The Grim Reaper Paradox, Hilbert’s Hotel, and the Case for a Finite Universe

The question of whether the past is infinite has long fascinated philosophers, theologians, and scientists. One of the most compelling arguments against an infinite past is the Grim Reaper Paradox, a thought experiment that reveals the logical impossibility of an infinite regress in time. Complementing this are other thought experiments, such as Hilbert’s Hotel, Thomson’s Lamp, and Infinite Dominoes, which expose the absurdities of actual infinities in reality. Coupled with insights from modern cosmology, historical philosophy, and theology, these arguments form a robust cumulative case for the necessity of a finite past and a transcendent cause for the universe.

The Grim Reaper Paradox

The Grim Reaper Paradox, popularized by Alexander Pruss, demonstrates the impossibility of completing an infinite causal sequence. Imagine an infinite number of Grim Reapers, each tasked with killing a person at a specific, progressively earlier time:

1. Grim Reaper 1 is scheduled to act at 1:00 p.m..

2. Grim Reaper 2 at 12:30 p.m..

3. Grim Reaper 3 at 12:15 p.m..

4. This sequence continues infinitely, with each Grim Reaper acting at half the interval of the previous one, converging toward noon.

By 1:00 p.m., the person must be dead because one of the Grim Reapers would have killed them. However, no specific Grim Reaper can be identified as the one who acted, since each has an earlier Grim Reaper that would have already killed the person. This creates a logical contradiction: the person cannot survive, yet no Grim Reaper is responsible for their death.¹

This paradox illustrates that actual infinities cannot exist in reality, as they lead to incoherent outcomes. The same reasoning applies to an infinite past, which would involve an infinite regress of temporal events. Such a sequence is not metaphysically possible, necessitating a finite beginning to time.

Hilbert’s Hotel: The Absurdity of Infinity

The Hilbert’s Hotel thought experiment, devised by mathematician David Hilbert, reveals the paradoxes of actual infinities in the physical world. Imagine a hotel with an infinite number of rooms, all of which are occupied. Despite this, the hotel manager can always accommodate new guests:

• If a single guest arrives, the manager shifts the occupant of Room 1 to Room 2, Room 2 to Room 3, and so on, freeing Room 1.

• If an infinite number of new guests arrive, the manager shifts the occupant of Room 1 to Room 2, Room 2 to Room 4, Room 3 to Room 6, and so forth, freeing all odd-numbered rooms.

While this scenario is consistent mathematically, it is metaphysically impossible. A hotel cannot be both “fully occupied” and simultaneously able to accommodate new guests. Applied to time, this paradox highlights the absurdity of an infinite past, which would require “completing” an infinite sequence of moments to reach the present.²

Thomson’s Lamp, Infinite Dominoes, and Other Absurdities

Additional thought experiments further expose the incoherence of actual infinities:

1. Thomson’s Lamp: Imagine a lamp that is switched on and off an infinite number of times in a finite period, with each interval halving in length (e.g., one second, half a second, a quarter second, etc.). At the end of the finite period, is the lamp on or off? The result is indeterminate, showcasing the impossibility of completing an infinite sequence within a finite time.³

2. Infinite Dominoes: Picture an infinite line of dominoes stretching back into the past. For the final domino to fall (representing the present), an infinite number of prior dominoes must have fallen. However, completing an infinite sequence is impossible, meaning the present moment could never arrive.⁴

3. The Infinite Bookshelf: Imagine a bookshelf with an infinite number of books, each labeled with a unique number. If you remove the first book, the sequence of books would remain infinite, yet paradoxically, it would still lack the first book. This further illustrates the absurdity of applying infinity to real-world scenarios.⁵

Philosophical and Historical Context

Philosophers have debated the impossibility of an infinite past for centuries:

• Aristotle distinguished between potential infinity (a process that can continue indefinitely) and actual infinity (a completed set). He rejected actual infinity in nature, aligning with the paradoxes discussed here.⁶

• Thomas Aquinas argued in his Summa Theologica that “there cannot be an infinite regress in efficient causes.” Without a first cause, the chain of causation collapses, as there would be no basis for subsequent causes.⁷

• Al-Ghazali, a medieval Islamic philosopher, emphasized the impossibility of traversing an infinite sequence to reach the present, a core tenet of the Kalam Cosmological Argument.⁸

Scientific Evidence for a Finite Past

Modern cosmology confirms the philosophical conclusion of a finite past:

1. The Second Law of Thermodynamics: The universe is moving toward a state of maximum entropy. If the universe were infinitely old, it would have already reached this state. Instead, the current low-entropy state indicates a finite beginning.⁹

2. The Borde-Guth-Vilenkin Theorem: Any universe that has been expanding throughout its history must have a beginning. As Alexander Vilenkin states, “There is no escape; they have to face the problem of a cosmic beginning.”¹⁰

3. Cosmic Microwave Background Radiation (CMBR): The discovery of the CMBR supports the Big Bang Theory, which indicates a singular starting point for space and time.¹¹

Theological Insights

The impossibility of an infinite past aligns with Scripture, affirming the finite nature of creation and the eternal nature of God:

• Psalm 90:2: “Before the mountains were brought forth, or ever you had formed the earth and the world, from everlasting to everlasting you are God.”

• Genesis 1:1: “In the beginning, God created the heavens and the earth.”

• Revelation 22:13: “I am the Alpha and the Omega, the first and the last, the beginning and the end.”

These passages underscore that the finite universe depends on an eternal Creator who exists outside of time.

Addressing Counterarguments

Critics often propose alternatives to a finite past, such as:

1. The B-Theory of Time (Eternalism): This view holds that all moments in time exist equally, negating the need for a beginning.

• Rebuttal: The B-theory contradicts our intuitive experience of time’s flow and undermines causality, which requires a sequential progression of events.¹²

2. An Oscillating Universe: Some suggest the universe cycles infinitely between expansion and contraction.

• Rebuttal: Such models fail due to entropy. Each cycle would lose usable energy, making infinite oscillations impossible.¹³

Conclusion

The Grim Reaper Paradox, Hilbert’s Hotel, and other thought experiments vividly demonstrate the absurdities of an infinite past. Supported by philosophical reasoning, historical insights, and scientific evidence, the case for a finite universe is overwhelming. This conclusion aligns with the Kalam Cosmological Argument, affirming the necessity of a transcendent first cause—one that many identify as God.

If time had a beginning, what does that mean for the universe’s ultimate cause? Could an eternal Creator provide the most reasonable explanation for existence itself?

Footnotes

¹ Alexander R. Pruss, Infinity, Causation, and Paradox (New York: Oxford University Press, 2018).

² David Hilbert, as discussed in William Lane Craig, The Kalam Cosmological Argument (London: Macmillan, 1979).

³ James F. Thomson, “Tasks and Super-Tasks,” Analysis 15, no. 1 (1954).

⁴ William Lane Craig and James D. Sinclair, “The Kalam Cosmological Argument,” in The Blackwell Companion to Natural Theology, ed. William Lane Craig and J.P. Moreland (Oxford: Wiley-Blackwell, 2009).

⁵ Alexander R. Pruss, Infinity, Causation, and Paradox.

⁶ Aristotle, Physics, trans. R.P. Hardie and R.K. Gaye, Book III, Part 6.

⁷ Thomas Aquinas, Summa Theologica, I.Q.2.Art.3.

⁸ Al-Ghazali, The Incoherence of the Philosophers, trans. Michael Marmura (Provo, UT: Brigham Young University Press, 1997).

⁹ Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe (New York: Vintage, 2007).

¹⁰ Alexander Vilenkin, quoted in John Barrow, The Universe That Discovered Itself (Oxford: Oxford University Press, 2000).

¹¹ Alan Guth, The Inflationary Universe (New York: Perseus Books, 1997).

¹² William Lane Craig, Time and Eternity: Exploring God’s Relationship to Time (Wheaton, IL: Crossway, 2001).

¹³ Ibid.