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Here is a formal system that captures the way that we increment (add 1 to) numerals given in base 2. I chose this system because you already know how to do this, and the system is small enough to describe thoroughly, but it's easy to see how it is a small fragment of something much more significant. I will describe the system in a conventional style used by mathematical logicians.
Because of all the funny symbols, I had to generate the discussion
of each formal system with a funky program called latex,
and the conversion to HTML is not completely reliable. If
there appears to be a problem with the HTML, you can choose
your favorite page format to view each system, as long as your
favorite is DVI, PostScript, or PDF.
The next formal system looks slightly more complicated than the binary numeral system. In fact, it is so powerful that in some sense it can do everything that any other formal system can do. The downside is that it doesn't look at all familiar.
tar -xzf
epsfigs.tgz.
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