You all had best be familiar with the following ten symbols {0,1,2,3,4,5,6,7,8,9}.. You should know how to use them for counting, and, elementary arithmetic. You ought to be able to denote fractions with them, and, if you are a bit sophisticated you might even be able to write out "decimals" for those fractions and maybe you pretend that you actually know what "3.14159265358979..." means, or even that it means anything at all !.. Ohh rapture Ohh joy .. Ohh.. what the hell do those things mean ??...I am willing to help you with that.. and to raise a few questions, easy to understand.. maybe not so easy to answer, however, they will make you think twice about the meaning of those symbols.. or other strange animals like 12345 or 234.346 whatever those could possibly mean !... So read on and become enlightened. I do promise that it won't hurt a bit ... not a bit at all !!!.. Now lets get started shall we ?..

Lets do some pretending, we will pretend that we have a bunch of little bags, and a bunch of marbles to put in them or take out of them and that such a thing is easy to do..
That's all you will have to think about.. marbles and bags and putting marbles in and taking marbles out !.. To keep things simple lets agree that all the marbles are the same size/color/shape and that if we were to put a marble in a bag and then follow by putting in another, then, we shan't be able to tell which was the first we put in. Okay with that said lets talk a bit ................ from here on forward we will forget the symbols {0,1,2,3,4,5,6,7,8,9} the idea {number} and the fact that we can count..

Suppose I take a bag and put a marble in it, I can look inside and think about what a see and say "I see a marble in a bag"...
If I take another marble and put it in the same bag, then, I look inside and see something different.. all I can say is "I see more marbles in the bag"
We see something different... but what has changed ?..
Now suppose I take another bag, put a marble in it, I can look inside and see what I see and then I say "I see a marble in the bag"
I then take my other bag and pour the contents into the bag I have just finished looking into..I say "There are now more marbles in this bag" ..
here is the problem with this..

i. when we look into this bag we see more marbles than were in the other bag, but, from our statement you couldn't ever tell that were the case...
It is clear that we percieve a difference between the contents of our bags. What is this difference ?.. I't can't be in the contents, we aren't able to tell one marble from another.. and hence it has someting to do with "more" and "less" and we shall call it "number" ... we would then say that we the difference in number between the contents of the one bag and the other... All this talk of bags is getting boring.. lets get more evolved and go on to using some sort of symbols. We will then agree that a bit of your screen is as good as any bag and that the mark "|" is as good as any marble is... so then we have the following things which each have "number"
{ || } { ||| } .. { |||| } ... "recall that you can't yet count"... is there a difference in number between { ||||||| } and { |||||||| } ?.. As long as we are aloud to use our imaginations lets forget about brackets and anything speical about the mark "|" and say that |,||,|||,... ect simply are numbers ... this way we can say that of our bags there was a bag which had || marbles and one which had ||| marbles ... finally we lets see if we can make a rule for telling just what is and what isn't a number ...
a string of symbols each being "|" ought to be a number, however, we must be able to ensure that it is infact a number and hence we make the rule
a string of |'s is a number if you can generate the string by pressing "Shift + \" repeatedly and at some point you stop and have generated the whole string.. Now you must stop and convince yourself that in this way we can surely check and/or generate any number and that this will satisfy any need we have of what is and what isn't a number..
so long as we recall that number is a property relating to the "amount" of "things" one can place in some sort of bag imaginary or otherwise, since there is no bag big enough to hold all the planets in our solar system but we would like to say there is some determined number of them... We shall call counting any process by which one determines the number of a collection of things... OKAY now here goes the simple question that you most likely won't be able to answer ...

Are you posative and without a doubt certain that your usual manner of denoting numbers with the symbols {0,1,2,3,4,5,6,7,8,9} will allow you to specify any number in the sense just defined ?..

Can you prove that this is true ?...

I can and it's a bit of what we call the theory of numbers... I have just introduced you to that theory ... now you should have all kinds of questions like
what does + mean.. how do I know that a + b = b + a .. what is a "prime" number.. what is a fraction .. what is an even or odd number ?.. how many numbers are there and is there a number for that ?.. what is multiplication and how do I know that a(b+c) = ab + ac ?.. what is a factor or divisor ?.. how many "prime" numbers are there ?.. is zero a number ... ect ... ect ... ect .. ? these questions have terribly simple answers .. the real interesting stuff comes later.. welcome to my world .. welcome to my questions ..