On Thursday, February 27, 2014 9:27:47 AM UTC-5, muec...@rz.fh-augsburg.de wrote:
> > > You could, however, get into trouble in your example if you attach multiple meanings to the same symbol. > > > > That would not be mathematics. But the other way round: you can attach different symbols like 1 and 1^1 and 1^1^1 to the same meaning. But that is not the most important failure. You do not get the natural numbers unless you specify that the differences must be units. >
If you don't want to use the usual axioms, I guess there is nothing to stop you from creating your own. For over a century now, I think Peano's axioms have been the standard, so you would have a quite a selling job ahead of you.