So we have done something interesting.. we have defined "numbers".. what a great thing for us to do !! .. but it seams that we have more to cover, in order that, we better know this beast we have uncovered...... We left off with numbers being these sort of things (||,|||,|||||||||,|||, |||||||||||||||||||||||||||||||||| ) ect ... which at least gives us some place to start. There is a problem with this though.. we can't tell very easily say if |||||||||||||||||||||||||, ||||||||||||||||||||||||||||| .. are different numbers. This makes it a very difficult thing to talk about numbers much bigger than ||||| ... Perhaps a look at some numbers might lead us to a solution of this problem.. First let's become very evolved and immediatly develop a super advanced concepts which took mankind millions of years to arrive at ...
(the S(n) notion is at most 100 years old and due to a fellow named Peano, read on to find out just what that idea is.. don't worry it won't hurt at all............)
VARIABLES and EQUALITY THEREOF...
We will call some symbols like (x,y,v,n,m,k,l...etc) variables and say that they "denote" numbers (whatever that is, they just have some relationship to numbers called denoting)... and we will use a symbol "=" and we will write
" x = y " to mean that " x and y denote the same number" and in that spirit we will write "x = ||" and say "x denotes the number || " and along the same lines if we pretend that the variable "n" is actually a number "we will often do this" then we will write "x = n" whenver "x denotes the number n" .. it is to be understood that variables denote numbers and only numbers and they don't "vary" we just like pedantic use of old words ... then we have the following things immediatly ...
1. for any n , ( n = n ) "Reflexive law"
2. for any n,m , ( n = m if and only if m = n ) "symmetric law"
3. for any n,m,k ( If n = m and m = k then n = k ) " transative law"
And these are "self evident truths" are they not ?...............................................................
Now with these powerful tools in hand we can do amazingly simple things!!...................................................................................................... To make everything more uniform, we will pretend that a variable is acutally the number it denotes. We just won't know what that number is, in this way we can talk about an "arbitrary" or nondeterminate number... which could be taken as talking about any given number whatsoever... and we will just say that ( x = n ) means that x,n are the same number and call them equal I.E. || = || ... etc.. Now we are highly evolved abstract mathematicians.. ohh we use magical "unknown" numbers and whatnot.. whoo joy rapture and glory haven't we come a long way toward being a godlike being !

Let us inspect a few numbers, which, are "small" enough to be immediatly recognizable as unequal .. {Note if we want to say that x,y are unequal or denote different numbers we may often write x =/= y } and suppose that looks like a crossed out equal sign........... okay so it is clear that | =/= || is a relationship between |,|| but of course we have the same relationship between |,||| ... and ||,|||| ... It appears though that there is a special relationship between the numbers |,|| .. ||,||| .. |||,|||| .. etc and this is a little more special than being unequal .. we would almost like to say that || is the smallest of our numbers which is unequal to the number |.. wouldn't we ??.. and likewise for
how ||| relates to ||. |||| to ||| etc.. We shan't say exactly that, because, we haven't thought about just what smallest/largest/bigger/smaller really mean in this case.. Instead we will just write that || = S(|) and call || the successor to one .. I.E. || follows directly after | ... and in the same manner ||| = S(||) etc.. We can't possibly go through all the numbers one by one and find the "successor" to that number... based on our definition though we know that we can type any number out by pressing "Shift + \" repeatedly for some amount of time.. it is not presumptuous to suppose tha we could then go ahead and press "Shift + \" again and call this the successor, which, we shall do !..
hence we have the following set of "self evident truths".. " you are free to think of S(k) as "k+1" even though the symbols "+,1" have no meaning yet.."
1. " | " is a number
2. If n is a number then so too is S(n)
3. if n =/= m then S(n) =/= S(m)

now we have a magic little way to get around with numbers ... we can even prove that || is a number and ||| and ||| and so on, just based on these three little "self evident truths".. fancy, amazing, fantastic and whooo whooo whoooo... now the question remains ... are these all the numbers ?, that is the ones which we can arrive at based on taking successior after successor and starting with | ..isn't that all of them ?.. well, "Outside note"... If we think about what counting is, we allways start at | and then goto the successor, so, every number you can "count" is sure one of the above type.. we assume that we are smart enough to count any number of times we press "Shift + /" so we will take this for true..." and we arrive at the following truth about numbers

4. If we have any collection of numbers containing | and this collection contains S(n) whenever it contains n, then the collection consists of all numbers.. So now we know something about numbers.. about all numbers !!! ... Super cool ! ........