## Wishes, Ultrawishes, and Infinity

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:

- 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.
- 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 –

- 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; - 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`

:

- 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`

. - 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`

.

Whoops.

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:

- If you measure as my professor suggested, you spend
`|N|`

wishes and have`0`

wishes remaining; - 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:

- spending all of the persons
`|N|`

remaining wishes in a single burst of*actually*infinite wish flux; - 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.

### Conclusions

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

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

Anyway – something to consider.