On Wednesday, March 5, 2014 11:03:06 PM UTC-5, John Gabriel wrote: > On Wednesday, 5 March 2014 20:32:58 UTC+2, Dan Christensen wrote: > > > > > A number is an element of the set N as defined by Peano's axioms as above. > > > > As I thought, you have no idea what is a number. Peano's rot does not define natural numbers, it defines the properties of natural numbers. Did we not agree to this earlier? Have you gone back to your original wrong ideas? :-) >
Play with words all you want, John Gabriel. If you want to do mathematical proofs about natural numbers, you need a list of their essential properties from which you can derive other properties. You seem never to have written ANY mathematical proofs, or you wouldn't be saying all these stupid things.
> > > You first said: > > > > 1. Peano's [axioms] defines natural numbers. >
They certainly do for the purpose of writing proofs.
> > > And then, > > > > 2. Peano's [axioms] defines the properties of natural numbers. > >
It amounts to the same thing. What is your problem?
> > > But this is just an informal comment. Construct your own more convoluted sentence if you don't like it. It doesn't really matter. > > > > Nonsense. I am not about to do any such thing. A great deal of ingenuity and thought went into the construction of number.
Peano's axioms are a marvel of intellectual progress, far surpassing your hero's attempt to define numbers, significant though it was in its day and for many centuries. In the present day, however, his work is of no more use than slings and arrows in modern warfare.
> > To begin formally deriving number theory, however, we just list Peano's axioms at the beginning of each proof an proceed, using the rules and axioms of logic and set theory. > > > > I have studied mainstream mathematics. I know very well what number theory is. :-)
Again, you seem to have no idea of how proofs are written or how formal definitions are used in practice.
> > I have been claiming for a few decades that I am the first in 2000 years to understand what the Ancient Greeks were trying to do.
Gosh! You must be smart! ;^b
> > Every professor of mathematics I have communicated with has NO IDEA what is a number or a magnitude. >
Unlike you, they do know something about writing proofs.
> > > In one sentence: > > > > "A number is the *measure* of a magnitude." > [snip]