In David Deutsch’s The Beginning of Infinity: Explanations That Transform the World there is a chapter about infinity in which he discusses many aspects of infinity. He also talks about the hypothetical scenario that David Hilbert proposed of an infinity hotel with infinite guests, infinite rooms, etc. I don’t know which parts of the hypothetical scenario are Hilbert’s original idea and which are Deutsch’s modifications/additions/etc.
In the hypothetical infinity hotel, to accommodate a train full of infinite passengers, all existing guests are asked to move to a room number that is double the number of their current room number. Therefore, all the odd numbered rooms will be available for the new guests. There are as many odd numbered rooms (infinity) as there are even numbered rooms (infinity).
If an infinite number of trains filled with infinite passengers arrive, all existing guests with room number n are given the following instructions: Move to room n*((n+1/2)). The train passengers are given the following instructions: every nth passenger from mth train go to room number n+n^2+((n-m)/2). (I don’t know if I wrote that equation correctly. I have the audio book and don’t know how it is written.)
All of the hotel guests’ trash will disappear into nowhere if the guests are given these instructions: Within a minute, bag up their trash and give it to the room that is one number higher than the number of their room. If a guest receives a bag of trash within that minute, then pass it on in the same manner within a half minute. If a guest receives a bag of trash within that half minute, then pass it on within the a quarter minute, and so on. Furthermore, if a guest accidentally put something of value to them in the trash, they will not be able to retrieve it after the two minutes. If they were somehow able to retrieve it, to account for the retrieval would involve explaining it with an infinite regress.
Some other things about infinity that he notes in the chapter:
It may be thought that the set of natural numbers involves nothing infinite. It merely involves a finite rule that brings you from one number to a higher number. However, if there is one natural number that is the largest, then such a finite rule doesn’t work (since it doesn’t take you to a number higher than that number). If it doesn’t exist, then the set of natural must be infinite.
To think of infinity, the intuition that a set of numbers has a highest number must be dropped.
According to Kant, there are countable infinities. The infinite points in a line or in all of space and time are not countable and do not have a one to one correspondence with the infinite set of natural numbers. However, in theory, the infinite set of natural numbers can be counted.
The set of all possible permutations that can be performed with an infinite set of natural numbers is uncountable.
Intuitive notions like average, typical, common, proportion, and rare don’t apply to infinite sets. For example, it might be thought that proportion applies to an infinite set of natural numbers because you can say that there an equal number of odd and even numbers. However, if the set is rearranged so that, after 1, odd numbers appear after every 2 even numbers, the apparent proportion of odd and even numbers would look different.
Xeno noted that there are an infinite number of points between two points of space. Deutsch said Xeno is misapplying the idea of infinity. Motion is possible because it is consistent with physics. (I am not sure I completely followed what he said the mistake Xeno made here was.)
This post reminds me of Ord’s mention in the The Precipice about the possibility of creating infinite value being a game changer.
In David Deutsch’s The Beginning of Infinity: Explanations That Transform the World there is a chapter about infinity in which he discusses many aspects of infinity. He also talks about the hypothetical scenario that David Hilbert proposed of an infinity hotel with infinite guests, infinite rooms, etc. I don’t know which parts of the hypothetical scenario are Hilbert’s original idea and which are Deutsch’s modifications/additions/etc.
In the hypothetical infinity hotel, to accommodate a train full of infinite passengers, all existing guests are asked to move to a room number that is double the number of their current room number. Therefore, all the odd numbered rooms will be available for the new guests. There are as many odd numbered rooms (infinity) as there are even numbered rooms (infinity).
If an infinite number of trains filled with infinite passengers arrive, all existing guests with room number n are given the following instructions: Move to room n*((n+1/2)). The train passengers are given the following instructions: every nth passenger from mth train go to room number n+n^2+((n-m)/2). (I don’t know if I wrote that equation correctly. I have the audio book and don’t know how it is written.)
All of the hotel guests’ trash will disappear into nowhere if the guests are given these instructions: Within a minute, bag up their trash and give it to the room that is one number higher than the number of their room. If a guest receives a bag of trash within that minute, then pass it on in the same manner within a half minute. If a guest receives a bag of trash within that half minute, then pass it on within the a quarter minute, and so on. Furthermore, if a guest accidentally put something of value to them in the trash, they will not be able to retrieve it after the two minutes. If they were somehow able to retrieve it, to account for the retrieval would involve explaining it with an infinite regress.
Some other things about infinity that he notes in the chapter:
It may be thought that the set of natural numbers involves nothing infinite. It merely involves a finite rule that brings you from one number to a higher number. However, if there is one natural number that is the largest, then such a finite rule doesn’t work (since it doesn’t take you to a number higher than that number). If it doesn’t exist, then the set of natural must be infinite.
To think of infinity, the intuition that a set of numbers has a highest number must be dropped.
According to Kant, there are countable infinities. The infinite points in a line or in all of space and time are not countable and do not have a one to one correspondence with the infinite set of natural numbers. However, in theory, the infinite set of natural numbers can be counted.
The set of all possible permutations that can be performed with an infinite set of natural numbers is uncountable.
Intuitive notions like average, typical, common, proportion, and rare don’t apply to infinite sets. For example, it might be thought that proportion applies to an infinite set of natural numbers because you can say that there an equal number of odd and even numbers. However, if the set is rearranged so that, after 1, odd numbers appear after every 2 even numbers, the apparent proportion of odd and even numbers would look different.
Xeno noted that there are an infinite number of points between two points of space. Deutsch said Xeno is misapplying the idea of infinity. Motion is possible because it is consistent with physics. (I am not sure I completely followed what he said the mistake Xeno made here was.)
This post reminds me of Ord’s mention in the The Precipice about the possibility of creating infinite value being a game changer.