[ukoto]

They are real numbers, for which we have just proven it is impossible to find a description that will match them. We have proven that no description will ever describe them.

Of course you can describe them. But describe them in terms of what? I think that's the key. Let's say you have a pencil and you want to describe its length, which will be a unit multiplied by a number. If I were to try to describe the true length in meters it will be: 0.0178(...) * meters. This would be an "indescribable" number the author talks about.

However, I can just describe it in terms of itself and call it one pencil long - we use the pencil itself as the basis of the unit so the number we multiply the unit to is just: 1 * pencil long.

[lotharbot]

> "describe them in terms of what?"

Describe them (completely) in terms of any finite series of symbols.

Some numbers can be described as "7". Some as "pencil length". Some as "the fourth zero of the third Bessel function of the first kind". These are all describable numbers.

Yet there are vastly more numbers that cannot be described in this way -- vastly more numbers that we miss than that we hit.

[ackien]

What you've shown is that that particular number (0.0178.../the length of the pencil in metres) is describable. This isn't the same as showing every real number is describable. There are some numbers which are not the length of anything (in metres), or the mass of anything (in kilograms), or the square root or sin of anything. You could try to capture more numbers by using descriptions like "the length of this pencil in feet" and other made-up units of measurement, but you'll never describe them all.

[cool-RR]

"describe them in terms of what?"

Decimal system :)

Interesting philosophical point though.

[ukoto]

Decimal system (well, numbers in general) is a great way to describe things as they provide infinite accuracy both in positive and in negative directions. But you must describe them relative to something - some kind of unit.

So your balloon, to fully described its length, you have chosen to describe it relative to meters. Here's the kicker - how well you can measure the length of the balloon depends also on how well you can measure a meter. In the real world, if you try to measure both, the decimal number that describes how relative a meter is to your balloon will only be limited by how accurate your methods of measuring are. You can develop better measuring technology that will forever approach but never reach the True value (only God knows). It is if you try to measure an analog signal, you can only get a digital estimation. But extracting knowledge requires energy, so you need greater computing power to measure an analog signal but all the computing power in the universe couldn't acquire the True value. This is the result of the real world being nondiscrete and containing infinite knowledge. You can measure things by proportion (such as tau (2*pi) being the relative constant of circle circumference to radius); these numbers are transcendental and we don't know their True value. Your "indescribable" numbers are as indescribable as transcendentals. So yes the decimal goes on forever, but you can still describe it in terms of itself and just give a constant name (like pi).