transfinitely

From transfinite +‎ -ly.

transfinitely (not comparable)

1. (manner) In a manner that requires a transfinite number of steps.
   • 1999, Thomas Strahm, “First Steps into Metapredictivity in Explicit Mathematics”, in S. Barry Cooper, John K. Truss, editors, Sets and Proofs, Cambridge University Press, page 399:

     We finish this section by briefly addressing how the reduction procedure for ${\displaystyle {\mathsf {EMU}}}$ described so far can be formalized in the transfinitely iterated fixed point theory ${\displaystyle {\widehat {\mathsf {ID}}}_{<\varepsilon _{0}}}$ of in order to yield conservativity of ${\displaystyle {\mathsf {EMU}}}$ over ${\displaystyle {\widehat {\mathsf {ID}}}_{<\varepsilon _{0}}}$ with respect to arithmetic statements.

   • 2003, Thomas Forster, Logic, Induction and Sets, Cambridge University Press, page 147:

     After we have performed the deletion operation infinitely many times, we can look at what is left and perform the deletion operation on that, and thereby continue the process transfinitely.

2. (degree) Of transfinite degree.
   • 2004, Lisa Robin Marks, The Axiom of Choice, Berkeley: University of California, page 35:

     The concern was voiced that Zermelo assumed transfinitely many successive choices, where one choice depends upon another, and that this was unacceptable.

   • 2014, Ali Sanayei, Otto E. Rössler, Chaotic Harmony: A Dialog about Physics, Complexity and Life, Springer, page 126:

     It was very radical because he also claimed that symmetry is a feature in mathematics and reality that is not just an infinitely accurate property, but a transfinitely accurate one. Like the modern idea of fractals—where we have this Mandelbrot type thinking of everything repeating itself all over again not just infinitely many times, but in such a way that the surface around it is transfinitely large.

• finitely
• infinitely
• transfinite