From Supertasks to Infinite Time

A Brief Sequel to My Strange Locations Argument for a Finite Past

In my last post, "Where is Waldo?", I explored a series of strange scenarios involving infinite sequences of events. Many readers pointed out that these scenarios involve supertasks: completing infinitely many actions within a finite span of time. A common response was this: Sure, supertasks seem absurd. But what if the same sequence of events is spread across infinite time? Wouldn’t that be different?

This is a thoughtful question, and I want to explore it further. My aim here is to offer a possible bridge from the impossibility of supertasks to the deeper impossibility of completing infinite sequences in infinite time.

Here is the first step on the bridge:

1. Principle of Independence: If a series of events is independent, so that no event depends on another, then the events can occur in any order and any temporal distance from each other.

If so, then if a supertask is possible, then its stretched-out version would also be possible. And if the compressed version is impossible, then the extended version would be too.

But:

2. Supertasks are impossible (including ones involving independent events).

In the Waldo argument, we saw that no final location is coherent. Any location he ends up in would either be reachable in a finite number of steps (which would contradict the infinity of steps) or would never be reached at all. The same problem arises regardless of how the steps are ordered or spaced.

At least for sake of argument, then, let’s suppose that a supertask (an infinite sequence in finite time) is impossible. A number of readers said they found the argument for at least this conclusion convincing. Where can we go from here?

From this, it follows that:

3. Completing an infinite sequence in infinite time is also impossible!

If it were possible, we could simply repackage it as a supertask by compressing the time between events and reordering them. But then we would be right back in the paradox.

This bridge, if sturdy, could help us walk from from one conclusion (the impossibility of supertasks) to a more fundamental conclusion (the impossibility of completing any infinite sequence). The idea is that the core difficulty is not limited to supertasks. The problem lies deeper. It lies in the structure of the infinite sequence itself.

I offer this bridge for further exploration, whether to walk across or to destroy by means of further analysis. One could also walk in the other direction: if you think an infinite sequence in infinite time is possible, then that will give you a reason to think a super task is also possible. Whatever the case, I hope the bridge serves all truth seekers in an effort to better understand infinity.