Here’s my thought process. Numbers are infinite, and decimals are infinite. Therefore units of measure are therefore infinite. A ruler may not be 1 foot long, it may be 1 + ( 1 × 10^-63836275783837472827263728) feet long.
To me, for a measurement to be exact, it has to be to the point where as you get to smaller and smaller decimals of the measurement, you get to the point where the additional decimals are infinitely 0. But if numbers are infinite, than the ability to continuously shrink down and get those decimals is infinite too. Doesn’t that mean it’s impossible to get these measurements?
And if these measurements are physically impossible to find, since the requirements for them are physically impossible to occur, doesn’t that mean that no object in the world has a physical set size? And moreso, that the space in between any two objects, as well as the size of those objects, are both by definition infinite?
In: Physics
The fact that digits are infinite has no bearing on physical reality. A ruler only has X atoms in it, no more, no less. So it does have a fixed, set size, because it has a fixed, set mass. Whether or not we can accurately describe that size using our numerical system has no bearing on the fact that the ruler, physically, has a fixed, set size.
If you want to look at it another way, pi is infinite, right? We will never find the last digit of pi. But we can confidently say that pi is less than 4, no matter how many digits you drill down. So a number with infinite decimals is not the same as an undefined or “infinite” number.
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