User blog comment:Eners49/The secret 0th hyper-operator?/@comment-35470197-20180726231426/@comment-31966679-20180730103339

I have actually been thinking about this since I made the level function (around 4 or 5 years ago), so I will say what I think about non-positive hyperoperators.

Also, for Eners49’s question, I am within the range 6 - 8 years old (that explains why I am so bad at math).

The 0th hyperoperator would be lvl(0, x, y)so following the definition of the lvl function (stated in my first blog post), we would get this:

lvl(1, x, y) = lvl(0, y, lvl(0, y, lvl(...)))

This doesn’t say anything about the 0th hyperoperator, so we need examples!

lvl(1, 1, 2) = lvl(0, 1, 1) according to the rules of my level function. If you have read my first blog post, then you should know that lvl(1, 1, 2) = 3

So now we know that lvl(0, 1, 1) = 3! In order to see a pattern in this, we need more examples. For simplicity’s sake, I’ll states lvl(0, x, y) as x?y.

0?0 = lvl(1, 0, 2) = 2 n?n = lvl(1, n, 2) = n + 2 n?n?n = lvl(1, n, 3) = n + 3 n?n?n?n = lvl(1, n, 4) = n + 4 In general, n?n?...?n with x ns = n + x n?(n + 2) = n?n?n = n + 3 n?(n + 3) = n?n?n?n = n + 4 In general, x?(x + y) = x + y + 1 However, we said that n?n = lvl(1, n, 2) = n + 2 n?(n + 0) = n + 0 + 1 = n + 1 However, n?(n + 0) = n?n, so n?n = n + 1, right? But we said that n?n = n + 2. Therefore, n + 1 = n + 2, then 1 = 2?! So in terms of the level function, there is no such thing as a 0th hyperoperator