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
asx -> 0
– in which case, we conclude the limit is1
. - We can make a sequence for the limit where we look at
0^x
asx -> 0
– in which case, we conclude the limit is0
.
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 have0
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.