This seems a silly question. You would think the answer is “bigger than that.”

But take a look at this:

All of these sequences go on “forever” (we’ll get back to that later)

So they are all infinite series. But, logically, the lower rows seem to have fewer numbers than the “all” row. These are all infinities and yet they clearly appear to have different sizes.

WRONG

Size simply is not a property of Infinity. Any more than perfume or colour.

Men are obsessed with “size”. Stop giggling at the back there!

As beings in this universe, physical dimensions are vital to us and to our thinking. It is very difficult for us, and I can probably speak for most lifeforms on this  planet, to get that imperative out of our heads.

It may be possible mathematically to compare infinities. But that takes place in Mathspace, a real but abstract phase space of all mathematics. It contains everything mathematical, possible, impossible, probable, improbable, correct and incorrect. In Mathspace, you can construct a mathematics where “2 + 2 = 5” is a correct statement. I don’t know what kind of consistent maths can be constructed on that premise. But it will exist somewhere as a zone in Mathspace.

We know a mathematics which seems to have a pretty close correlation with what we observe in the sector of Realspace, where we live. Oh dear, forgot. “Realspace” is the phasespace of all possible universes and realities. Ours is in there somewhere. Along with one where five-legged elephants are real.

So, for mathematical operations, we dip in and out of the zone of Mathspace which seems to relate best, do our calculations and then apply the result to our zone of Realspace to explain observed events.

Now we come to a different problem.

As far as I can see, if we are to use a number, we need to define it.

This seems to be stating the obvious to an infinite degree, but please bare with me.

You are sitting in a room with a clock, a screen and your desk. You have paper and pencil and your job is to perform all the possible arithmetic operations with the numbers you can see on the screen

You can add, subtract, multiply, divide.  To keep it simple, you are not allowed to divide by 0, that gets too complicated.

The clock ticks and the screen shows 0.

You write:

• 0 + 0 = 0
• 0 – 0 = 0
• 0 x 0 = 0

And that’s about it.

The clock ticks and the number 1 appears besides the 0

• 1 – 1= 0
• 1 x 1 = 1
• 0 ÷ 1 = 0
• 1 + 1 = ?           Can’t do that yet, 2 has not been invented and tested before release into general use.

The clock ticks and 2 appears.

Hurrah!

• 1 + 1 = 2
• 2 – 1 = 1
• 2 – 2 = 0
• 2 ÷ 1 = 2
• 2 ÷ 2 = 1

The list of operations that have to be performed before the number is defined and can be used gets longer and longer. The clock ticks get further apart.

If this is correct, and a number has to be defined before it can be applied to Realspace, then eventually, the time to define a new number will exceed the lifetime of the universe.

Certes it will be a very big number. But it will not be infinite.

It doesn’t matter what shorthand is used to express that number, if it has to have a unique name and symbol, then sooner or later we will run out of time and our series of possible numbers in the universe is not infinite.

Now the biggie. Does the universe use numbers?

If it does, then every point in the universe should have a set of coordinates in space and time expressed as numbers.

That would mean the furthest point is limited by the largest number yet released that can be used in its coordinates.

That would mean the rate of expansion of space and time is limited by the rate of defining and release of new numbers.

Wish I hadn’t started this. You wish you hadn’t started reading it

Both need coffee.