> what could the result of measuring anything to an infinite precision possibly look like?

Depends on what you're measuring. To illustrate why that isn't a facetious response, consider the difference between 'measuring' pi, 'measuring' a meter and 'measuring' the mass of a proton. (Or, for that matter, the relative mass of three of something to one of it.)

You'd need to somehow record refinements endlessly? I don't get what you're getting at.

How do you measure pi?

By repeatedly throwing a needle on a striped pattern: [1]. Obviously, you will need an infinite number of throws for an infinitely precise measurement of pi.

[1] https://en.wikipedia.org/wiki/Buffon%27s_needle_problem

> you will need an infinite number of throws

It's worse than that: you also need an unambiguous way of determining whether the needle is overlapping a stripe.

That would affect only a few borderline trials and would average out with subsequent throws. It would be much more worrisome that the length of a needle or the width of a stripe is not infinitely precise, that would consistently affect all the trials.

> How do you measure pi?

Pick your method. It’s the ratio of a circle’s circumference to its diameter.

Considering that we don't know the value of pi (not that we could write it out nor read it), I'm not sure your definition of "measure" is the same as mine or most people's.

I think your definition of "know" is unreasonably strict. Especially because we can write out pieces of algebra that are exactly pi.

I think it's reasonable to say we can't truly measure pi, though.

And you can neither know nor measure a random real.

Hmm, I should say "the numerical value of pi in base 10" (or really any rational base), even if we were to weaken that with the qualifier "a to arbitrary degree of precision". We know pi in the sense of "a unique real number satisfying many useful properties".

Isn't "3.14" pi to an arbitrary degree of precision? Or am I misreading that.

> We know pi in the sense of "a unique real number satisfying many useful properties".

We know it a lot better than that. We have efficient programs that output the numerical value of pi for as many digits as you want.

There's a bunch of real numbers we can identify that are far harder to make use of or approximate, and don't have easy exact description of their value.

> we can write out pieces of algebra that are exactly pi

Sure, but how would you compare those against a measurement?

That depends on how you're measuring. But the second paragraph of that post already says you can't truly measure pi.

You can calculate or 'measure' an arbitrary approximation of that ratio by various methods, but calculating all of it takes infinite time, which I don't have and thus can't do it.