Pt 1. Given as input the tuple: (word, natural number)–does there exist a verifier, that runs faster than a fixed enumerator, that can accept if the word is in the position of the natural number in the enumeration by the fixed enumerator, and reject if it is not?
I suspect the answer is no. In that case…
Pt 2. Is there a way to get around this by adding a “tagging” function to the enumerator so that a verifier can be given something like (word#tag, natural number) so that it almost immediately knows, from the tag, if the word is in the position of the natural number (do there exist such ‘modified languages’) AND IT CAN’T BE FOOLED! Of course you can stick a natural number onto a word, but how can you do this so the verifier can BE SURE you aren’t lying…? Trying to figure out a way to tag the words (or generate a description for the words in a language (modify the language being enumerated/the enumerator)) so that stuck on to the words enumerated is the positions their the enumeration, and this information is reliable and can be verified quickly.
Any ideas? 🙂