User blog comment:Ubersketch/What are some ways to map a well-order to the natural numbers without using a Godel coding/@comment-39541634-20190909210146

Here is an order-preserving map of ω+1 to the natural numbers:

'''2,3,4,5, ... (all the following numbers in seqeunce) ... , 1'''

You can see that we have an infinite list of numbers (2,3,4,5,...) followed by a single number (1), which is exactly what "ω+1" means.

Formally, there are two ways define the above order:

(a) Explicitly state which ordinal (up to ω+1 and not including ω+1 itself) maps to which number:

0 → 2

1 → 3

2 → 4

In general, for any finite ordinal x: x → x+2

ω → 1.

Now we write all the ordinals below ω+1 in their usual order:

0,1,2,3,4, ..., ω

and replace each ordinal with the corresponding natural number

2,3,4,5,6, ..., 1

And you can see that we - indeed - got the exact same ordering given in the beginning of this comment.

(b) Directly define an ordering on the natural numbers that will give you the desired structure. In other words, we need to define a new version of "x is greater than y" which will give us the desired order.

To differentiate between our new version and the usual meaning of "x>y" we will use a new symbol: "x⊳y". This will mean "the number x comes after the number y on our list".

And we will define it like this:

(1) If x>1 and y>1 then x⊳y if and only if x>y

(or in english: for numbers greater than 1, our new ordering is the same as the usual order of natural numbers)

(2) For all x, 1⊳x

(or in english: In our new ordering, the number 1 comes after all the other numbers)

By looking at the english translations, you should have no problem verifying that we - indeed - end up with the ordering mentioned in the beginning of this comment:

2,3,4, ..., 1

Was that clear? If you have any more questions, don't hesitate to ask.

I'll be happy to give you more examples for higher ordinals, but I want to make sure that we're on the same page first.