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In Hilbert’s infinite hotel, a room can always be found for newly arriving guests.
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Posted by Keith Tidman
‘No vacancies. Rooms available’. That might as well be the contradictory sign outside the Hilbert Hotel of legend. Yet, is the sign really as nonsensical as it first seems?
The Hilbert Hotel paradox was made famous by the German mathematician David Hilbert in the 1920s. The paradox tells of an imaginary hotel with infinite rooms. All the rooms were occupied by an infinite number of guests.
However, a traveller wondered if a room might still be available, and approached the receptionist. The receptionist answered that the hotel could indeed accommodate him. To make the solution work, the receptionist asked all the current guests simply to move to the next room, making it possible to assign the new guest to Room 1.
This was a scalable maneuver, accommodating any number of new lodgers, whether a hundred, a hundred million, or far more. Because of the infinite rooms, importantly there was no last room; the receptionist could therefore keep moving the current guests to higher room numbers.
But the challenge was to get a bit harder. What showed up next was an infinitely large coach occupied by an infinite number of vacationers. To accommodate these guests, the receptionist shifted people so that only the infinite even-numbered rooms were occupied.
Increasingly complex scenarios arose. Such as when an infinite number of coaches, each carrying infinite travellers, pulled into the hotel’s infinite parking lot. But we don’t need, for our purposes here, to delve into all the mathematical solutions. Suffice it to say that any number of new travellers could be lodged.
The larger significance of Hilbert’s thought experiment was that an ‘actual infinite’ is indeed logically consistent, even if on the surface it’s counterintuitive. As with Hilbert’s hotel, the infinite exists. Infinity’s logical consistency has further consequence, tying the thought experiment to the cosmological notion of an infinite past.
That is, a beginningless reality. A reality in which our own universe, like infinite other universes, is one bounded part. An unlimited reality that extends even to the ‘far side’ of the Big Bang that gave rise to our universe almost fourteen billion years ago. A universe located within the continuum of the infinite. And a universe in which change gives us the illusion of time’s passage.
A common argument in cosmology (origins) is that the string of causes must start with the Big Bang. Or, rooted in a theological origin story, that it must start with a noncontingent divine creator, or so-called ‘first cause’. The claim of such arguments is that reality doesn’t reach back indefinitely into the past, but has a starting point.
‘Our minds are finite, and yet even in these circumstances of finitude,
we are surrounded by possibilities that are infinite’.
Alfred North Whitehead, philosopher and mathematician
There are no grounds, however, to believe that our universe, with its time-stamped beginning (the Big Bang) and its one-way life-cycle toward net disorder, is the entirety of existence. Rather, an infinite history before the Big Bang, or beginningless reality, does make sense. As does the other bookend to that reality, an endless future, where infinity describes both before and after the fleetingly present moment, or what we might think of as the ‘now’. Nothing rules out or contradicts that unlimited scope of reality.
With an unchangeable beginningless reality, there is no need to evoke the concept of ‘something coming into being from nothing’; there is no need to interrupt the different laws of physics, or of time, governing each universe’s own separate reality; there is no time zero or insupportable moment of all creation. It’s infinity ‘all the way down’, to paraphrase British philosopher Bertrand Russell’s whimsical reference to infinite regress.
We ought to avoid conflating the emergence of things within our bounded universe (like the making of new galaxies, of which there is a finite number) with the emergence of things within the infinite (like the formation of new universes, of which there is an infinite variety, each with its unique properties, life cycle, and natural laws).
‘No other question [than the infinite] has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect’, declared David Hilbert. Our bounded universe is simply one part of that infinite, that is, part of beginningless reality. Our universe existing among infinite others, like the infinite rooms in Hilbert’s hotel.
