[RajBhai]

I would say that any function that implicitly favors a single number must be explicitly stated, and thus, if used for this game, be the number 2. So all uses of the radical must state which root (2). Dirac's solution then wouldn't work because the use of 2 is O(n).

Logs would also need to state the base. No implicit use of e or 10, and lg wouldn't be allowed in place of log2.

I haven't said much other than logs and roots are binary operators with one of the operands usually implicit in the notation, so if we don't have special notation for powers and exponentiation, then we shouldn't allow the same for their inverse operations.

[necovek]

Why not?

Why is it ok to use "22" = 2 * 10^1 + 2 (when it could be a number in base 3 — 2 * 3^1 + 2 = 8 decimal — or any other base)? This implies base 10, just like root implies base 2, or ln means e.

As I said, this is a game, and trying to imply certain artificial constraints will be really hard with how abstract maths is.

Again, mention of successor function is apt: everything else is built from 1, succ() and another axiom, definition or so. So everything else can be reduced to this.

[RajBhai]

I said that this implicit use of 10 or some other number shouldn't be allowed. So log, ln, lg (i.e base 2) shouldn't be allowed, but log_b(x) where b and x are states is OK, just as 10^x, e^x, and 2^x require you to explicitly expose the base (and for this puzzle, disallow 10 and e since neither is a 2).

Successor is essentially s(n) = n + 1, so that shouldn't be allowed either.