Real Numbers Computability

because even e or pi or any other real number constructed by an equationNot true. Most real numbers cannot be constructed at all [1] [2]. There is some interest in replacing the mathematics of real numbers with computable numbers, but this adds additional complexity to analysis (requiring a computable modulus of convergence).[1] https://en.wikipedia.org/wiki/Computable_nu

Also ironic is that real numbers may not be "real" (i.e., exist) at all. The uncountable part which is THE part that completes rationals to continuity cannot be described or generated in any way since there are only COUNTABLY many different computer programs (or mathematical formulas).

There are real numbers that can't be represented in an "infinite" computer either.

Stupid question form a ignorant person (me): Are reals actually real? Do they even exist?Almost all reals (besides some countable subsets, which are irrelevant compared to the many many more reals that aren't part of those sets) have infinite many non repeating decimal places, and what's even more annoying, almost all of them aren't computable.The physical universe we can experience is finite. How can more than infinite many "unspeakable" things "exist&

The 'problem' with the reals is that there are numbers that cannot be constructed.Every number that we can construct can be constructed in a finite amount of symbols. For example sqrt(2) is an unambiguous description of itself. Without use of the sqrt function, we can also call it the number x such that x*x=2. However, every description is a finite string constructed from a finite alphabet. We can easily show that the set of all such descriptions is countably infinite. However, we c

Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite precision aren't a problem".

what the author means to allude to, through some nonsensical rambling, are the incomputable numbers [1]the cardinality of all real numbers that can be described by a terminating computer program to some accuracy is a countable set (since the number of such programs is countable) however, the cardinality of the reals is uncountable.hence most real numbers cannot be computed beyond a certain accuracy.Edit (additionally): the set of computable numbers forms a field (if a,b are computable,

You could make the same argument against the real numbers. Almost all of them cannot even be written down or described, so it's hard to say how they map to the real-world. But the concept of real numbers is extremely useful for a large number of mathematical proofs.

Yes, but the real numbers that can be computed to arbitrary precision with an algorithm are countable: this means almost all the reals contain an infinite amount of information. The real numbers are useful to reason and construct proofs, but as far as the physical world go, only very few of them are actually meaningful.

You are talking about rationals and algebraic numbers, which are just a tiny subset of the real numbers. Your point about computable vs non-computable numbers is valid, but I think the author meant that you will never encounter a real number that is non-computable, since those are impossible to express in the first place. Real numbers can contain an infinite amount of information.