Strong Indescribable Ordinals

Well, my first Indescribable ordinals isn't defined above Pi^1_w so I have to start over again. Let A will be a ordinal and let A- will be a ordinal such that A- is the biggest ordinal that can be correct with {A-|A>A-}.

Arrays on Cardinals

First, W(1,0) will define that works like W but replace w^A with W_A. then W(A,B) will define that works like W but replace w^C with W(A-,W(A,B-)+C). And W_A is shorthand for W(0,A).

  • psi_W(1,0)(0)[0] = 1
  • psi_W(A,0)(0)[0] = 0
  • psi_W(A,B)(0)[0] = W(A,B-)
  • psi_W(A,B)(C)[0] = psi_W(A,B)(C-)+1
  • psi_W(A,B)(C)[D] = W(A-,psi_W(A,B)(C)[D-1])
  • psi_A(B)[C] (if W(D,E)+w > A > W(D,E)) = W_A-^psi_A(B)[C-1] and W_A works like W but replace psi with psi_A.

We can continue like this: W(R,a+1,b+1) works like W(1,0) in W(R,a,W(R,a+1,b)+_), W(R,a+1,0,O,b+1) works like W(1,0) in W(R,a,_,O,W(R,a+1,O,b)+1), W(R,a,O,b+1)[n] = W(R,a[n],O,W(R,a,O,b)+1), and W(R,a)[n] = W(R,a[n]), where R is the array of ordinals and O is array of zeroes.

Mahlo Cardinals

We will do similar psi function to cardinals instead of ordinals, by changing psi to chi and W to M. And definition of chi(n) changes:

  • chi(0) = W.
  • chi(n) is defined as an uncountable cardinal after chi(m), where m is biggest ordinal exists such that n > m.
  • chi(F(M)) works like W(1,0) in chi(F(_)).

Now define W_(a+1) is a next cardinal of W_a. If M is a mahlo cardinal then W_(M+1) is a next cardinal of M and M_(a+1) is next mahlo cardinal of M_a (it would uses W_(M_a+A) instead of W_A).

Oh, I forgot, chi_A(B) = W_(M_A+B).

Weakly Compact Cardinals

Before we continue, let Psi_W_(1+N) have the same function of Psi_W_N and Psi_M_(1+N) have the same function of chi_N.

If chi(M) works like W_A in chi(_), chi(F[M]) works like W_A in chi(F[_]), and M_(a+1) works like M in W_(M_a+_), then Psi_Xi(n+1)(F[Xi(n+1)]) works like Xi(n) in Psi_Xi(n+1)(F[_]) and Xi(n+1)_(m+1) works like Xi(n+1) in Xi(n)_(Xi(n+1)_m+_).

If M works like W-to-M map in W_(_) and Xi(n+1) works like W-to-M map in Xi(n)_(_), then Xi(F[K]) works like W-to-M map in Xi(F[_]).

Let W = Xi(0), M = Xi(1), Xi = Xi(2).