Comments on Everyone Else is Crazy: Countable Ordinals in Haskell

Hi, it is me again. I wrote this for jared and any...

2008-02-06T02:29:22.993-08:00

Hi, it is me again. I wrote this for jared and anyone else that might be interested. As my post explains, in the cae of ordinals, the cruicial question is: which operations on ordinals should be computable. For example, is the order relation computable for your representation?

The V=L axiom has little to do with what you are t...

2008-02-06T00:02:06.673-08:00

The V=L axiom has little to do with what you are talking about. You make it sound as if V=L means that "all is countable". In fact, even if you assume V=L there will still be plenty of very large sets around.

Also, it is a bit inaccurate to say that you are representing all countable ordinals. At best you are representing ordinals below the Church-Kleene ordinal.

An Jared said that it is "impossible for a computer to represent uncountable sets". This is false, see e.g., Klaus Weihrauch's book "Computable Analysis" for an explanation about why this is not the case. Actually, I am going to go now and write a blog post about this, since "non-professionals" seem to be a bit misguided about this.

It is indeed impossible for a computer to represen...

2008-02-05T19:39:56.631-08:00

It is indeed impossible for a computer to represent an uncountable set. Such a representation would be a function from the natural numbers to the set, which would constitute a proof that the set was countable.