My BEAF extension

I also made an extension of Bowers' exploding array function (BEAF) that goes up to the Bachmann-Howard ordinal level, based off Bird's array notation (BAN). This extension is still work in progress (WIP) and is not formalized yet. Here I explain expansions for several examples at n = 2, 3, 4 given by me.

This expansion uses slashes and square brackets. Slash is a shorthand for [0]. Note that zeroes are omitted by default before parentheses () for dimensions and square brackets [] for superdimensions. Dimensions from "0, 1" to "0 / 1" are usually excluded since they consult the same pattern as with the precedent. 

1. {a, n (0 / 1) 2}

   • n = 2: {a, 2 (0, 1) 2}

   • n = 3: {a, 3 ((0, 1) 1) 2}

   • n = 4: {a, 4 (((0, 1) 1) 1) 2}

2. {a, n (0 / 1) 3}

   • n = 2: {a, 2 (0, 1) 2 (0 / 1) 2}

   • n = 3: {a, 3 ((0, 1) 1) 2 (0 / 1) 2}

   • n = 4: {a, 4 (((0, 1) 1) 1) 2 (0 / 1) 2}

3. {a, n (0 / 1)(0 / 1) 2}

   • n = 2: {a, 2 (0 / 1)(0, 1) 2}

   • n = 3: {a, 3 (0 / 1)((0, 1) 1) 2}

   • n = 4: {a, 4 (0 / 1)(((0, 1), 1), 1) 2}

4. {a, n (1 / 1) 2}

   • n = 2: {a, 2 (0 / 1)(0 / 1) 2}

   • n = 3: {a, 3 (0 / 1)(0 / 1)(0 / 1) 2}

   • n = 4: {a, 4 (0 / 1)(0 / 1)(0 / 1)(0 / 1) 2}

5. {a, n (2 / 1) 2}

   • n = 2: {a, 2 (1 / 1)(1 / 1) 2}

   • n = 3: {a, 3 (1 / 1)(1 / 1)(1 / 1) 2}

   • n = 4: {a, 4 (1 / 1)(1 / 1)(1 / 1)(1 / 1) 2}

6. {a, n (0, 1 / 1) 2}

   • n = 2: {a, 2 (2 / 1) 2}

   • n = 3: {a, 3 (3 / 1) 2}

   • n = 4: {a, 4 (4 / 1) 2}

7. {a, n ((0 / 1) 1 / 1) 2}

   • n = 2: {a, 2 ((0, 1) 1 / 1) 2}

   • n = 3: {a, 3 (((0, 1) 1) 1 / 1) 2}

   • n = 4: {a, 4 ((((0, 1) 1) 1) 1 / 1) 2}

8. {a, n ((0 / 1) 2 / 1) 2}

   • n = 2: {a, 2 ((0, 1) 1 (0 / 1) 1 / 1) 2}

   • n = 3: {a, 3 (((0, 1) 1) 1 (0 / 1) 1 / 1) 2}

   • n = 4: {a, 4 ((((0, 1) 1) 1) 1 (0 / 1) 1 / 1) 2}

9. {a, n ((0 / 1)(0 / 1) 1 / 1) 2}

   • n = 2: {a, 2 ((0 / 1)(0, 1) 1 / 1) 2}

   • n = 3: {a, 3 ((0 / 1)((0, 1) 1) 1 / 1) 2}

   • n = 4: {a, 4 ((0 / 1)(((0, 1) 1) 1) 1 / 1) 2}

10. {a, n (((0 / 1) 1 / 1) 1 / 1) 2}

    • n = 2: {a, 2 (((0, 1) 1 / 1) 1 / 1) 2}

    • n = 3: {a, 3 ((((0, 1) 1) / 1) 1 / 1) 2}

    • n = 4: {a, 4 (((((0, 1) 1) 1) / 1) 1 / 1) 2}

11. {a, n (0 / 2) 2}

    • n = 2: {a, 2 ((0 / 1) 1 / 1) 2}

    • n = 3: {a, 3 (((0 / 1) 1 / 1) 1 / 1) 2}

    • n = 4: {a, 4 ((((0 / 1) 1 / 1) 1 / 1) 1 / 1) 2}

12. {a, n ((0 / 2) 1 / 2) 2}

    • n = 2: {a, 2 (((0 / 1) 1 / 1) 1 / 2) 2}

    • n = 3: {a, 3 ((((0 / 1) 1 / 1) 1 / 1) 1 / 2) 2}

    • n = 4: {a, 4 (((((0 / 1) 1 / 1) 1 / 1) 1 / 1) 1 / 2) 2}

13. {a, n (0 / 3) 2}

    • n = 2: {a, 2 ((0 / 2) 1 / 2) 2}

    • n = 3: {a, 3 (((0 / 2) 1 / 2) 1 / 2) 2}

    • n = 4: {a, 4 ((((0 / 2) 1 / 2) 1 / 2) 1 / 2) 2}

14. {a, n (0 / 4) 2}

    • n = 2: {a, 2 ((0 / 3) 1 / 3) 2}

    • n = 3: {a, 3 (((0 / 3) 1 / 3) 1 / 3) 2}

    • n = 4: {a, 4 ((((0 / 3) 1 / 3) 1 / 3) 1 / 3) 2}

15. {a, n (0 / 0, 1) 2}

    • n = 2: {a, 2 (0 / 2) 2}

    • n = 3: {a, 3 (0 / 3) 2}

    • n = 4: {a, 4 (0 / 4) 2}

16. {a, n (0 / 1, 1) 2}

    • n = 2: {a, 2 ((0 / 0, 1) 1 / 0, 1) 2}

    • n = 3: {a, 3 (((0 / 0, 1) 1 / 0, 1) 1 / 0, 1) 2}

    • n = 4: {a, 4 ((((0 / 0, 1) 1 / 0, 1) 1 / 0, 1) 1 / 0, 1) 2}

17. {a, n (0 / 0, 2) 2}

    • n = 2: {a, 2 (0 / 2, 1) 2}

    • n = 3: {a, 3 (0 / 3, 1) 2}

    • n = 4: {a, 4 (0 / 4, 1) 2}

18. {a, n (0 / 0, 0, 1) 2}

    • n = 2: {a, 2 (0 / 0, 2) 2}

    • n = 3: {a, 3 (0 / 0, 3) 2}

    • n = 4: {a, 4 (0 / 0, 4) 2}

19. {a, n (0 / (1) 1) 2}

    • n = 2: {a, 2 (0 / 0, 0, 1) 2}

    • n = 3: {a, 3 (0 / 0, 0, 0, 1) 2}

    • n = 4: {a, 4 (0 / 0, 0, 0, 0, 1) 2}

20. {a, n (0 / (2) 1) 2}

    • n = 2: {a, 2 (0 / (1)(1) 1) 2}

    • n = 3: {a, 3 (0 / (1)(1)(1) 1) 2}

    • n = 4: {a, 4 (0 / (1)(1)(1)(1) 1) 2}

21. {a, n (0 / (0, 1) 1) 2}

    • n = 2: {a, 2 (0 / (2) 1) 2}

    • n = 3: {a, 3 (0 / (3) 1) 2}

    • n = 4: {a, 4 (0 / (4) 1) 2}

22. {a, n (0 / (0 / 1) 1) 2}

    • n = 2: {a, 2 (0 / (0, 1) 1) 2}

    • n = 3: {a, 3 (0 / ((0, 1) 1) 1) 2}

    • n = 4: {a, 4 (0 / (((0, 1) 1) 1) 1) 2}

23. {a, n (0 / (0 / 2) 1) 2}

    • n = 2: {a, 2 (0 / (0 / 1) 1) 2}

    • n = 3: {a, 3 (0 / ((0 / 1) 1 / 1) 1) 2}

    • n = 4: {a, 4 (0 / (((0 / 1) 1 / 1) 1 / 1) 1) 2}

24. {a, n (0 / (0 / (0 / 1)) 1) 2}

    • n = 2: {a, 2 (0 / (0 / (0, 1) 1)) 2}

    • n = 3: {a, 3 (0 / (0 / ((0, 1) 1) 1)) 2}

    • n = 4: {a, 4 (0 / (0 / (((0, 1) 1) 1) 1)) 2}

25. {a, n (0 / 0 / 1) 2}

    • n = 2: {a, 2 (0 / (0 / 1) 1) 2}

    • n = 3: {a, 3 (0 / (0 / (0 / 1) 1) 1) 2}

    • n = 4: {a, 4 (0 / (0 / (0 / (0 / 1) 1) 1) 1) 2}

26. {a, n (0 / 1 / 1) 2}

    • n = 2: {a, 2 ((0 / 0 / 1) 1 / 0 / 1) 2}

    • n = 3: {a, 3 (((0 / 0 / 1) 1 / 0 / 1) 1 / 0 / 1) 2}

    • n = 4: {a, 4 ((((0 / 0 / 1) 1 / 0 / 1) 1 / 0 / 1) 1 / 0 / 1) 2}

27. {a, n (0 / 2 / 1) 2}

    • n = 2: {a, 2 ((0 / 1 / 1) 1 / 1 / 1) 2}

    • n = 3: {a, 3 (((0 / 1 / 1) 1 / 1 / 1) 1 / 1 / 1) 2}

    • n = 4: {a, 4 ((((0 / 1 / 1) 1 / 1 / 1) 1 / 1 / 1) 1 / 1 / 1) 2}

28. {a, n (0 / 0 / 2) 2}

    • n = 2: {a, 2 (0 / (0 / 0 / 1) 1 / 1) 2}

    • n = 3: {a, 3 (0 / (0 / (0 / 0 / 1) 1 / 1) 1 / 1) 2}

    • n = 4: {a, 4 (0 / (0 / (0 / (0 / 0 / 1) 1 / 1) 1 / 1) 1 / 1) 2}

29. {a, n (0 / 0 / 0 / 1) 2}

    • n = 2: {a, 2 (0 / 0 / (0 / 0 / 1) 1) 2}

    • n = 3: {a, 3 (0 / 0 / (0 / 0 / (0 / 0 / 1) 1) 1) 2}

    • n = 4: {a, 4 (0 / 0 / (0 / 0 / (0 / 0 / (0 / 0 / 1) 1) 1) 1) 2}

30. {a, n (0 / 0 / 0 / 0 / 1) 2}

    • n = 2: {a, 2 (0 / 0 / 0 / (0 / 0 / 0 / 1) 1) 2}

    • n = 3: {a, 3 (0 / 0 / 0 / (0 / 0 / 0 / (0 / 0 / 0 / 1) 1) 1) 2}

    • n = 4: {a, 4 (0 / 0 / 0 / (0 / 0 / 0 / (0 / 0 / 0 / (0 / 0 / 0 / 1) 1) 1) 1) 2}

31. {a, n ([1] 1) 2}

    • n = 2: {a, 2 (0 / 0 / 1) 2}

    • n = 3: {a, 3 (0 / 0 / 0 / 1) 2}

    • n = 4: {a, 4 (0 / 0 / 0 / 0 / 1) 2}

32. {a, n (0 / 1 [1] 1) 2}

    • n = 2: {a, 2 (([1] 1) 1 [1] 1) 2}

    • n = 3: {a, 3 ((([1] 1) 1 [1] 1) 1 [1] 1) 2}

    • n = 4: {a, 4 (((([1] 1) 1 [1] 1) 1 [1] 1) 1 [1] 1) 2}

33. {a, n (0 / 0 / 1 [1] 1) 2}

    • n = 2: {a, 2 (0 / (0 / 1 [1] 1) 1 [1] 1) 2}

    • n = 3: {a, 3 (0 / (0 / (0 / 1 [1] 1) 1 [1] 1) 1 [1] 1) 2}

    • n = 4: {a, 4 (0 / (0 / (0 / (0 / 1 [1] 1) 1 [1] 1) 1 [1] 1) 1 [1] 1) 2}

34. {a, n ([1] 2) 2}

    • n = 2: {a, 2 (0 / 0 / 1 [1] 1) 2}

    • n = 3: {a, 3 (0 / 0 / 0 / 1 [1] 1) 2}

    • n = 4: {a, 4 (0 / 0 / 0 / 0 / 1 [1] 1) 2}

35. {a, n ([1] 3) 2}

    • n = 2: {a, 2 (0 / 0 / 1 [1] 2) 2}

    • n = 3: {a, 3 (0 / 0 / 0 / 1 [1] 2) 2}

    • n = 4: {a, 4 (0 / 0 / 0 / 0 / 1 [1] 2) 2}

36. {a, n ([1] 0 / 1) 2}

    • n = 2: {a, 2 ([1] ([1] 1) 1) 2}

    • n = 3: {a, 3 ([1] ([1] ([1] 1) 1) 1) 2}

    • n = 4: {a, 4 ([1] ([1] ([1] ([1] 1) 1) 1) 1) 2}

37. {a, n ([1] 0 / 0 / 1) 2}

    • n = 2: {a, 2 ([1] 0 / ([1] 0 / 1) 1) 2}

    • n = 3: {a, 3 ([1] 0 / ([1] 0 / ([1] 0 / 1) 1) 1) 2}

    • n = 4: {a, 4 ([1] 0 / ([1] 0 / ([1] 0 / ([1] 0 / 1) 1) 1) 1) 2}

38. {a, n ([1][1] 1) 2}

    • n = 2: {a, 2 ([1] 0 / 0 / 1) 2}

    • n = 3: {a, 3 ([1] 0 / 0 / 0 / 1) 2}

    • n = 4: {a, 4 ([1] 0 / 0 / 0 / 0 / 1) 2}

39. {a, n ([2] 1) 2}

    • n = 2: {a, 2 ([1][1] 1) 2}

    • n = 3: {a, 3 ([1][1][1] 1) 2}

    • n = 4: {a, 4 ([1][1][1][1] 1) 2}

40. {a, n ([3] 1) 2}

    • n = 2: {a, 2 ([2][2] 1) 2}

    • n = 3: {a, 3 ([2][2][2] 1) 2}

    • n = 4: {a, 4 ([2][2][2][2] 1) 2}

41. {a, n ([0, 1] 1) 2}

    • n = 2: {a, 2 ([2] 1) 2}

    • n = 3: {a, 3 ([3] 1) 2}

    • n = 4: {a, 4 ([4] 1) 2}

42. {a, n ([(0 / 1) 1] 1) 2}

    • n = 2: {a, 2 ([(0, 1) 1] 1) 2}

    • n = 3: {a, 3 ([((0, 1) 1) 1] 1) 2}

    • n = 4: {a, 4 ([(((0, 1) 1) 1) 1] 1) 2}

43. {a, n ([([1] 1) 1] 1) 2}

    • n = 2: {a, 2 ([(0 / 0 / 1) 1] 1) 2}

    • n = 3: {a, 3 ([(0 / 0 / 0 / 1) 1] 1) 2}

    • n = 4: {a, 4 ([(0 / 0 / 0 / 0 / 1) 1] 1) 2}

44. {a, n ([0 / 1] 1) 2}

    • n = 2: {a, 2 ([(0 / 1) 1] 1) 2}

    • n = 3: {a, 3 ([([(0 / 1) 1] 1) 1] 1) 2}

    • n = 4: {a, 4 ([([([(0 / 1) 1] 1) 1] 1) 1] 1) 2}

45. {a, n (0 / 1 [0 / 1] 1) 2}

    • n = 2: {a, 2 (([(0 / 1) 1] 1) 1 [(0 / 1) 1] 1) 2}

    • n = 3: {a, 3 ((([(0 / 1) 1] 1) 1 [(0 / 1) 1] 1) 1 [(0 / 1) 1] 1) 2}

    • n = 4: {a, 4 (((([(0 / 1) 1] 1) 1 [(0 / 1) 1] 1) 1 [(0 / 1) 1] 1) 1 [(0 / 1) 1] 1) 2}

46. {a, n ([([0 / 1] 1) 1] 1 [0 / 1] 1) 2}

    • n = 2: {a, 2 ([([(0 / 1) 1] 1) 1] 1 [0 / 1] 1) 2}

    • n = 3: {a, 3 ([([([(0 / 1) 1] 1) 1] 1) 1] 1 [0 / 1] 1) 2}

    • n = 4: {a, 4 ([([([([([0 / 1] 1) 1] 1) 1] 1) 1] 1) 1] 1 [0 / 1] 1) 2}

47. {a, n ([0 / 1] 2) 2}

    • n = 2: {a, 2 ([(0 / 1 [0 / 1] 1) 1] 1 [0 / 1] 1) 2}

    • n = 3: {a, 3 ([([(0 / 1 [0 / 1] 1) 1] 1 [0 / 1] 1) 1] 1 [0 / 1] 1) 2}

    • n = 4: {a, 4 ([([([(0 / 1 [0 / 1] 1) 1] 1 [0 / 1] 1) 1] 1 [0 / 1] 1) 1] 1 [0 / 1] 1) 2}

48. {a, n ([0 / 1] 0 / 1) 2}

    • n = 2: {a, 2 ([0 / 1] ([0 / 1] 1) 1) 2}

    • n = 3: {a, 3 ([0 / 1] ([0 / 1] ([0 / 1] 1) 1) 1) 2}

    • n = 4: {a, 4 ([0 / 1] ([0 / 1] ([0 / 1] ([0 / 1] 1) 1) 1) 1) 2}

49. {a, n ([0 / 1][0 / 1] 1) 2}

    • n = 2: {a, 2 ([0 / 1][([0 / 1] 0 / 1) 1] 1) 2}

    • n = 3: {a, 3 ([0 / 1][([0 / 1][([0 / 1] 0 / 1) 1] 1) 1] 1) 2}

    • n = 4: {a, 4 ([0 / 1][([0 / 1][([0 / 1][([0 / 1] 0 / 1) 1] 1) 1] 1) 1] 1) 2}

50. {a, n ([1 / 1] 1) 2}

    • n = 2: {a, 2 ([0 / 1][0 / 1] 1) 2}

    • n = 3: {a, 3 ([0 / 1][0 / 1][0 / 1] 1) 2}

    • n = 4: {a, 4 ([0 / 1][0 / 1][0 / 1][0 / 1] 1) 2}

51. {a, n ([0 / 2] 1) 2}

    • n = 2: {a, 2 ([([0 / 1] 1) 1 / 1] 1) 2}

    • n = 3: {a, 3 ([([([0 / 1] 1) 1 / 1] 1) 1 / 1] 1) 2}

    • n = 4: {a, 4 ([([([([0 / 1] 1) 1 / 1] 1) 1 / 1] 1) 1 / 1] 1) 2}

52. {a, n ([0 / 0 / 1] 1) 2}

    • n = 2: {a, 2 ([0 / ([0 / 1] 1) 1] 1) 2}

    • n = 3: {a, 3 ([0 / ([0 / ([0 / 1] 1) 1] 1) 1] 1) 2}

    • n = 4: {a, 4 ([0 / ([0 / ([0 / ([0 / 1] 1) 1] 1) 1] 1) 1] 1) 2}

53. {a, n ([[1] 1] 1) 2}

    • n = 2: {a, 2 ([0 / 0 / 1] 1) 2}

    • n = 3: {a, 3 ([0 / 0 / 0 / 1] 1) 2}

    • n = 4: {a, 4 ([0 / 0 / 0 / 0 / 1] 1) 2}

54. {a, n ([[0 / 1] 1] 1) 2}

    • n = 2: {a, 2 ([[([0 / 1] 1) 1] 1] 1) 2}

    • n = 3: {a, 3 ([[([[([0 / 1] 1) 1] 1 / 1] 1) 1] 1] 1) 2}

    • n = 4: {a, 4 ([[([[([[([0 / 1] 1) 1] 1] 1) 1] 1 / 1] 1) 1] 1] 1) 2}

55. {a, n ([[[0 / 1] 1] 1] 1) 2}

    • n = 2: {a, 2 ([[[([[0 / 1] 1] 1) 1 / 1] 1] 1] 1) 2}

    • n = 3: {a, 3 ([[[([[[([[0 / 1] 1] 1) 1] 1] 1] 1) 1 / 1] 1] 1] 1) 2}

    • n = 4: {a, 4 ([[[([[[([[[([[0 / 1] 1] 1) 1] 1] 1] 1) 1] 1] 1] 1) 1 / 1] 1] 1] 1) 2}

56. {a, n ([L] 1) 2} (limit of my extended BEAF)

    • n = 2: {a, 2 ([0 / 1] 1) 2}

    • n = 3: {a, 3 ([[0 / 1] 1] 1) 2}

    • n = 4: {a, 4 ([[[0 / 1] 1] 1] 1) 2}

Example:

{3, 3 ([[0 / 2] 2] 1) 2}

• {3, 3 (0 [0 [0 / 2] 2] 1) 2} (*to make the solving steps more convenient)

• = {3, 3 (0 [0 [0 (0 [0 [0 (0 [0 [0 / 1] 1 [0 / 2] 1] 1) 1 / 1] 1 [0 / 2] 1] 1) 1 / 1] 1 [0 / 2] 1] 1) 2}

• = {3, 3 ([[([[([[0 / 1] 1 [0 / 2] 1] 1) 1 / 1] 1 [0 / 2] 1] 1) 1 / 1] 1 [0 / 2] 1] 1) 2}

•