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If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego bricks. It's also conceivable that I could get any real number. My question is, can there mathematically exist a universe in which these ratios are not real numbers? How do we know that the real numbers are all the numbers, and that they dont have "gaps" like the rationals?


I want to know if what I (or most people) intuitively think of as length of an idealized physical object can be a non-real number. Is it possible to have more then a continuum distinct ordered points on a line of length 1? Why do mathematicians mostly use only R for calculus etc, if a number doesnt have to be real?

By universe I just mean such a thing as Eucildean geometry, and by exist that it is consistent.

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    Your question is predicated on an idea of "measurement" that seems much stronger than any I am familiar with in the real world. Any form of physical measurement I know will (at best) give the following answer: the quantity $X$ you are trying to measure has value $A < X < B$, where $A$ and $B$ are rational numbers. With such a form of measurement it is not even possible to tell whether $X$ is rational or irrational real (at least not by any finite sequence of measurements, and physically speaking I don't know any other kind). So what do you have in mind here?2011-07-14
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    I knew physicist Bob Mills (as in Yang-Mills). Once I suggested doing physics using some number system other than the real numbers. He considered that to be a worthless suggestion (even though he was too polite to say it that way).2011-07-14
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    Also, speaking as a mathematician, I am utterly baffled at what "can there mathematically exist a [physical, presumably] universe" might mean. We are not in the business of deciding whether universes can exist or not!2011-07-14
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    @vittulainen: It is possible to *imagine* that a mathematical structure that involves infinitesimals will be useful to physicists. In a certain sense, it already happened (maybe Newton, certainly the Bernoullis, many others). Objects that are distant from simple notions of measurement, such as Hilbert space, are certainly useful to physicists.2011-07-14
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    The rationals don't have gaps, they are dense.2011-07-14
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    What is your question ?2016-04-28

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