successor ordinal in a sentence

1. For infinite cardinals, the successor cardinal differs from the successor ordinal.
2. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.
3. If ? is a successor ordinal, then \ aleph _ { \ beta } is a successor cardinal.
4. If \ lambda were a successor ordinal, then \ aleph _ \ lambda would be a successor cardinal and hence not weakly inaccessible.
5. In set theory, a "'limit ordinal "'is an ordinal number that is neither zero nor a successor ordinal.
6. It's difficult to find successor ordinal in a sentence.
7. A pointclass is said to be " self-dual " if it is even successor ordinal, or a limit ordinal of countable cofinality.
8. 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.
9. 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 ).
10. 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 ).
11. 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 ?.
12. 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.

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