Infinite Hotel

Originally posted by Anthem
What mathematician ever defined "same amount" via the existence of a bijection? That sounds more like the definition of cardinality to me.

Cardinality is the measure of the number of elements in a set.

Originally posted by Anthem
What mathematician ever defined "same amount" via the existence of a bijection? That sounds more like the definition of cardinality to me.

All mathematicians, when working with infinities, define "same amount" as "equal cardinality".

Richard

Originally posted by Palynka
A finite number of elements can always be added, but you need to change the mapping else there would be no available rooms.

What doesn't make any sense is your blatant disregard for set theory and cardinality. If two sets are linked by a bijection, they have the same cardinality, and sets with the same cardinality have the same number of elements.

Haha, ok, sorry for blaspheming against set theory.

I'm sure you don't disagree with me that infinity is not a number and you can't use it as one; it doesn't make any sense. If you use infinity as a number, laws of numbers and mathematics start to break.

I just think when you take a concept like a "hotel with infinitely many rooms" and "infinitely many guests", then yes you can form a bijection such that for each guest, they are mapping to exactly one room. This means that if you started adding guests to rooms, then for every guests that is next in line, that guest has an open room.

But you can't then "fill all the rooms" any more than you can count to infinity. You will never get to the last guest, and therefore you can never fill the last room.

Originally posted by forkedknight
Haha, ok, sorry for blaspheming against set theory.

I'm sure you don't disagree with me that infinity is not a number and you can't use it as one; it doesn't make any sense. If you use infinity as a number, laws of numbers and mathematics start to break.

I just think when you take a concept like a "hotel with infinitely many rooms" and "infinitel ...[text shortened]... You will never get to the last guest, and therefore you can never fill the last room.

Sorry for the provocation. 🙂

Let's ask the question: What does a fully booked infinite-roomed hotel mean?

Does it mean:
1) There are no unoccupied rooms.
Answer: The hotel is fully booked.

2) We can accommodate previously unroomed guests if:
2.1) Moving guests to different rooms is allowed
Answer: The hotel is not fully booked.

2.2) Moving roomed guests is not allowed.
Answer: The hotel is fully booked.

The iterative response ('does the next in line have a room'😉 cannot answer the questions whether all guests can be roomed, so I think it's misleading. If guests are the real numbers and the hotel rooms are integers, then for every guest in line there is an open room. But we know that it is not possible that all guests have a room because you cannot find a bijection between the two sets.

Originally posted by Shallow Blue
All mathematicians, when working with infinities, define "same amount" as "equal cardinality".

Richard

Some other math constructs that stand in for "amount": in analysis the Lebesgue and other measures, order type (ie ordinal numbers) for ordered sets, in various constructions it is important to distinguish between infinite sets that are cofinite and those that are not (or cocountable/not). And there are many others.

Yes, cardinality is probably the most ubiquitous and important, but my point is that natural language words are not unambiguously mathematically defined. The concept of "amount" is far more multifaceted than any one mathematical construct can capture. If we ignore this and simply say "amount" is "cardinality" then we lose most of the richness of the original concept, which has inspired far more math than just the definition of cardinality.