तात्पुरते,

transfinitely's Usage Examples:

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

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

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

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

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

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

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

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

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.

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.

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