Hopefully someone better educated than me can answer this - several of the definitions in the link feel constructivist, i.e. they describe constructions of of real numbers. It seems easy to think of methods of constructing non-rational numbers, by e.g. using infinite sequences, by taking roots, or whatever.
It seems harder to prove that every real number can be constructed via such a method.
Is there a construction-based method that can produce ALL real numbers between, say, 0 and 1? This seems unlikely to me, since the method of construction would probably be based on some sort of enumeration, meaning that you would only end up with countably many numbers. But maybe someone else can help me become un-confused.
I came up with a different definition that is a kind of inverse of Dedekind cuts. It is the idea that a real number is the set of all rational intervals that contain it. Since this is circular, there are properties that I came up with which say when a set of rational intervals qualifies to be called a real number in my setup. I have an unreviewed paper which creates a version that is a bridge between numerical analysis and the theoretical definition of a real number. Another unreviewed paper shows the equivalence between my definition and Dedekind cuts. You can read both at [1].
There is a long tradition of using intervals for dealing with real numbers. It is often used by constructivists and can be thought of viewing a real number as a measurement.
Its interesting. When I first encountered complex numbers when starting high school it was very difficult to wrap my head around how they could be actual numbers.
I no longer have that problem, ever since I truly understood how all numbers are simply abstract tools for reasoning. In a way, it's interesting that complex numbers seem more "real" than the real numbers themselves.
I remember listening to a radio show where a physicist discussed the link between quantum mechanics and complex numbers, and thus how they were fundamental to reality [1], whereas we don't know whether real numbers actually describe physical reality.
[1] If I remember correctly, one argument was that although a common use of complex numbers is an alternative number system for making trigonometric/polar calculations simpler, they underpin quantum mechanics in a way that cannot be alternatively formulated in terms of real number numbers
A lot of physics equations describe real quantities like E=mc2. We just kind of take it for granted. You can formulate quantum mechanics without complex numbers but they seem kind of fundamental to it in a similar way to how real quantities like energy seem fundamental to reality.
I am no mathematician, but the idea of real numbers as the limit of rational numbers that don't belong to the set of all rational numbers blows my mind. And to top, the set of real numbers is so much bigger than the set of all rational numbers!
11 comments
[ 10.3 ms ] story [ 283 ms ] threadIt seems harder to prove that every real number can be constructed via such a method.
Is there a construction-based method that can produce ALL real numbers between, say, 0 and 1? This seems unlikely to me, since the method of construction would probably be based on some sort of enumeration, meaning that you would only end up with countably many numbers. But maybe someone else can help me become un-confused.
There is a long tradition of using intervals for dealing with real numbers. It is often used by constructivists and can be thought of viewing a real number as a measurement.
1: https://github.com/jostylr/Reals-as-Oracles
I no longer have that problem, ever since I truly understood how all numbers are simply abstract tools for reasoning. In a way, it's interesting that complex numbers seem more "real" than the real numbers themselves.
I remember listening to a radio show where a physicist discussed the link between quantum mechanics and complex numbers, and thus how they were fundamental to reality [1], whereas we don't know whether real numbers actually describe physical reality.
[1] If I remember correctly, one argument was that although a common use of complex numbers is an alternative number system for making trigonometric/polar calculations simpler, they underpin quantum mechanics in a way that cannot be alternatively formulated in terms of real number numbers
https://leanprover-community.github.io/mathlib-overview.html
https://binary.dw.cash/binaryuniverse/T27-3-zeckendorf-real-...