User blog:Mh314159/A new notation for large numbers

This set of functions is based on the vertical bar symbol, which stands for a series of functions indicated by ordinals to its left and arguments to its right.

n|11 = n|1 = n + 1

1|1x = 1|x = A(x) = Ack(x,x)

n|mx = n|(n|m-1x)

n|x = (n-1)|nx

2|2 = 1|32 = A(A(A2) = A(A(7))

2|3 = 1|23 = AA(A(7))3. This is much greater than Graham's number and also greater than Friedmann's n(4).

2|4 = 1|24 = the number you get when you recursively apply the A function 2|3 times. This is greater than 3>3>3>3 in Conway right arrow notation.

3|2 = 2|42 = 2|2|2|2|2 (right associative) = 2|2|2|(A(A(7))

where 2|(A(A(7)) = a functional recursion pyramid of 1| functions (A(A(7)) levels high with 2|1 at the top. Second level from top = 2|2 = 1|32 = A(A(7)). Third level from top = 2|3 (see above). Fourth level from top = 2|4 (see above). Each level involves applying the 2| function a number of times equal to the value of the previous level. After determining the number generated at the bottom of (A(A(7)) levels, this becomes the argument M in 2|M and when this is evaluated as some number P, the value of 3|2 is 2|P

9|9= 8|99 = a recursion pyramid 9 layers high with a left foot of 8 constant except for the top layer where it is 9, and a right foot of 9 counting down to 1. The top two layers: 8|92 = 8|102 = 8|8|8|8|8|8|8|8|8|8|2 right associated. Since the last 8|2 = 7|92 = 7|7|7|7|7|7|7|7|7|2 we have 8|8|8|8|8|8|8|8|8|7|7|7|7|7|7|7|7|7|2 etc. And when we have worked this down to 1| functions and evaluated right to left we will have evaluated the second layer from the top.

 1,


 * c indicates c consecutive bars

1|cundefineda = <a|c-1undefineda|c-1undefined... a with a instances of a, left associative

1|||a= ((a||a)||a)...a with a instances of a, left associative

etc.

a|cundefinedb = (a-1)|cac(b-1)b
a|cmb = a|c(a|c...a|cb) with m instances of a, right associative

1||3 = (3|3)|3

2||2 = 1||22 = 1||(1||(1||2)) = 1||(1||(2|2)) = 1||(<a|a|...a) with a = 2|2 occuring 2|2 times and left associative.

2||3 = 1||23 = 1||(1||(...1||3)) with 2||2 instances

4||2 = 3||42 = 3||52 = 3||(3||(3||(3||(3||2))))

4||4 = 3||44

1|||4 = <4||4||4||4

3|||4 = 2|||34

etc.

Next would come a sequence of symbols beyond the vertical bar. I have notes on this also, if anyone is interested, and it takes the numbers to extremely higher values. But first I'd like to hear if anyone has any thoughts on how fast the bar functions grow.