(That was trolling.)
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.
It gets weirder. What is a set? For finite sets, we know it intuitively. But consider the Axiom of Choice. There is a consistent mathematics in which a choice set is a set, and one in which the same meta-mathematical object is not a set. (Unless, of course, ZF is inconsistent.)
Another subtle detail is that while it's true that every real number corresponds to (and can be represented by) a Cauchy sequence of rationals, the very sequence itself might be undefinable.
The decimal numbers, for example, can be viewed as an infinite converging sum of powers of ten. Theoretically one could produce a description, but only a countable number of those could be written down in finite terms (some kind of finite recipe). So those finite ones could fall in a constructivist camp, but the ones requiring an infinite string to describe would, as far as I understand constructivism, not fall under being constructivist. To be clear, the finite string doesn't have other be explicit about how to produce the numbers, just that it is naming the thing and it can be derived from that. So square root of 2 names a real number and there is a process to compute out the decimals so that exists in a constructivist sense. But "most" real numbers could not be named.
If you are a constructivist, then you will supply direct proofs for your results as you reject indirect proof, proof by contradiction, law of excluded middle, and things of this nature.
The converse does not necessarily hold. Providing a direct construction of an object satisfying the field and completeness axioms (e.g. the Dedekind construction) does not mean that one is a constructivist. Indeed, one can use the Dedekind construction and still go on to prove many more results on top of it that still do rely on indirect proof and reductio ad absurdum.
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.
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
thatguysaguy•2h ago