transfinitely's Usage Examples:

obtained by starting with the open subsets of X {\displaystyle X} , and transfinitely iterating the operations of complementation and wellordered union.

which cannot be reached “from below” using the Veblen function of transfinitely many variables) is sometimes known as the “large” Veblen ordinal.

number of times, and to the extent that the model has dependent choice, transfinitely.

ordinals show that there are very few ordinals below the ordinal of transfinitely iterated Π 1 1 {\displaystyle \Pi _{1}^{1}} -comprehension where the.

in the context of the von Neumann universe V, which is built up by transfinitely iterating the powerset operation, which collects together all subsets.

psi functions one can extend this definition easily to the ordinal of transfinitely iterated Π 1 1 {\displaystyle \Pi _{1}^{1}} -comprehension (see Analytical.

The Mahlo operation can be iterated transfinitely as follows: M0(X) = X Mα+1(X) = M(Mα(X)) If α is a limit ordinal then.

elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set operation.

replacement in an essential way, in order to iterate the powerset axiom transfinitely often.

operator taking X to the set of n satisfying the formula) can be iterated transfinitely along any countable well ordering starting with any set.

\varepsilon _{\alpha }=\alpha } , but since the Greek alphabet does not have transfinitely many letters it is better to use a more robust notation: define ordinals.

It is possible to iterate this construction transfinitely: given an ordinal α, define a new theory by adding an equivalence relation.

notations ϕ ( … ) {\displaystyle \phi (\ldots )} predicatively using transfinitely-but-predicatively-many variables), the limit ψ ( ε Ω + 1 ) {\displaystyle.

of S 0 {\displaystyle S_{0}} validity, and alternatives to it, use transfinitely long revision sequences.