Countability of Options ♾️

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Countability of Options ♾️

“I have no idea how you came up with that, but it was just the right thing.”

That’s what one of my clients told me after a session recently.  Maybe it seemed like magic, but in that moment I knew exactly why I thought of what I did.

She was struggling through something very challenging.  It was time to make a decision and it had been a long time coming.  She struggled with the decision for a handful of reasons, and the best option was not clear.

I think that there were so many moving pieces and emotional elements that the problem seemed infinite and overwhelming.

Do you ever have challenges like that?

I think we all do from time to time.  If you feel a sense of discouragement or paralysis that is likely what is happening.

But, here’s the interesting thing.

Some parts of reality are truly infinite, or might as well be.

But, others are not. They are quite limited and countable.

Often, what seems infinite is simply something we wish not to look at or acknowledge.  

Definite.  Finite.

Lately I have been fascinated by the mathematical concept of infinity.

You know it, right?  It’s an abstract symbol for an unbounded quantity.  Symbolized by that number eight laying on its side.

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It’s a strange concept.  We can’t imagine it, but we use it in conversation and thinking.

 A placeholder for something transcendent that we can’t quite glimpse.

Here’s what’s interesting though.  Did you know there are different kinds of infinity?  It’s true.  And some are “bigger” than others.

How can that be?

Well, it was mathematically proven by a German fellow named Georg Cantor in the 1870s.  Cantor had this sensibility that tied together mathematics and theology.  He had visions of infinity that may have been religious.  He was convinced God had given him this insight into the infinite.  He also may have been bipolar, and the volatility of the political and economic climate did not help his mental state.

His idea that infinities came in different sizes was initially met by the mathematical community as you might expect.  What?  Crazy.  Isn’t infinity infinity?  Unbounded is unbounded.  How can you compare them?

But Cantor was persistent.  And he proved that it was true.

He proved that some infinities are bigger than others.

Essentially, if you can map a set of anything discretely to the natural numbers (1, 2, 3, 4, 5…), then it is countable.  And if you can’t, then it is not countable.  And uncountable is considerably more vast than countable.

He used a method called the diagonal proof to show that, for example, fractions are countably infinite.  Because we can systematically construct

0*

1/1

1/2

1/3, 2/3

1/4, 2/4 (=1 /2), 3/4

1/5, 2/5, 3/5, 4/5

…

*(0 can be expressed as 0/x where x ≠ 0)

See what we’re doing there?  The diagonal will grow infinitely, but it’s all quite discrete.

(And, interestingly, the fractions furthest to the right in each new row will converge on but never quite hit 1 as the series reaches toward infinity and the difference between the numerator and denominator becomes less and less significant.  This is where we get calculus limits, sig figs in chemistry, and many other numerical concepts.  But, I digress.  Seriously though.  Aren’t numbers mystifying?!)

The same thing happens with irrational polynomials, but in a higher dimension, and with more embedded diagonals:

x^2 + 1 = 0, x^2 + 2 = 0, x^2 + 3 = 0…

2x^2 + 1 = 0, 2x^2 + 2 = 0, 2x^2 + 3 = 0…

3x^2 + 1 = 0, 3x^2 + 2 = 0, 3x^2 + 3 = 0…

x^3 + x^2 + 1 = 0, x^3 + x^2 + 2 = 0, x^3 + x^2 + 3 = 0

x^3 + 2x^2 + 1 = 0, x^3 + x^2 + 2 = 0, x^3 + x^2 + 3 = 0

…

That one is a bit harder to represent in this format and much easier to lose track of I notice.  Still infinite but countable though.  Buildable with discreet terms.  Technically the polynomials are a “bigger” set than the rationals, even though they are both infinite.

And it all boils down to whether we can capture it in a systematic expression.

All rationals are x/y where x and y are integers and y ≠ 0

All polynomials are some form of f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

I’m getting out of my depth with this, but there are other numbers that are transcendental, which means they are also irrational, but can’t be reliably compressed in a similar way.  They are uncountably infinite because there’s no way to systematically start the numbering.

(And by the way, making lists like this reminds me very much of mapping out possible harmony options in music theory.  Music students learn a similar kind of mathematical thinking, although it is not often presented as countable like this.  When I taught university music theory a few years back I think my students were unusually mentally clear about the structure of harmonic progression thanks to this sensibility.)

So, infinities of different sizes.  Pretty crazy.  I never knew this, but now I can’t unknow it.

And I think about what can be discreetly symbolized and counted all the time now.

As I listened to my client it seemed infinite.

And then I realized, no, it’s not.

There were exactly 5 options.  And 2 of them were subsets of one larger option.

So I said, let’s number these:

1.

2A.

2B.

3.

4.

There.  Those are the options.  Now, let’s talk through each.

And we did.  And 2 remained.  And then she talked to other people that day and worked through those 2 options and only 1 remained.  And then the decision was clear, and she could move on.

So, there’s all these numbers up in the abstract heavens.  Infinitely many.  Orders of infinity.  That’s where mathematicians play.  And then we get to use their discoveries (or inventions - jury is out on that!).

And I was contemplating countable vs. uncountable infinities.  It’s really the difference between finding discrete symbolic paths vs. a cloud of overwhelming noise.

And often our lives feel like that cloud of overwhelming noise.

But actually they’re more like natural numbers, or fractions, or polynomials.  There are discrete options that are countable.

And not even infinite.

Quite finite.

5 options, with 2 as a subset of 1, so actually 4 options.

That was all.

And then she had clarity, and a path forward.

So, is there a hidden, discrete countable structure to your problems?

There probably is.  Or for your family, or your organization, or your community.

There’s a way through the overwhelm, and perhaps a simpler one that you might expect.

Give me a call and let’s find your options so you can move into the future with clarity and confidence.