# successor ordinal in a sentence

- For infinite cardinals, the successor cardinal differs from the
*successor ordinal*. - Every ordinal number is either zero, or a
*successor ordinal*, or a limit ordinal. - If ? is a
*successor ordinal*, then \ aleph _ { \ beta } is a successor cardinal. - If \ lambda were a
*successor ordinal*, then \ aleph _ \ lambda would be a successor cardinal and hence not weakly inaccessible. - In set theory, a "'limit ordinal "'is an ordinal number that is neither zero nor a
*successor ordinal*. - It's difficult to find
*successor ordinal* in a sentence. - A pointclass is said to be " self-dual " if it is even
*successor ordinal*, or a limit ordinal of countable cofinality. - Note that if ? is a
*successor ordinal*, then ? is compact, in which case its one-point compactification ? + 1 is the disjoint union of ? and a point. - H . Friedman has shown that for every countable
*successor ordinal* \ beta, every stationary subset of \ omega _ 1 contains a closed subset of order type \ beta ( Friedman ). - The epsilon number \ varepsilon _ { \ alpha + 1 } indexed by any
*successor ordinal* ? + 1 is constructed similarly, by base ? exponentiation starting from \ varepsilon _ \ alpha + 1 ( or by base \ varepsilon _ \ alpha exponentiation starting from 0 ). - Using this definition, ? + 3 can be seen to be a
*successor ordinal* ( it is the successor of ? + 2 ), whereas 3 + ? is a limit ordinal, namely, the limit of 3 + 0 = 3, 3 + 1 = 4, 3 + 2 = 5, etc ., which is just ?. - If we use the Von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal ( and this is a fitting observation, as " cardinal " derives from the Latin " cardo " meaning " hinge " or " turning point " ) : the proof of this fact is done by simply showing that every infinite
*successor ordinal* is equinumerous to a limit ordinal via the Hotel Infinity argument.