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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?

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  • $\begingroup$ There are definable numbers that may or may not be computable. $\endgroup$
    – Asaf Karagila
    Commented May 7, 2021 at 21:04
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    $\begingroup$ The question is, how exactly do you define "finite definition"? $\endgroup$ Commented May 7, 2021 at 21:04
  • $\begingroup$ It would be weird to have only countably many real numbers... I suppose that's one way to cheat the continuum hypothesis. $\endgroup$ Commented May 7, 2021 at 21:09
  • $\begingroup$ 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. $\endgroup$ Commented May 7, 2021 at 21:16
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    $\begingroup$ @MaliceVidrine: What about things like left/right continuity, intermediate value theorems, etc.? $\endgroup$
    – Asaf Karagila
    Commented May 7, 2021 at 21:35

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If we interpret "has a finite description" as "is computable," then in fact there are very stark differences between classical analysis and its computable analogue. For example (contra your claim), there are computable monotonically increasing bounded sequences of rationals whose suprema are not computable; such sequences are called Specker sequences. The general study of analysis from a computable perspective is, appropriately enough, called computable analysis and there are a number of textbooks and papers on the topic.

One possible response to this is to broaden the notion of "finitely describable," but at this point we need to be very careful about our intuitions around this notion. For example, we could restrict attention to the real numbers which are parameter-freely first-order definable over some large set (e.g. some "big enough" level of the cumulative hierarchy); this would guarantee lots of nice analytic properties, but to me far overshoots the notion of "finitely describable" that we're going for here.

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