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Wittgenstein (PR 181) talks about a criterion of completeness for the irrationals. I am trying to understand what this might mean.

Completeness of the reals, in the decimal number system, is the statement that every infinite decimal expansion represents a real number. It follows, I believe, that any two decimal expansions must be comparable, although I can't quite spell this out in terms of the notion forestalling a possible "gap" in the reals.

Analogously, one would expect completeness of the irrationals to be the statement that every infinite decimal expansion that is non-terminating and non-periodic represents an irrational number.

Finally, a "criterion" is presumably a method for determining when completeness holds. Understood in a verificationist sense, this would have to be an effective procedure. But I'm not sure how to state the criterion. Would it suffice for any two such decimal expansions to be comparable (in an effective way)?

Is there a way of stating it in terms of the idea of determining every possible "gap"?

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    $\begingroup$ PR= Philosophical Remarks. Stanford Encyclopedia of Philosophy: "he opposes foundationalism and especially its concept of a 'gapless mathematical continuum.'" $\endgroup$ Commented Jun 30, 2018 at 22:24
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    $\begingroup$ Is Wittgenstein really that perverse? That "completeness" is something about decimals? $\endgroup$ Commented Jun 30, 2018 at 23:34
  • $\begingroup$ @GeraldEdgar I didn't mean to suggest that Wittgenstein is supposing completeness to be $essentially$ a fact about decimal expansions. But his concern in this context seems to be with Cantor's diagonal method of proof, and the latter makes relevant the formulation of completeness in terms of decimal representations. $\endgroup$ Commented Jul 1, 2018 at 2:33

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It think that he is referring to the Completeness of the real numbers, i.e. the idea that :

there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line.

See the discussion of Dedekind's cuts into :

Can the idea of a cut now be said to have led us from the rational to the irrational numbers? Are we for example led to $\sqrt 2$ way of the concept of a cut? Now what is a cut of the real numbers? Well, a principle of division into an upper and a lower class. Thus such a principle yields every rational and irrational number. [...] But now Dedekind's idea is that the division into an upper and lower class (under the known conditions) is the real number.

The thing now is to prove that no other numbers except the real numbers can perform such a cut.

This is very similar to Philosophical Remarks, 180, where Wittgenstein discusses of "cut[ing] at a place where there is no rational number".

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