Il giorno sabato 1 marzo 2014 07:49:26 UTC+1, John Gabriel ha scritto:
> 1. Zero is a natural number.
> So, the first so-called "axiom" contains two undefined and unqualified terms. You shall see how rigorous and sound John Gabriel's axioms are at the end of this comment.
Your mind on this subject is really confused. What Peano is doing here is "defining" the natural numbers. So he is not assuming the knowledge of "natural numbers". Zero is just the name of an element in the set.
In practice he is saying: This set that I'm defining will be called natural numbers and it contains at least one element, which I call zero.
I find amazing (disturbing) that you are not understanding the meaning of Peano axioms and still you are criticizing them...
> And now, for the sound construction of numbers from scratch and the new axioms:
> Construction of rational numbers: > > 1. A magnitude is the idea of size of extent. We can either tell that two magnitudes are equal or not. If we can tell they are not equal, then we know which is smaller or bigger, but we can't tell how much bigger or smaller. This is called qualitative measurement (without numbers).
It seems to me that there are a lot of undefined terms here. Idea ? Size ? Extent ? bigger or smaller ? Do you really think this is rigorous? I'm assuming you are joking and this is a sort of parody of mathematics.