User blog comment:Superman37891/Explain Rayo please/@comment-30754445-20170622112528/@comment-30754445-20170622151148

Your two questions aren't really related.

The answer to your first question is yes. There are functions which cannot be defined in FOST.

For example, Rayo's function itself cannot be defined in FOST. Otherwise we would hit Barry's Paradox again. The same is true for the few known functions that beat Rayo (BIG FOOT and the like).

So one obvious way to "beat" Rayo is to create a new language (say) FOST-2 which includes FOST + a special new "word" for Rayo(n). This will be stronger than Rayo, but jury is still out on whether this should be considered a naive extension or an actual improvement (personally I consider it a naive extension).

You could then repeat this process as many times as you like, to get FOST-3 and FOST-4 and so on. But really, a better way to go would be to encode this entire process into an even more powerful language that allows you to create things like FOST-n on the fly.

At any rate, this should not be attempted by anyone who doesn't understand exactly what they're doing. A few months ago we had a bunch of newcomers who tried to beat Rayo in his own game by creating ill-defined languages of their own. The result was an ugly mess, made worse by the fact that the creators of these systems didn't realize that they were talking nonesense.

So as they say on TV: don't try that a home :-)

As for your second question:

"will there ever be a faster growing function that's based on a completely different idea than "take n letters to define the biggest number in language X"

Nobody knows.

I find it somewhat unlikely, since any such concept would have to be - by definition - beyond the scope of any formal language. So how are we going to speak about it or even define it?

Then again, maybe this is just an example of our current lack of imagination.

After all, Rayo itself would seem like an impossibility for a 19th century mathematician. How on earth can we have a function that grows faster than any step-by-step process? That's rediculous... yet Rayo is exactly such a function.

So perhaps we should be weary of predicting what will be possible in the future.