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There are uncountably many reals. However, there are only countably many definable numbers. Thus, almost all reals are undefinable. Undefinable means that the shortest representation of that number requires infinite bits of information. This seems very strange because anything representing infinite information fundamentally contradicts physics. So I wonder: Are all undefinable reals "fake" because they have to be irrelevant to everything in our universe?

Furthermore, I wonder if there are any applications in physics of "real" reals? Does it make any difference if you build a theory of (quantum) physics solely with numbers that represent at most finite amounts of information?

LinusK
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    "...fundamentally contradicts physics". How? – jacob1729 Mar 09 '22 at 17:01
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    Correct me if I am wrong, but AFAIK there's only finite amounts of information in the universe. If I understand correctly, there's a minimum amount of energy required to represent a bit of information. Thus, infinite information requires infinite energy. – LinusK Mar 09 '22 at 17:05
  • can't put my finger on it, but I think the primary assumption here is that information is ordered but AFAIK information is entropy. See: Shannon Entropy – Yorik Mar 09 '22 at 17:27
  • How is physics relevant to the validity of mathematical entities? – PM 2Ring Mar 09 '22 at 17:48
  • @PM2Ring My questions regards the opposite direction. How is the validity of mathematical entities relevant to physics? – LinusK Mar 09 '22 at 18:01
  • Maybe TOE could make a physics without fixed parameters. So all the universal constants would be - at least in theory - calculable by it. It would be very funny, if TOE would only prove that this calculation exists, but it would also prove that it can not be calculated by a finite algorithm. So, there would be undefinable constants. – peterh Mar 09 '22 at 18:15
  • Related: What is the 'complete' number system to measure objects we see in reality? – Sandejo Mar 10 '22 at 02:49

3 Answers3

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Numbers are logic objects, like operators ("add five to the following value, then square it"), operations ("add the next value to the previous value"), and so on. Numbers can work like operators when used as coefficients on variables ("multiply the following value by 5").

Mathematicians care about logic objects.

Physicists care about physical processes. Numbers are logic objects that we can use to manipulate symbolic representations of physical processes or their characteristics. This lets physicists infer the nature of other physical processes when we translate back from our symbolic representational abstracted approximate model of the universe to statements about the real physical universe.

There is no such thing in physical reality as a $5$, just like there is no such thing as a $+$ or a $\int$. Thus it doesn't matter how much energy a $5$ has vs how much energy a $5.01$ or a $5 + 10^{-10^{100}}$ or "the smallest undefinable number larger than 5" has. None of them are physical processes, so none of them have any amount of energy, not even $0$.

g s
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    @MahirLokvancic a thought about a number is no more itself a number than is a drawing of a number. – g s Mar 09 '22 at 18:28
  • I get what you mean but my point is that 5 is a concept that can be useful to describe physical processes. In contrast, every undefinable number has to be irrelevant for describing any finite universe. – LinusK Mar 09 '22 at 19:17
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    @LinusK Why? When we integrate along a path, we assume a priori that there are a continuum of positions (not necessarily distance-valued positions) between endpoints - which includes an uncountable infinity of undefinable number valued states, and integration works great for describing and predicting the universe. Of course by the time we're integrating, we're no longer working with the universe, but with our abstracted model. – g s Mar 09 '22 at 19:45
  • @LinusK There also isn't a guarantee the universe is finite, we don't know the answer to that question though we tend to assume it isn't. – Triatticus Mar 09 '22 at 20:08
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This seems very strange because anything representing infinite information fundamentally contradicts physics.

Mathematicians over the past couple of centuries have had a lot of fun exploring the implications of real numbers and real analysis, but none of that existed until physicists invented it. They invented it because they needed it to describe natural phenomena. I don't think you can reasonably argue that something "contradicts physics" when so much of physics is built upon it.

Maybe, some day, somebody will come up with a convincing argument that space and time actually are discrete phenomena at some finer scale than we have probed to date. But, that's not where we are today, and even if it does turn out that way, real analysis most likely will continue to be a valuable tool for understanding things on the same scales that we observe today.

Solomon Slow
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  • I don't doubt real analysis here. However, isn't space and time considered to be discrete phenomena in terms of Planck length and time? I thought that matches well with a discrete set of definable numbers. – LinusK Mar 09 '22 at 19:03
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    @LinusK Planck length and time are dimensionful constants defined with the other fundamental ones, it doesn't imply discreteness of space and time, this is a common misconception. – Jeanbaptiste Roux Mar 09 '22 at 19:10
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If every number you can write or express, like 0.3432 and π are definable, then yes, you don't need the undefinable numbers. But who knows, they may be defined in the future.

Venkat Sarma
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