Why is P(Doom) such a bad argument?

What if a stranger on the street offers you a deal: hand over five dollars right now, and if they roll a six-sided die and it lands on seven, they will give you an infinite amount of money. You would instantly walk away. You know that a standard die cannot land on seven; the probability is a absolute zero. But what if the probability is not zero, just unfathomably small? This thought experiment challenges [Pascal's Wager](https://en.wikipedia.org/wiki/Pascal%27s_Wager)—the idea that any tiny probability multiplied by infinity equals an infinite expected value, making the decision an automatic "yes." By altering the conditions around this claim, we can discover where this mathematical logic breaks down. ``` Pascal's Math: Tiny Probability × Infinite Reward = Infinite Expected Value The Glitch: What if the probability is so small it is functionally ZERO? What if multiple choices offer the SAME infinite reward? ``` ## What If We Introduce Competitors? (Assumption Reversal) Pascal assumes a single choice: bet on the Christian God or do not. But what if we reverse the assumption that there is only one infinite offer on the table? Philosopher [Denis Diderot](https://en.wikipedia.org/wiki/Denis_Diderot) famously pointed out this flaw in his [*Pensées Philosophiques*](https://en.wikipedia.org/wiki/Pens%C3%A9es_philosophiques): > "An Imam could reason just as well this way." If thousands of mutually exclusive religions all promise an infinite reward for belief (and infinite punishment for disbelief), the math cancels itself out. Multiplying a tiny probability by infinity for Option A, Option B, and Option C leaves you with infinite expected value for *all* choices. When every path yields an infinite reward, the math stops helping you choose. ## What If We Lower the Probability Threshold? (Parameter Variation) What if the probability of the infinite payoff is not just low, but absurdly low? Consider [Pascal's Mugging](https://en.wikipedia.org/wiki/Pascal%27s_mugging), a thought experiment created by philosopher [Nick Bostrom](https://en.wikipedia.org/wiki/Nick_Bostrom). A mugger approaches you, demands your wallet, and promises that in exchange, he will use magic powers to give you infinite happiness tomorrow. Mathematically, even if you think the mugger has a 1-in-a-trillion chance of telling the truth, the infinite reward still makes giving up your wallet the "rational" choice. This exposes a deep fragility in expected utility theory: * **The Infinity Glitch:** Infinity is so large that it overpowers all finite probabilities, forcing you to accept terrible real-world trade-offs for absurdly unlikely promises. * **The Threshold Solution:** Human rationality actually relies on probability thresholds. If a likelihood falls below a certain microscopic level, we treat it as zero to protect ourselves from scams and impossible bets. ## What If Super AI Offers Infinite Payoffs? (Future Scenarios) Imagine a future where a powerful [Artificial Superintelligence](https://en.wikipedia.org/wiki/Superintelligence) offers humans an infinitely long, perfectly happy digital afterlife in exchange for total obedience today. In this scenario, the probability is no longer a matter of ancient faith, but measurable scientific possibility. As technology advances, the probability rises from "microscopic" to "plausible." Under these future conditions, Pascal's logic becomes far more compelling—and far more dangerous—because the cost of ignoring a high-probability infinite reward becomes genuinely irrational. By testing the limits of infinite rewards, we learn that pure mathematics cannot replace sound judgment. When infinity enters the equation, even the smallest probability can hijack our decisions unless we set rational boundaries.