On Saturday, 22 March 2014 19:57:44 UTC+1, Dan Christensen wrote: > On Friday, March 21, 2014 2:41:17 AM UTC-4, muec...@rz.fh-augsburg.de wrote: > > > On Thursday, 20 March 2014 22:35:55 UTC+1, Dan Christensen wrote: > > > > > > > > > > > > > The real numbers can also be constructed from Peano's axioms.
> > > That was not the question. > > > > > It should be > > > possible to prove than no irrational number is equal to a natural number. > > > > That was not the question. > > You wanted to know if it was possible to prove, starting from Peano's axioms, that pi (or any other irrational number?) is not equal to a natural number.
No, here you can read the original question: > > > The question was: If the five truncated Peano axioms, not more and not less, define the natural numbers, as you said, what part of the axioms excluded that pi is a natural number? >
> You are grasping at straws, WM. Again, the Peano axioms are just the 5 essential properties of the natural numbers from which it seems all others can be derived.
Of course everything can be derived by adding suitable definitions (like "0 = 1" or like "the difference between two successors is always the same and is called a unit"). But you said that your five truncated axioms, ***not more and not less***, are sufficient to define the natural numbers. Now defend that assertion or drop it.