On Sunday, 16 June 2013 17:38:56 UTC+2, fom wrote: > On 6/16/2013 5:47 AM, email@example.com wrote:
Sorry, it is really impossible for me with my Brouwser to answer your long text. Therefore only few points.
>> "Eventually, most mathematicians came to accept that definability should not be required, partly because the axiom of choice leads to nice results, but mostly because of the difficulties that arise when one tries to make notion of definability precise." (Andreas Blass) > >> That is a real surprise to me. Which mathematicians accepted that and when?
> I must concede to you on this one. Tarski wrote a paper on definability in which he made the observation that mathematicians are not too keen on the subject.
Definability is not a logical but a practical notion. Every child knows what is meant.
In 2009 a reviewer of FOM (I think it was Blass) wrote: "He's right that some statements about definability don't need details of the language, but I think he's wrong to infer (in his first sentence) "an absolute meaning of undefinability." (Does he perhaps think there's a particular real number that is undefinable in any (countable) language?)"
That shows that Blass (if he was it) at that time did not yet know that there are undefinable real numbers (or that there has been a resolution of a majority to accept that).
>> Wouldn't a set with undefined elements contradict the Axiom of Extensionality: If every element of X is an element of Y and every element of Y is an element of X, then X = Y. How could that be decided for undefined elements? >
> For that I have no answer. This is, in fact, where I have non-standard views.
Why? Standard views also cannot explain that. They prefer to delete the questions.
> In my version of foundations, definability is fundamental.
Then you should consider the fact that the definitions are a subset of a countable set.
> But, do you have a link to where you discuss/define "subcountable" with the intention of the interpretation you give above?
Sorry, I don't. I think it was somewhere here that I read it. But have forgotten context and possibly the intended meaning. (In my question I used that only in a rhetoric way - since every subset of a countable set is countable or empty and by no means uncountable. It was in fact not a real question.)