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That was very interesting as an idea. Indeed, if you take the list of all descriptions of irrationals (use alphabetic-dictionary ordering), then you can construct a new description of an irrational which does not coincide with any of the ones you just listed descriptions of - simply let it differ from the n'th member of your list in the n'th decimal place (by a deterministic method like adding one to that digit, modulo ten).

Doesn't that describe an irrational? It would be a contradiction if it did, since it differs from all describable irrationals somewhere. So what's wrong?

Well, we buttumed that there were countably many describable irrationals. Isn't that right? Well, we know there are countably many descriptions, because we can list them in order (we list all strings and chuck out those that don't make syntactic sense, and "making syntactic sense" is a decidable parsing procedure). So aren't there ****ably many descriptions of rational numbers?

Maybe not - do we have a way of telling if a description describes a rational number? What is such a thing? It's a computational procedure (in constructivist universes) for computing its digits. Can we tell when a description is a procedure? Yes - it looks like a computer program (say, in fortran). Can we tell what a procedure computes? No - we can't tell if a procedure will churn out digits forever or stop afer a finite number or get hung up without progress.

In fact, the argument you inspired and which I detailed above SHOWS that there is no procedure for deciding if a procedure goes into an infinite loop without progress, or not (i.e. continues churning out digits forever, or stops happily saying "that's it"). I.e. in particular, there is no procedure for deciding if a procedure describes a number or not.

Curious.

Is the real Turing argument like that? It's been so long that I've forgotten! I thought it was a fixpoint argument using a contraction mapping in some metric space. Hmm .. I suppose my argument above is "taking a limit" in some sense.

Apparently in constructivist space, one can't just "take a subset" by SAYING (i.e. via "comprehension"). The subset has to be defined via a procedure.

Peter