Any Rayo string can be written as either:
1. xi∈xj
2. xi=xj
3.(¬A)
4. ∃xi(A) (where A is a Rayo string)
5. (A∧B) (where A and B are Rayo strings).
And their lengths:
Type 1: 3
Type 2: 3
Type 3: L(A)+3
Type 4: L(A)+4
Type 5: L(A)+L(B)+3
From the above it is easy to see that the only possible lengths below 10 are: 3,6,7 and 9.
And thus we can write a complete list of all Rayo strings whose length is less than 10:
Length 3:
1. xi∈xj
2. xi∈xi
3. xi=xj
4. xi=xi
Length 6:
5. ¬(xi∈xj)
6. ¬(xi∈xi)
7. ¬(xi=xj)
8. ¬(xi=xi)
Length 7:
9. ∃xi(xi∈xj)
10. ∃xj(xi∈xj)
11. ∃xk(xi∈xj)
12. ∃xi(xi=xj)
13. ∃xj(xi=xj)
14. ∃xk(xi=xj)
15. ∃xi(xi∈xi)
16. ∃xj(xi∈xi)
17. ∃xi(xi=xi)
18. ∃xj(xi=xi)
Length 9:
19. ¬(¬(xi∈xj))
20. ¬(¬(xi∈xi))
21. ¬(¬(xi=xj))
22. ¬(¬(xi=xi))
23-82. 4*Bell(4)=60 expressions of the form (A∧B) where A and B are of one of the forms 1-4 (more on those - later).
Now we'll show that none of the above expressions can be a Rayo name of any number:
1. Let X={ {∅} } and Y={X}. Note that neither sets represent a number.
2. #1 and #7 are satisfied by xi=X, xj=Y. So it can't be a Rayo name of any number.
3. #2 and #8 are never satisfied.
4. #4 amd #6 are always satisfied
4. #3 and #5 are satisfied by xi=xj=X.
5. #11, #14, #16 and #18 are equivalent to #1, #2, #3 and #4 respectively, and are therefore redundant.
6. #10, #12 and #13 and #17 are alway satisfied.
7. #9 is satisfied for xi=xj=X.
8. #15 is never satisfied.
This leaves us with the the 60 expressions of the form (A∧B).
We'll organize these in groups of 4, by the indices of the x's, giving us 15 groups in all:
iiii, iiij, iiji, iijj, iijk,
ijii, ijij, ijik, ijji, ijjj,
ijjk, ijki, ijkj, ijkk, ijkl
Of these, we can eliminate every group that has:
1. Two identical indicies in either positions 1 & 2 or positions 3 & 4. This is because "xi=xi" is a tautology and "xi∈xi" is a contradiction.
2. A repeated pair of indices. Such an expression is of one of two forms: (A∧A) which is equivalent to A, or (xa∈xb∧xa=xb) which is a contradiction.
This leaves us with the following 6 groups of 4 expressions each:
ijik, ijji, ijjk, ijki, ijkj, ijkl
Now, each one of the expressions A and B is either an equality or an inclusion relation:
1. If both A and B are equalities, then the entire expression would be satisfied by setting all the variables to (say) X.
2. If A is an equality and B is an inclusion, then one of the following happens:
(a) B is never satisfied, and the entire expression is never satisfied.
(b) B is satisfied by setting all the variables in B to either X or Y (as given by our analysis of expressions #1-#4).
In case b, (A∧B) will be satisfied by setting the remaining variables of A to either X and Y (since A is an equality). Hence the entire expression is satisfied by setting all the variables to either X and Y, and it is not a Rayo name of any number.
3. If B is an inclusion and A is an equality, we follow the previous argument while swapping A and B.
This leaves us with a total of 6 expressions:
1. (xi∈xj∧xi∈xk)
2. (xi∈xj∧xj∈xi)
3. (xi∈xj∧xj∈xk)
4. (xi∈xj∧xk∈xi)
5. (xi∈xj∧xk∈xj)
6. (xi∈xj∧xk∈xl)
Let Z={Y} and we have:
1. #1 and #3 are satisfied by xi=X, xj=Y, xk=Z.
2. #2 is never satisfied.
3. #4 and #5 are satisfied by xi=Y, xj=Z, xk=X.
4. #6 is satisfied by xi=xk=X, xj=xl=Y.
This concludes the verification that all 82 Rayo forms with 9 characters or less are not a Rayo name for any number.
Hence Rayo(n)=0 for all n<10.
QED