{"id":13,"date":"2024-08-15T10:26:29","date_gmt":"2024-08-15T09:26:29","guid":{"rendered":"https:\/\/blog.samuelgill.net\/?p=13"},"modified":"2025-05-19T12:10:56","modified_gmt":"2025-05-19T11:10:56","slug":"maths","status":"publish","type":"post","link":"https:\/\/blog.samuelgill.net\/index.php\/2024\/08\/15\/maths\/","title":{"rendered":"Lets Build the Reals."},"content":{"rendered":"\n\n\n

The real numbers are probably what you think about when you think about numbers, but what are they? How do we get all those numbers from the ground up?<\/p>\n\n\n\n

Von Neumann Ordinals<\/h2>\n\n\n\n

So we first want to create the naturals (and 0), and our tool to make these? Sets. However, we do have a problem, we don’t have anything to put in our set yet, so lets start by defining 0.<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[0 := \{\}\]\"<\/p><\/p>\n\n\n\n

Now we have our first number defined we can start to inductively define the next numbers, given n, the successor of n, S(n) (Also known as n+1) is defined as<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[S(n) := \{n, \{n\}\}\]\"<\/p><\/p>\n\n\n\n

And now we have defined all the natural numbers! With <\/p>\n\n\n\n

  <\/span>   <\/span>\"\[n := \{0,1,2,...,n-1\}\]\"<\/p><\/p>\n\n\n\n

So why do we define it this way?<\/h2>\n\n\n\n

There are other ways of defining the naturals, one could define<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[S(n) := \{n\}\]\"<\/p><\/p>\n\n\n\n

This type of ordinals are called the Zermelo Ordinals, why are they not used? Well the reason is not that complex, its mostly down to the utility of the Von Neumann Ordinals, With Zermelo Ordinals, the cardinality (the number of elements in the set) is always one, however for Von Neumann Ordinals, the cardinality of a number is the number itself (in the way you’d normally understand it, not from the definition, that would be a bit circular), This means that we can say a set has a cardinality of n if there is a bijection from a set to n.<\/p>\n\n\n\n

Now if we repeat the generative process above for all finite sets, this will give you the naturals and 0 \"\mathbb{N}_0\" (or \"\mathbb{N}\" We are going to be letting \"0\notin\mathbb{N}\"). Note you can extend these ordinals to transfinite size, this gives us \"\omega\" with cardinality \"\aleph_0\", which you may have heard of.<\/p>\n\n\n\n

Addition and multiplication<\/h2>\n\n\n\n

Now that we have the naturals (and 0) defined, lets define how to add and multiply numbers, as you’ve seen before we’ve already defined \"n+1\" as \"S(n)\", so now we can inductively define \"n+m\" as<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[n+m := n + \underbrace{1 + 1 +... + 1}_{\text{$m$ times}}\]\"<\/p><\/p>\n\n\n\n

And now we can work on multiplication, we can also define that inductively with<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[nm := n\times m := \underbrace{n + n + ... + n}_{\text{$m$ times}}\]\"<\/p><\/p>\n\n\n\n

With \"n+0 := n\" and \"n\times 0 := 0\"<\/p>\n\n\n\n

Now you may wonder why we don’t define subtraction and division yet, they are actually not that bad to define, we just say they are the inverses of addition and subtraction, however we cant define them yet as our current number systems are not closed under them, as \"4-6\notin \mathbb{N}\cup\{0\}\" or \"\frac{4}{3}\notin \mathbb{N}\cup\{0\}\"<\/p>\n\n\n\n

Time to get rational<\/h2>\n\n\n\n

Now that we have the naturals defined, lets start moving onto the rationals, you may wonder why we are not going onto the integers as they seem like the next step, they can actually be defined now, however the way they are defined is very similar to how we will define the reals from the positive reals later, the working here should not need that much editing to fit it here, but I personally think its nicer to stick with positive numbers (and 0) for now.<\/p>\n\n\n\n

To define the rationals we need to first need to understand something called an equivalence class, so lets start there.<\/p>\n\n\n\n

Equivalence Relations<\/h2>\n\n\n\n

We can define a relation \"\sim\" as the subset of a set \"X \times Y\", think about it as question that takes in two inputs and gives you true or false about it (whether it is in \"X \times Y\" or not), we write this as \"x \sim y\", pronounced as \"x\" relates to \"y\", note that when \"X = Y\" we call the relation homogeneous, we shall only use homogeneous relations from now on. One example of a relation would be \"<\" on \"\mathbb{N}\" so we could say \"3<4\" but not \"6<2\" or \"1<1\"<\/p>\n\n\n\n

An equivalence relation is a specific type of relation that has various properties, these are<\/p>\n\n\n\n