The Infinite Monkey Theorem is a famous thought experiment in probability theory that suggests that given an infinite number of monkeys, typewriters, and time, one of those monkeys will eventually type out a complete work like Shakespeare’s Othello. In theory, every possible sequence of characters will eventually emerge because, with infinite resources, even the most improbable outcomes are guaranteed.
However, what if this assumption doesn’t hold up when we move from theoretical infinity to practical infinity?
The Infinity Exhaustion Paradox
In a theoretical run of infinite possibilities, every event, no matter how remote, is guaranteed to happen because infinity ensures that every outcome will be actualized. It’s a mathematical abstraction where probability and time are treated as limitless resources.
But in a real, actual run of infinite possibilities, it’s possible that some extremely remote possibilities may never occur. This is because certain outcomes might require an infinite amount of time to appear—even if infinite time is granted. For example, the monkey typing Othello might never actually happen in practice, as the time needed for such an event could itself stretch into infinity.
This presents a significant critique of the Infinite Monkey Theorem, which I call the “Infinity Exhaustion Paradox.” It suggests that even with infinite resources, certain rare outcomes may never manifest because they are infinitely improbable in any meaningful, practical sense.
Refuting the “Infinity Exhaustion Fallacy”
The “Infinity Exhaustion Paradox” can be seen as a counterpoint to the “Infinity Exhaustion Fallacy” often implied in theoretical discussions of probability. The fallacy assumes that infinite time and possibilities automatically lead to the actualization of all potential events. But the paradox highlights that even in infinite scenarios, some events could be so remote that they require an infinite amount of time to occur, making them practically impossible.
Theoretical vs. Actual Infinity
This distinction touches on deep areas of mathematics and philosophy, including set theory, modal realism, and the philosophy of probability. It’s similar to ideas in constructive mathematics, where the focus is on what can actually be realized, as opposed to abstract infinite assumptions.
In conclusion, the Infinity Exhaustion Paradox challenges the classical interpretation of infinite probability by introducing the possibility that even in an infinite world, certain outcomes may remain forever out of reach. If this concept has not yet been formally described, I propose this name to encapsulate the idea that infinite time may not guarantee the realization of all possibilities.
This post explores the nuances between theoretical and practical infinities and offers a potential framework for discussing the limits of infinite probability. Whether or not this idea has been explored in full, the “Infinity Exhaustion Paradox” presents an intriguing new angle on the Infinite Monkey Theorem.