Another is that Thomas’s question didn’t make it clear that each sentence covered had to be called out individually, as opposed to constructing some description that covers exactly the right sentences. Another is that even if we suppose our natural language augmented by allowing infinite sentences, it’s not clear that it should allow non-computable infinite sentences.
Another is that Thomas’s question didn’t make it clear that each sentence covered had to be called out individually, as opposed to constructing some description that covers exactly the right sentences. Another is that even if we suppose our natural language augmented by allowing infinite sentences, it’s not clear that it should allow non-computable infinite sentences.
The whole discussion, plus the posted problem, plus the (re)defining the problem itself—is more interesting than the problem alone.
As I see it, these are my assertions:
an infinite sentence is possible whenever the infinity is permitted
with such an infinite sentence you can uniquely describe all the finite sentences
even if the infinite sentence is selfdescribing
with such an infinite sentence you can uniquely describe countably infinite number of infinite sentences, too
there are non-countably many such infinite sentences
some of them can be described by some finite sentence
every finite or infinite sentence can be uniquely described by some infinite sentence and also by non-countably many of them
there may be some finite self describing sentences in English
By selfdescribing I mean the Quine type of a sentence. A self reproducing sentence.
Funny, but the whole set of infinite sentences can be described by just one finite sentence.