We prove more results for Ordinal and Cardinal arithmetic. Including showing that we can apply division with remainders to ordinals. We use that to prove Cantor’s Normal Form Theorem. Finally, we give a proof of Konig’s Theorem for Cardinals.
In this entry we cover more basic results about ordinal arithmetic. We also prove the so-called “Fundamental Theorem of Cardinal Arithmetic”, and then we finish with a short discussion about cofinality of cardinals.
Here we dig into ordinal and cardinal arithmetic a little bit. We also establish the famous result (originally due to Cantor) that |S| < |P(S)|, i.e. that the power set of any set is larger than the set.
In this entry, we establish more results about ordinals, including proving that every well-ordered set has the order type of an ordinal. We introduce transfinite induction, Cardinality, and use the Axiom of Choice to prove that every set can be well-ordered, among other results.
