Infinity and Time
What is "Hilbert's Hotel" and the problems it raises, and how does Craig use it against the eternal universe?
This question takes us to the heart of one of the most powerful contemporary philosophical arguments against the eternity of the universe, developed by William Lane Craig based on a famous mathematical paradox formulated by mathematician David Hilbert. Understanding this argument and its applications is necessary for anyone who wants to engage with the contemporary debate about the beginning of the universe.
Inadequate responses to avoid
From some defenders of the kalām argument:
"Hilbert's Hotel proves the impossibility of the infinite, and God is beyond this." This is a logical contradiction. If the infinite is absolutely impossible, how can we describe God with infinite attributes (infinite power, infinite knowledge)? Craig himself distinguishes between the actual infinite and the potential infinite, and does not claim the impossibility of all types of infinity.
"Mathematical paradoxes prove the falsity of atheism." This is an unjustified leap. Paradoxes of infinity raise deep philosophical questions, but they do not "prove" a metaphysical position directly. The transition from mathematics to metaphysics requires careful argumentative bridges.
"Mathematicians reject the actual infinite." This is a historical error. Since Georg Cantor in the nineteenth century, modern mathematics has dealt with actual infinities systematically (set theory). The disagreement is not mathematical but philosophical.
From some critics:
"Hilbert's Hotel is merely a thought experiment with no relation to reality." This is misleading oversimplification. The paradox reveals real tensions in the concept of actual infinity when applied to the material world. Dismissing it as a "game" ignores the genuine philosophical problem.
"Modern physics proves the possibility of infinity in the universe." This confuses levels. Mathematical models in physics may use infinity as a computational tool, but this does not mean that actual infinity exists in physical reality. Even physicists disagree on interpreting these models.
"Craig applies double standards to God and the universe." This criticism has partial validity but needs precision. Craig distinguishes between temporal and atemporal existence. His argument is that actual infinity is impossible in temporal series, not in atemporal existence. This distinction is open to criticism but is not naive "double standards."
Why these responses are inadequate
They share a failure to understand the precise logical structure of Craig's argument. The argument is not "infinity is impossible therefore God exists," but rather a series of precise inferences connecting the impossibility of actual infinity in temporal series with the necessity of a beginning for the universe. Understanding these details is necessary for serious engagement with the argument.
What is "Hilbert's Hotel"?
German mathematician David Hilbert (1862-1943) formulated this paradox to illustrate the puzzling nature of actual infinity:
Imagine a hotel with an infinite number of rooms, all occupied. A new guest arrives. In an ordinary hotel, there is no room for him. But in Hilbert's Hotel, the manager asks each guest to move to the next room (from room n to room n+1). The guest in room 1 moves to 2, the one in 2 to 3, and so on to infinity. Now room 1 is empty for the new guest!
Even stranger: if a bus with an infinite number of new guests arrived, they could all be accommodated! How? Each guest moves from room n to room 2n (from 1 to 2, from 2 to 4, from 3 to 6...). Now all the odd-numbered rooms (1, 3, 5...) are empty - and they are infinite in number!
Philosophical problems
The paradox reveals several problems:
The part-whole problem. In ordinary mathematics, the part is smaller than the whole. But in Hilbert's Hotel, the set of even-numbered rooms (a part) equals the set of all rooms (the whole). This violates our basic intuition about the relationship between part and whole.
The addition and subtraction problem. In a completely full hotel, one can add a guest (even infinity guests) without removing anyone. And if an infinite number of guests leave, the hotel might remain full or empty or half full - depending on who leaves! (∞ - ∞ = ?)
The equality problem. In Hilbert's Hotel, number of rooms = number of even rooms = number of odd rooms. But even rooms + odd rooms = all rooms. So does 2 × ∞ = ∞?
How does Craig use this?
William Lane Craig transforms this paradox into a philosophical argument against the possibility of an actual infinite existing in the material world:
First step: From mathematics to metaphysics. Craig says: These are not mere mathematical games. If an actual infinite existed in reality (like a real Hilbert's Hotel), it would lead to metaphysical contradictions. For example: books in an infinite library with half red and half black - the number of red books = number of black books = total number!
Second step: From spatial to temporal infinity. If actual infinity is impossible in space, it is impossible in time as well. An infinite series of past events = a realized actual infinite.
Third step: Impossibility of infinite past. If the universe were eternal, the series of past events would be actually infinite. But this leads to paradoxes: for example, the number of past days = number of past years, even though a year = 365 days!
Fourth step: Necessity of beginning. Therefore the universe has a temporal beginning. And what has a beginning needs a cause (principle of causality). This cause must be outside time - otherwise it would itself need a cause.
Strengths of Craig's argument
Conceptual clarity. Craig carefully distinguishes between: potential infinite - a series that can continue without end, and actual infinite - a set already realized with infinite members. His argument is against the latter only.
Internal coherence. The argument does not rely on religious text or vague intuition, but on philosophical analysis of the concept of infinity. Even a philosophical atheist can follow and evaluate the argument.
Intuitive force. The paradoxes Craig points to have intuitive force. It is difficult to imagine a real library with infinite books, where removing all odd-numbered books does not reduce the total number!
Serious criticisms and Craig's responses
Criticism: Cantorian mathematics. "Cantor's set theory solved these paradoxes. Infinities have different sizes (ℵ₀, ℵ₁...) and the rules are clear."
Craig's response: Mathematics is one thing and reality is another. Cantor developed a consistent mathematical system, but this does not mean that actual infinity is possible in the material world. Even Cantor himself was cautious about applying his theory to physical reality.
Criticism: Arbitrary distinction. "Why is infinity impossible in the universe but possible in God's attributes?"
Craig's response: The difference between temporal and atemporal existence. An infinite temporal series means events realized in succession - an actual infinite. God's attributes (like infinite knowledge) are not a temporal series but an atemporal existential state. (This response is controversial even among believing philosophers).
Criticism: Cosmological models. "Many physical models assume infinite time or a cyclical universe."
Craig's response: Mathematical models are not necessarily descriptions of reality. Most cosmologists today accept the Big Bang model which gives the universe a definite beginning (~13.8 billion years). Even cyclical models face the entropy problem.
Balanced critical assessment
Craig's argument from Hilbert's Hotel is strong but not decisive. Its strengths:
- It reveals real difficulties in conceiving an actual material infinite
- It is consistent with modern cosmology (Big Bang)
- It provides a philosophical argument independent of religious texts
Its weaknesses:
- The transition from "intuitively strange" to "metaphysically impossible" is debatable
- The distinction between temporal and atemporal infinity needs stronger justification
- Some philosophers (like Quentin Smith) have developed coherent responses