The *Horizon* programme “To Infinity and Beyond” was repeated on BBC4 recently. I didn’t know in advance, but this showing brought another crop of emails, to most of which I have tried to reply.

Alif Mirza pointed me to chapter and verse in the holy Quran where, he says, the Big Bang is discussed. I haven’t replied in detail yet (apologies, Alif), since I have been a bit preoccupied lately and haven’t had time to read the verses. In any case, I don’t claim any special knowledge about the cosmology part of the programme, though I have tried to react to particular questions I get.

Pete Searle asked about Hilbert’s hotel: “To my mind, a hotel with an infinite number of rooms and an infinite number of guests is full.” I tried to explain how we view this: if only the even-numbered rooms are occupied, there are an infinite number of guests but still plenty of empty rooms for more guests who could arrive.

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.

Paul Brookes was puzzled about whether Infinity × Zero = One, on the grounds that

1 × 1 | = | 1, |

10 × 0.1 | = | 1, |

100 × 0.01 | = | 1, |

1000 × 0.001 | = | 1, |

and so on. Mathematicians would not agree with the deduction, since by slightly varying the calculations we can make Infinity × Zero come out to be anything we like, including infinity or zero, or not have a defined value at all. So we would say that it is not possible to give meaning to the calculation.

However, his question had a metaphysical basis, which I don’t think I quite properly came to grips with.

Paul also delighted me by pointing me to the record of his coast-to-coast walk on his blog.

Harcharan Singh was delighted with the mathematical part of the programme, and hoped that I would do more public exposition. Well, I would love to!

Finally, a correspondent I won’t name (since the conversation is ongoing) tried to persuade me that he has refuted Cantor’s diagonal argument and shown that the sets of natural numbers and real numbers have the same number of elements. Needless to say, he hasn’t convinced me yet!

Mathematics is, *par excellence*, the subject in which we don’t have to take anyone’s word for anything unless it is backed by a convincing argument. But the word “convincing” is important there!

“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?

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 2

kcorrespond to the numberk, for each natural numberk. This definition doesn’t care what kind of things the elements of the sets are.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.

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.

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?

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

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?

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?

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!