> On 2/28/2014 2:01 AM, email@example.com wrote: >> On Thursday, 27 February 2014 22:52:54 UTC+1, fom wrote: >> >>>> The question was whether Peano defines the natural numbers. He fails. >>> >>> >>> >>> Why do you say that? >>> >>> >> I say that because it is widely assumed that Peano defined the natural >> numbers. People assume the natural numbers and find that Peano is >> rigth. But the other way dos not work. Assuming the Peano axioms does >> not yield N. I emphasize this because it has been hundred years taught >> falsely. >>> >>> >>> It would be correct to say that he assumes the >>> >>> natural numbers and places identity criteria >>> >>> onto his denotations to restrict their >>> >>> interpretations. In that sense he does not >>> >>> define them. But I do not think this is what >>> >>> you mean. >> >> It is precisely what I mean. And I am in particular happy that there >> are some like you who have not yet been completely perverted by the >> study of mathematics but can understand that some topics have been >> taught wrong. >> >> > Ok. Yes, Peano's actual axioms are different from the > recharacterizations according to the needs of computer systems or the > first-order axioms. > > Thanks.
There is exactly one natural numbers structure as defined by Peano's axioms, up to isomorphism, with second order logic. And more than "up to isomorphism" doesn't make any sense. Mückenheim is obviously unable to grasp this, especially the clause "up to isomorphism". I am unable to grasp how one can take a person like Mückenheim serious in these matters. Mückenheim's idiotic babble has nothing to do with the non-uniqueness in first order logic, because a non-standard model of first order Penao arithmetic is a beast totally diofferent from Mückenheim's idiotic garbage.