Cardinal arithmetic: Difference between revisions
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== Successor cardinal == |
== Successor cardinal == |
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If the axiom of choice holds, then every cardinal κ has a successor, denoted κ<sup>+</sup>, where κ<sup>+</sup> > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ<sup>+</sup> such that \(κ^+≰κ\). For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal. |
If the axiom of choice holds, then every cardinal κ has a successor, denoted κ<sup>+</sup>, where κ<sup>+</sup> > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using Hartogs' theorem, it can be shown that for any cardinal number κ, there is a minimal cardinal κ<sup>+</sup> such that \(κ^+≰κ\)). For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the successor ordinal. |
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== Cardinal addition == |
== Cardinal addition == |
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