User blog:Eners49/The secret 0th hyper-operator?

The 1st, 2nd, 3rd operations are addition, multiplication, and exponentiation, respectively. You can also continue with tetration, pentation, and more up-arrows. But what about the 0th hyperoperator (I'm going to denote the 0th hyperoperator as x?y from here on)? Is it really the successor function ? What's more, how can we evaluate things such as the -1th hyperoperator?

The Googology Wiki article on the successor function states that it is the 0th hyperoperation, being f(x) = x+1. This makes perfect logical sense, since addition is the repeated act of adding 1, but we have a few things to consider, since if the 0th hyperoperation is the successor function, some of the things we already know about hyperoperations do not hold true anymore.

We know that 2^^^^3 = 2^^^4.

Continuing, we can also get: So if we use the 0th hyperoperation on 2 and 4, we get 5! Not quite what we expected. But, if we do the same thing we do for up-arrows: Logically, it follows that 2?2 = 4, since 2{n}2 = 4 for all n, and this seems to work for the 0th hyperoperation too. But let's try a few more examples: Strangely, we notice that 3?3 is equal to 2?4.
 * 2^^^3 = 2^^4 = 65,536
 * 2^^3 = 2^4 = 16
 * 2^3 = 2*4 = 8
 * 2*3 = 2+4 = 6
 * 2+3 = 2?4 = 5
 * 2^^^^3 = 2^^^2^^^2
 * etc.
 * 2+3 = 2?2?2 = 2?4 = 5
 * 3^^^^2 = 3^^^3 = tritri
 * 3^^^2 = 3^^3 = 7,625,597,484,987
 * etc.
 * 3+2 = 3?3 = 5

Something else: We can say that 3+3 = 3?3?3 = 6, and we already know what 3?3 is. Thus, 3?5 = 6. But HANG ON! If 3?3 is 5, and 3?5 is 6, then WHAT IS 3?4 ? We have yet to find out. (I probably screwed up somewhere)

I'm not going to cover anything else on this topic because I really don't know what else to say, and I probably already terribly messed up. Let me know what you guys think in the comments!