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”

Ordinals Multiplication

After we’ve defined properly the action of ‘addition’ on ordinals, it’s time to move on to the next action – the multiplication. The process is going to be very similar: we are are going to define the product of two ordinal as an order type of some well-ordered set. How exactly? let’s find out: TheContinue reading “Ordinals Multiplication”

After we’ve met the order type of a well ordered set, we are finally ready to discuss about ordinal arithmetic. In this post I’ll present the term of addition. As I said in the previous post – It might not be as easy as you think. “Almost” addition Suppose that are ordinals. I am nowContinue reading “Ordinal addition”

Order type

After we’ve properly defined the natural numbers, it’s time do define arithmetics on ordinals – such as addition, multiplication and exponentiation. However, we need to do a little bit of work first. However, what I am going to discuss about today is improtant regardless my goal to define arithmetics. I am going to show aContinue reading “Order type”

The Natural numbers

What are the natural numbers? and yes, I do mean . What kind of question is it? Everyone knows what the natural numbers are, even my 6 years old brother, which is currently in the first grade, knows what the natural numbers are. However, how can you define them? The natural numbers seem so fundementalContinue reading “The Natural numbers”

Lower sets

In the last post, I’ve introduced the term of an ordinal recall that an ordianl is a well ordered (By the order ”) and -trasnsitive set . We’ve also seen some interesting results on ordinals, such as : If is an ordinal, then so as . is an ordinal and is transitive is an ordinal.Continue reading “Lower sets”

Ordinals

In the last post we prepared the ground by defining Well-ordered sets,-transitive sets, and talked about the order . What is it? It is now finally time to define what ordinals are: A set is called an ordinal if it is: -transitive well ordered by the order ” () Ok, in first sight, this definitionContinue reading “Ordinals”