Comments on: Infinite reactions, 5

By: dratman

dratman — Sun, 25 Mar 2012 23:41:33 +0000

A travel agent wanted to know if the hotel is set up for Continuum (some kind of holiday club, I think). I assured her they take EVERYBODY, but she insisted this was different. Something about a “diagonal” scheme. I would just ask the manager, but ever since I inquired about the percent occupancy, he refuses to take my calls. Said it was none of my damned business, and hung up!

By: Peter Cameron

Peter Cameron — Sun, 25 Mar 2012 15:31:42 +0000

I am sure that, if the manager kept infinitely many rooms free, (s)he would get into trouble from management for under-utilising the resources. Though again, following on from Paul Brookes’ question, if infinitely many of the infinite number of rooms are full, what is the percentage occupancy?

By: Sune K Jakobsen

Sune K Jakobsen — Sun, 18 Mar 2012 21:11:34 +0000

The lobby could be unbounded and the hotel could have infinitely many employees! This would make it possible for infinitely many people to check in, in a finite amount of time. I don’t think you have to be able to contact all the guests in finite time, as long as the information travels much faster than the guests, this shouldn’t be a problem. I think the biggest problem would be that most guest would have to move really far. Perhaps a better strategy would be never to fill the hotel, and always keep infinitely many rooms free?

By: dallas simpson

dallas simpson — Sat, 17 Mar 2012 17:21:07 +0000

Thanks for your reply. I’m curious about mathematical ways of describing things and their reality. I think I’m modelling things in ways that mathematicians don’t. These are very difficult ideas to express as I can’t use equations. In my understanding the attribute ‘oddnumber’ is a pure realm of quality, an abstract descriptor, a realm of unexpressed potentiality of number – a ‘dimension-like’ realm.

While the set of odd numbers is an expresssed potentiality – an infinite set populated by a particular (particle-like or quantized) ‘set’ of number. So, from my particular viewpoint, an infinite expressed set of odd numbers can be considered to be an expressed set of number ‘bounded by quality’, that is the quality of ‘oddnumberness’ and not bounded by magnitude.

So what is the difference between my description of ‘oddnumber’ as a pure dimension-like realm of quality and the ‘empty set’? Probably one of definition! And ultimately properties.

In my description, as a number becomes larger and ‘approaches infinity’ it becomes more dimension-like and less particle like.

I think its as you say – I’m using a different definition where attribute and magnitude affect the property, whereas ‘(your) definition doesn’t care what kind of things the elements of the sets are.’

I’m sorry if I’m completely off topic, its just that I find such considerations fascinating. Thank you for your time in responding.

By: Yiftach Barnea

Yiftach Barnea — Sat, 17 Mar 2012 17:16:51 +0000

Of course, it is possible that the guests and the rooms are getting smaller and smaller!

By: Peter Cameron

Peter Cameron — Sat, 17 Mar 2012 16:19:03 +0000

I agree completely. In fact, I think there is a more serious problem. If all the guests could be contacted in a finite amount of time, they would all have to lie within a bounded distance of the lobby. Then, for sure, the whole hotel would collapse into a black hole …

Maybe this is what caused the Big Bang?

By: dratman

dratman — Sat, 17 Mar 2012 16:11:33 +0000

it suddenly occurs to me that Hilbert’s Hotel could not not really (funny word) have been as easy to manage as claimed, because an infinite number of guests would have been out and unreachable (no cell phones then) when it was time to switch rooms. For that matter, an infinite number of them would have been ill, even dead, or maybe just uncooperative. But even a finite number of such snarls would (conjecture) cause the recommend transfers to fail for an infinite amount of time. An infinite number of them would cause the blood pressure of the manager to rise to a… what should I say… would cause his or her or … would cause it to go way up.

By: Peter Cameron

Peter Cameron — Sat, 17 Mar 2012 14:15:08 +0000

I'm sorry, but I don't understand you. Why should an infinity have a boundary? Is there a difference between "the attribute `oddnumber'" and the set of odd numbers? A correspondence is, simply, a way of matching up the two sets so that each element of one set is mapped with a unique element of the other set. We match the even natural numbers with all natural numbers by letting the even number 2k correspond to the number k, for each natural number k. This definition doesn't care what kind of things the elements of the sets are.

By: dallas simpson

dallas simpson — Sat, 17 Mar 2012 13:38:34 +0000

“Counting is, basically, just establishing a correspondence between two sets. Nowadays one of these sets is usually a set of numbers, and the other the unknown set we want to count. There is archaeological evidence from Iraq that an innumerate shepherd would match up his flock with a set of pebbles or clay balls, and the numerate owner of the flock (or his accountant) could then count the balls and know how many sheep he had. The argument I presented in the programme shows that we can match up the set of even numbers with the set of all counting numbers; so these two infinite sets contain the same number of elements, even though one is a proper subset of the other.”

The ‘problem’ here, as I see it, is what defines the boundary of the infinity, exactly what we mean by ‘correspondence’, and indeed the nature of ‘elements’. Counting is indeed ‘just establishing a correspondence between two sets.’ But at another level of reality the attribute ‘oddnumber’ is not the same as the attribute ‘allnumber’, as indeed stones are not sheep. In number theory mathematics these attribute distinctions are apparently not of any significance, but I wonder whether there are situations where the differing attributisation of the infinities being ‘in correspondence’ is relevant to the point of invalidity?