Properties of exponantiation of ordinals

In the last post, we worked pretty hard to define ordinals exponentiation, however, This hard work revealed some great properties: If is a successor then: If is a limit ordinal then; Those properties are great – the first two are pretty intuitive, and the third is also a pretty comfortable definition of . We canContinue reading “Properties of exponantiation of ordinals”

Ordinals Exponentiation

After defining ordinals addition and multiplication, it’s now time for the next action – ordinals exponentiation. Before I’ll define it properly, I do neet to make some preparations. In contrast to the previous actions, here the preperations may seem a little unrelated and not intuitive al all, but I’ll try to make it as intuitiveContinue reading “Ordinals Exponentiation”

Transfinite idnduction in action

After presenting the method of proof called transfinite induction, I want to show some pretty awesome stuff we can prove with it. So let’s start then: Comparing order types My goal here is to prove that if are both well-ordered sets and Is an order-preserving map, then: This is a quite strong statement, it givesContinue reading “Transfinite idnduction in action”

Transfinite induction – The ‘infinite’ induction

If you are a math major, you probably proved theorems using induction countless of times. And that’s not suprising – induction is one of the most powerful tools a mathematician has in his toolkit. However, induction has one major problem – It is finite. For example, if you managed to prove that union of twoContinue reading “Transfinite induction – The ‘infinite’ induction”