Math Is Fun Forum

I've just discovered the mathisfunforum.  Seems awesome.

I'm trained as a chemical engineer, so I've used plenty of math, especially calculus, but always as a tool.  Nevertheless I'm really interested in math for its own sake.  Right now I'm trying to learn about information theory, which has prompted a review of discrete math.

But I often find myself really trying to understand beyond the practical explanations given in text books.  Generally, things are explained in a degree of shall we say reasonable rigor, because too much precision is hard for people who just need the practical to understand.  This leaves me invariably with questions.  I wonder about the fundamental things, like what, exactly, do we mean by a "number".  What is a number!?  Similarly, one finite text brought up the idea of alphabets which are supposed to be sets of symbols.  What is a symbol!?  And how does that relate to the fact that math consists of lines of symbols!?

Does anybody else wonder about these things?

Actually, I know people do.  Having been interested in this stuff, I did come across things like Godel's incompleteness theorem (I understand the juiced version, but dare not try to understand the actual papers).  And I think some of this is dealt with in "elementary number theory".  Any leads on how to get into that.  I find myself wading through gobbledeygook.  I think somebody didn't know what elementary means!

Know of any gentle, readerly intros to these ideas?  I know that this stuff is hard.  That's ok.  It would just be nice to actually have terms defined before their used, etc.  Know what I'm saying?

I'd love to talk to other people who think about these things.

I just got a decent discrete math text and am learning about sets.  They've just introduced "alphabets" as a type of set that contains "symbols".  No definition is given for a symbol, which is interesting.  It is interesting because the following could be seen as a set of sets, or as an alphabet:

Of course, it depends on whether A, B, and C are sets or symbols.  This is strange because, if I am not mistaken, math is written in symbols.

Next the text defines a "word" or "string" as an ordering of symbols.

So words are constructed by placing symbols in ordered sequences.

My question is subtle, I hope you won't mistake it for dumb:

Is a symbol a word of length 1, or is it not a word at all?

Why does it matter?  Well, I'm wondering if a set of words of length 1 is an alphabet.  Perhaps symbols are not words.  There is a difference, after all, between an ordered sequence containing one symbol, and a symbol, is there not?

Think of it this way.  Imagine that to create a word, the notation was a little more precise, and we had to use, say braces around the symbols.  Then we could make words like

and the null word,

The null word is an object because it is an ordered list of symbols containing zero symbols, just like the null set is an unordered collection of zero elements.  On this basis, the word