On 16 Jun., 18:07, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > Herman Jurjus <hjm...@hetnet.nl> writes: > > Also many classical mathematicians appreciate this as an example > > showing that the extensional notion 'Turing computable' is a slight > > distortion of the intuitive notion 'computable'. > > Possibly, but I don't think this is quite the right diagnosis. The issue > is more subtle.
The issue is very simple. A real number is computable, if its place on the real axis can be established, i.e., trichotomy. Otherwise it does not deserve the name number but at most number form or interval (like 0.1x means 0.10 to 0.19 in decimal). > > It's a well known phenomenon that many classically minded mathematicians > who have had little practice in constructive thinking are unwittingly > inconsistent in their reading of intuitionistic quantifiers. It's an > equally striking phenomenon that classically minded mathematicians in > certain contexts naturally adopt an intuitionistic reading of classical > quantifiers. In addition to the example provided by Virgil it's not > uncommon for people to mistakenly think, in a classical context, that > countable sets come equipped with a designated bijection witnessing > their countability.
The proof of countability of S does not require a bijection of S with N but only an injection of some superset of S into N.
This is established by my list
0 1 00 01 ...
where every line may be enumerated by an element of the countable set omega^omega^omega (and, if required, finitely many more exponents for alphabets, languages, dictionaries, thesauruses, and further properties)