Who said anything about recording? What would the subjective experience of measuring something with infinite precision possibly be like?
>> Just because you can't record something...
>>> Who said anything about recording?
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.)
Pick your method. It’s the ratio of a circle’s circumference to its diameter.
I think it's reasonable to say we can't truly measure pi, though.
And you can neither know nor measure a random real.
Sure, but how would you compare those against a measurement?
> 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.
It's worse than that: you also need an unambiguous way of determining whether the needle is overlapping a stripe.