Why are the Real Numbers indecomposable according to Wikipedia?

Asked Aug 31 '20 at 17:01

Active Aug 31 '20 at 17:07

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It follows from the indecomposability principle that any property of real numbers that is decided (each real number either has or does not have that property) is in fact trivial (either all the real numbers have that property, or else none of them do).

Is the property $x>0$ not a counterexample of this? Is wikipedia's definition of indecomposability lacking details?

edited Aug 31 '20 at 17:07

Michael Hardy

asked Aug 31 '20 at 17:01

JMC

4

How familiar are you with intuitionistic logic, constructive mathematics, and the rejection of the law of the excluded middle? – Arthur Aug 31 '20 at 17:07
Very unfamiliar, to be honest. – JMC Aug 31 '20 at 17:07
As am I. But the concept of indecomposability belongs over there somewhere. – Arthur Aug 31 '20 at 17:09
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The problem is that you're not reading the article in the context of constructive mathematics. $\qquad$ – Michael Hardy Aug 31 '20 at 17:09
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It's not true constructively that every real number $x$ is either $>0$ or $\leq 0$ - you need the law of the excluded middle for that. – Matthew Towers Aug 31 '20 at 17:09

Why are the Real Numbers indecomposable according to Wikipedia?

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