When I was in Intro to Advanced Mathematics (first 300-level course), there was a problem about infinities that always bothered me.

The Description

Imagine you encounter a math genie that gives you the following offer: at time 0, 0.5, 0.75, … she’ll give you 10 wishes (that is, an infinite number of times in 1 second), which you can save or spend at each step however you want.

My professor offered two choices:

  1. At each step, save the 10 wishes and spend the next wish – so at step one, just save. Then step two, save the next ten wishes (now twenty total) and spend the first wish. Etc.
  2. At each step, spend 9 wishes and save 1 wish. Every time.

Her argument was that you should choose option 2 because it’s better, according to the following logic:

Count the number of wishes which you don’t spend at some time step –

  1. in the first case, every wish is spent in some time step and so after 1 second you are left with zero wishes remaining and have spent |N| wishes;
  2. and in the second case, there is no point at which wish n mod 10 = 0 is spent and so after 1 second you are left with |N| wishes remaining and have spent |N| wishes.

This makes sense, in so far as it goes.

Context Interlude

However, this makes an assumption about the “right” way to measure our wishes.

In many cases this isn’t a problem, but for limits, they only exist if the different ways to measure our limit agree.

The traditional example of this is given in calculus with x^x as x -> 0:

  1. We can make a sequence for the limit where we look at x^0 as x -> 0 – in which case, we conclude the limit is 1.
  2. We can make a sequence for the limit where we look at 0^x as x -> 0 – in which case, we conclude the limit is 0.


There isn’t a limit for x^x as x -> 0 because different ways of measuring what the limit disagree on the value.

Depending on the context, you can choose different values – but there can’t be a coherent limit, because there isn’t a consistent way to measure.

The Problem

Which brings us back to our math genie.

Option 2 is “consistent”, in that there isn’t a question about what the outcome is: you both spend |N| wishes and have |N| because the wishes are cleanly partitioned at the end of the wishes, and so you can make a “smooth” transition back at t=1 when the genie stops granting wishes.

However, that’s not the case for option 1:

  1. If you measure as my professor suggested, you spend |N| wishes and have 0 wishes remaining;
  2. but, if you measure by wishes remaining, then you spend |N| wishes and have… |N| wishes remaining!

What happened??

Well, the total of wishes saved diverges as t=1: at every step, you have more wishes! The sequence of wishes saved diverges as 10, 19, 28, ...!

How do “which wishes did I not spend?” and “how many wishes do I have?” not agree on the outcome??

Introducing Ultrawishes

The answer lies in the context above: we have one way of measureing wishes that gives us 0 and one way that gives us |N|. That’s as fundamental a disconnect in measurement as with x^x when we got 0 and 1.

No matter what we choose, at t=1 for option 1, we have an ULTRAWISH! A wish with infinite wish potential in a single instance, either:

  1. spending all of the persons |N| remaining wishes in a single burst of actually infinite wish flux;
  2. or, despite spending every wish in those final moments, leaving the wisher with |N| wishes remaining.

Much like the above discontinuity in potential limits for x^x requires a choice, we require a “wish discontinuity” or ULTRAWISH (arbitrarily doing one of the above) to solve the problem of different measures producing different results.


There’s a few conclusions to take away from this:

  1. My professor, Janet Mills was right that infinity is strange (the real point to her story);
  2. and even genies are subject to mathematics.

Anyway – something to consider.