/undergraduate/arts-and-humanities/philosophy/ind...

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Category:

Education

Tags:

• Mathematician
• room
• space

[oriocoookie]

the paradox is that infinite sets are proper subsets of themselves

[StrikaAmaru]

@IsraeliXdude Well, it's not a paradox proper, it's a detail of counting infinite sets, like, for instance, the set of all positive whole numbers, N.

The catch with the hotel is that, if you were to count all numbers in N, then all the even positive ints, and all odd positive ints, you'll reach the same number (because you can match each 2n with an n, and each 2n+1 with an n, ad infinitum)... despite the two sets being a partition of N. Basically, infinity + infinity = the original infinity.

[TheEmperorGulcasa]

What most people fail to realize is that infinite sets are NOT ALL EQUAL. Take, for example, all the numbers in-between 0 and 1. this is an infinite series that is smaller than the numbers between 0 and 2. Originally the hotel had two equal infinite sets: guests and rooms. The initial statement of "No Vacancies" indicates that the room and guest sets (although infinite) are fixed. The reason the idea of the hotel works is because the hotel is cheating by expanding its room set.

[Athrun000]

Apparently... infinity have different values.

[MrSharewndws]

you cannot increase the size of infinity without being its equal. we knows that is not possible. but an infinity can absorb the number 1. my precious solution.

[MrSharewndws]

you don't add to infinity. infinity adds to you.

[sniped101]

whats the point of this one?

[mmasny]

@Swolf943 Not exactly pointless. It should be said that finite mathematics is possible and probably cultivated by a few mathematicians, but it is much less useful than mathematics with infinities. Ask any physicist. The universe they propose is finite, yet infinities turn out to be indispensable in the explanation of it.

[mmasny]

@oriocoookie No, that is not the definition. The definition of a Dedekind-infinite set is that it has a proper subset of the same cardinality. The notion of a set which its own proper subset is absurd. If A is a proper subset of B, then in particular A is not equal B. If A were a proper subset of A, then A would have to be different from A, which would violate the axiom of extensionality.

[oriocoookie]

@mmasny i thought that that was the definition of an infinite set ... it is a proper subset of itself .... check cantor

[mmasny]

@oriocoookie No. That isn't true. No set can be a proper subset of itself. However, infinite sets do have proper subsets which are equinumerous to them.

[MiriamLeis]

I love your Channel!