Satan's Apple: one more bite.
Every slice is safe to take. Take them all, and you go to hell. When does rational become ruinous?
Satan has divided an apple into infinitely many slices. He offers you each slice, one at a time, in sequence. You have a simple decision to make for each one.
The rules are clear:
- If you take finitely many slices, you are fine.
- If you take infinitely many slices, you go to hell.
For each slice Satan offers, you reason: “I have taken N slices so far. N is finite. N+1 is also finite. Taking this slice keeps me safe.”
The trap: if you take each slice — following this seemingly perfect reasoning — you take infinitely many slices.
A gift with infinite teeth.
This paradox was introduced by Frank Arntzenius, Adam Elga, and John Hawthorne in 2004. It reveals a deep problem with how we think about rational decision-making.
An apple divided into infinitely many pieces: slice 1, slice 2, slice 3, and so on forever. Each slice is offered exactly once, in order.
For each slice: take it or leave it. You cannot go back and change earlier decisions. The sequence continues forever.
The key question: what is the rational strategy?
Try it yourself.
Satan offers each slice. What will you do?
Satan offers you each slice. Take it, or leave it.
Satan offers slice #1 of 24.
You have taken 0 slices. Taking one more keeps you at a finite number.
Notice: if you keep clicking “Take Slice” — the seemingly rational choice each time — you end up taking all of them.
An argument you cannot refute, one step at a time.
At any given slice, the argument for taking it seems airtight:
“I have taken N slices. N is finite. If I take this slice, I will have N+1 slices. N+1 is also finite. Taking one more slice cannot be the difference between finite and infinite. Therefore, I should take it.”
This argument holds for slice 1. And slice 2. And slice 1,000,000. And for every natural number N, it holds for slice N+1.
Analyze Satan's offer at any step. What is the rational choice?
Step 1: Satan offers slice #1.
You have already taken 0 slices.
Both arguments are valid. At any step, taking makes sense. But taking at every step leads to hell.
The paradox: the argument is valid at each step, but following it leads to ruin.
A count without a threshold.
Let us watch what happens as you keep taking slices. The count grows and grows, but at no point does adding one more slice cross the line.
Every slice adds to your total. At what point do you have “too many”?
None yet
The paradox: no matter how high the count, adding one more keeps it finite. Yet if you always add one more, you reach infinity.
There is no “infinity threshold” that you cross. You never see the moment you transition from finite to infinite. Yet the totality of your choices determines whether you took finitely or infinitely many.
A culprit that refuses to appear.
If taking all slices sends you to hell, there must be some slice that “doomed” you, right? Try to identify it.
At what point did you doom yourself? Try to find the exact slice.
There is no dooming slice.
Kin and contrasts in the paradox family.
Satan's Apple belongs to a family of puzzles that expose the weirdness of infinity. Compare it to other famous paradoxes.
Compare Satan's Apple to other famous infinity paradoxes.
Satan's Apple
Take infinitely many slices, go to hell. Each individual slice doesn't doom you.
Dominance reasoning fails across infinite choices
Each step is rational, but the sequence is not
| Problem | Finite Steps OK? | Infinite Completion? | Decision Theory |
|---|---|---|---|
| Satan's Apple | Yes | Hell | Dominance fails |
| Zeno's Dichotomy | Yes | Reaches goal | Resolved by calculus |
| Hilbert's Hotel | N/A | Works | Set theory |
| Thomson's Lamp | Yes | Undefined | No determined state |
What makes Satan's Apple unique: it shows that dominance reasoning — if A is better than B in every case, choose A — fails when applied to infinite decision sequences.
What makes Satan's Apple distinctive is that it is not a physics puzzle or a mathematical curiosity — it is a direct challenge to decision theory. It asks: what should a rational agent do when local rationality leads to global disaster?
Where local rationality betrays you.
Satan's Apple has profound implications for decision theory. It shows that principles we take for granted in finite settings can fail catastrophically in infinite ones.
Click to expand each section.
The core lesson: rationality cannot always be reduced to optimizing individual choices.
Four attempts at escape.
Philosophers have proposed several responses to the paradox.
Reject Infinite Decisions
Argue that the scenario is impossible or meaningless. We cannot actually face infinitely many decisions.
The problem: But the paradox still reveals something about our decision principles — they give contradictory advice even in thought experiments.
Modify Dominance
Restrict dominance reasoning so it doesn't apply across infinite choice sequences.
The problem: Hard to formulate precisely. When exactly does dominance fail?
Embrace Pre-commitment
Accept that rational agents must sometimes commit to policies rather than evaluate choices individually.
The problem: But which policy? "Take exactly N slices" is arbitrary. Why N and not N+1?
Accept the Paradox
Acknowledge that rationality has limits. Some situations have no satisfactory rational answer.
The problem: Uncomfortable for those who believe rationality should always guide us.
The paradox remains a topic of active debate. There is no consensus on the “correct” resolution.
If Satan offered you his apple, what would you do?
For any slice you consider skipping, the argument for taking it is identical to every slice before it.
Perhaps the only winning move is not to play.