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
> Numbers are infinite, and decimals are infinite.
Don’t confuse the map for the territory. Numbers are abstract concepts that we use as an idealization of things in the real world. Numbers aren’t themselves physical objects and they don’t have to correspond precisely to physical reality. We can only measure anything in reality approximately. And for most purposes we only need to know things to within a certain accuracy. For example if you want to build a table to fit in a certain space, it’s good enough to know that the space is approximately 1.35m wide – knowing that it’s actually closer to 1.352383746m wide doesn’t make it any easier to build the table at all.
> doesn’t that mean that no object in the world has a physical set size?
Once you get down to small enough scales everything is an indeterminate size anyway. Most objects grow or shrink very slightly depending on factors such as temperature, and quantum-scale objects such as atoms and electrons don’t really have a well-defined size.
> And moreso, that the space in between any two objects, as well as the size of those objects, are both by definition infinite?
You mean because there are infinitely many points between them?
In terms of physical reality, we can’t really say that there are infinitely many points there. How could we demonstrate that there are? We can’t exactly do an experiment in which we move an object to infinitely many different positions.
In terms of the mathematical idealization of space, it’s perfectly possible to define a mathematical space in which there are infinitely many points between 0 and 1 and yet the distance between 0 and 1 is still just 1. The concept of distance has nothing to do with the number of intervening points.
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