# Defining real numbers to exclude incomputable numbers

The real numbers are normally constructed via Dedekind cuts or similar approaches, which result in incomputable numbers: numbers that no finite algorithm can produce to arbitrary precision. This is in contrast with the computable reals, such as $$\pi$$, which, while irrational, can be computed to arbitrary precision via a finite algorithm (e.g. a Turing machine).

Would an alternate approach, that required a set, sequence, or number, whether finite or infinite, to have a finite definition, work?

It would seem to me that Dedekind cuts would still apply (as the sets of rationals need to be definable, e.g. "the set of rationals that, when squared, are less than the limit of $$\sum \frac{x^n}{n!}$$"), Completeness Property would still be met (any set with an upper bound has a least upper bound; as all sets need a finite definition, there is no set with an uncomputable bound).

Is there any inconsistency with defining sets, sequences, and the real numbers this way? Would it violate any of the principles of real analysis?

And, if not, what reason do we have to include sets which lack finite definitions, and result in incomputable numbers?

• There are definable numbers that may or may not be computable. May 7, 2021 at 21:04
• The question is, how exactly do you define "finite definition"? May 7, 2021 at 21:04
• It would be weird to have only countably many real numbers... I suppose that's one way to cheat the continuum hypothesis. May 7, 2021 at 21:09
• Can you construct an unmeasurable set in this thin subset of the reals? The usual construction fails because there aren't enough cosets of the reationals. May 7, 2021 at 21:16
• @MaliceVidrine: What about things like left/right continuity, intermediate value theorems, etc.? May 7, 2021 at 21:35