This article is rated Stub-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
So I read this:
What does this mean? The unit interval - that's a continuum, right? Consider all points and all points . Those are two non-intersecting sets that partition the unit interval into two parts. I'm guessing wildly, but perhaps the precise definition meant to say: the continuum cannot be partitioned into two non-empty open sets that have empty intersection? Something like that? 67.198.37.16 ( talk) 17:56, 27 November 2017 (UTC)
Is "defined" in all points? For example, can you distinguish all points from rational or irrational numbers?-- SilverMatsu ( talk) 15:18, 4 July 2021 (UTC)
This is an issue in constructive mathematics, not intuitionistic logic: I can construct theories axiomatised in intuitionistic logic where decomposability holds. One can simply take an axiomatisation of the intuitionistic continuum with reals expressed as equivalence classes of Cauchy-convergent sequences of rationals and add trichotomy as a theorem: the theory becomes nonconstructive, but the theory will express formulae to which the principle of the excluded middle will not apply, so its logic remains intuitionistic. The problem with the article is that it fails to do the spadework to make clear the significance of the thesis.
For example, if I define a representation of the reals to be either a rational number (represented as a pair of coprime numbers) or a Cauchy-convergent sequence that does not converge to a rational number (represented as function from naturals to rationals), then every rational has a representation that is entirely arithmetic. But if we try to work with this representation, we can't decide what representation to use in general for the sum of two irrationals, since it is undecidable whether they are a rational.
I'd recommend merging this to either intuitionistic analysis or computable analysis, but neither of those articles have the required background either. There is the rather better article computable number, but it still assumes without discussion one particular representation of computable reals. Our coverage of the intuitionistic continuum sucks, unfortunately. — Charles Stewart (talk) 18:48, 5 July 2021 (UTC)
This article is rated Stub-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
So I read this:
What does this mean? The unit interval - that's a continuum, right? Consider all points and all points . Those are two non-intersecting sets that partition the unit interval into two parts. I'm guessing wildly, but perhaps the precise definition meant to say: the continuum cannot be partitioned into two non-empty open sets that have empty intersection? Something like that? 67.198.37.16 ( talk) 17:56, 27 November 2017 (UTC)
Is "defined" in all points? For example, can you distinguish all points from rational or irrational numbers?-- SilverMatsu ( talk) 15:18, 4 July 2021 (UTC)
This is an issue in constructive mathematics, not intuitionistic logic: I can construct theories axiomatised in intuitionistic logic where decomposability holds. One can simply take an axiomatisation of the intuitionistic continuum with reals expressed as equivalence classes of Cauchy-convergent sequences of rationals and add trichotomy as a theorem: the theory becomes nonconstructive, but the theory will express formulae to which the principle of the excluded middle will not apply, so its logic remains intuitionistic. The problem with the article is that it fails to do the spadework to make clear the significance of the thesis.
For example, if I define a representation of the reals to be either a rational number (represented as a pair of coprime numbers) or a Cauchy-convergent sequence that does not converge to a rational number (represented as function from naturals to rationals), then every rational has a representation that is entirely arithmetic. But if we try to work with this representation, we can't decide what representation to use in general for the sum of two irrationals, since it is undecidable whether they are a rational.
I'd recommend merging this to either intuitionistic analysis or computable analysis, but neither of those articles have the required background either. There is the rather better article computable number, but it still assumes without discussion one particular representation of computable reals. Our coverage of the intuitionistic continuum sucks, unfortunately. — Charles Stewart (talk) 18:48, 5 July 2021 (UTC)