#### Write4U

**Valued Senior Member**

Thank you for that comprehensive response. Much food for thought.Yes that's true. There are real numbers for which we can not have any symbols or descriptions or ways of specifying them.

This is a bit off-topic from infinitesimals. There are the non-computable numbers. These are real numbers that can not be described by any algorithm or finite-length string of symbols. There are many noncomputable real numbers. If you pick a random real, it will almost certainly be noncomputable. It's a little mysterious.

By the way "measurable" is a technical term in math that means something else, so best to avoid it.

If I'm understanding you, you're making the distinction between a number and any of its many possible representations. For example 2 + 2, 4, and 10 (base 4) are different expressions that both point to the same number. The number itself is an abstraction.Much food for thought.

Yes in this case that's true. But most real numbers don't have closed-form representations. All the familiar numbers we use do have closed-form representations though. For example the digits of pi or sqrt(2) can be cranked out by a computer program, and a program is a finite string of symbols. We can think of a program as a closed form for the number whose digits it cranks out. But since there are fewer programs than real numbers, there are real numbers without closed forms.

To be fair, nothing in this thread is about the physical world. We're talking about the mathematical abstraction of the real numbers. As far as we know, real numbers can't be instantiated in the physical world because it takes an infinite amount of information to specify most real numbers.

It's always important to distinguish math from physics.