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Recently, I tried creating a Peano-based googology model. I started with 10^3003, which I understand is named a millillion. I then defined a millillillion - or, for short, a thousand 2ill (there are 2 ills, because mille means thousand) - as 10^millillion. I then defined a thousand 3ill, then so on for a thousand nill. When you reach a thousand millillionill, 10^thousand millillionill is a first-order 1. The first-order successor function is 10^x. So first-order 2 is 10^first-order 1, etc. We'll use 1o for first-order to be short. 1o addition is defined as '1ox + 1oy = 10^10...^10^1oy, where there are x 10s. 1o multiplication is defined to the same way as 1o addition as normal multiplication is to normal addition. In the same way, we define 1o exponentation, etc. So we then reach, using the same method as above, 1o thousand millillionill, then 10^1o thousand millillionill = 2o1. Generally, the xo successor function is (x-1)o 10^xoy. So to get to, say, 100th-order 1 to 100th-order 2, you do 99o10^100o1. We can then reach a millillionth-order 1, then a thousand millillionillth-order 1. We can then reach second-period first-order 1, which is the number after a thousand millillionillth-order 1. My problem is that, since xo addition changes, I can't define how to get from xp1 to (x+1)p1. Please comment if you don't understand. Can somebody figure out the generalised formula for getting from xp1 to (x+1)p1?

Mark Smith
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    This question needs major revisions before it is ready for public consumption. A non-exhaustive list of my concerns begins… The way the font renders zero and lowercase O makes the post completely unreadable. Some paragraph breaks would be welcome also. In any case, I don’t understand the notation at the end; 2p1 means “second period 1” or “second period first order 1”? If the latter, where is the order in this notation? Either way, is there a first period? What do you mean by “xth order addition changes”; if you are changing it with the period, don’t you have the flexibility to choose how? – Eric Nathan Stucky Sep 07 '23 at 04:38
  • More seriously, your actual query feels odd to me. Is there some property we’re supposed to think of this p function as satisfying? It reads as though you are asking “I just built a new room for my house and it was a lot of fun! What’s the next room I should build for my house?” I don’t know mate, it’s your house. – Eric Nathan Stucky Sep 07 '23 at 04:46

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