>> I'm not sure what you are saying. The fact is, we can prove >> that for every real r_n on the list, d is not equal to r_n. > >Of course. Every real r_n belongs to a finite initial segment of the >list. >That does not yield any result about the whole list
On the contrary, the definition of "d is on the list" is that "there exists a natural number n such that r_n = d". We proved "forall n, r_n is not equal to d". So that means "there does not exist a natural number n such that r_n = d", so that means "d is not on the list".
We have thus proved something about the whole list.
>> That means that d is not on the list. There is no extrapolation >> involved. > >Look here: We can prove for any finite segment >{2, 4, 6, ..., 2n} >of the ordered set of all positive even numbers that its cardinal >number is surpassed by some elements of the set. > >Nevertheless this appears not be a proof that the cardinal number of >the whole set is less than some elements of the set.
So there we have an example of an illegitimate extrapolation. If you prove Phi(n) for an arbitrary natural number n, then you are allowed to conclude:
forall natural numbers n, Phi(n).
So you can conclude:
forall natural numbers n > 0, the set of all even numbers less than or equal to 2n has a cardinality less than 2n.
That's true. That's a legitimate proof. On the other hand, it is not legitimate to conclude:
The set of all even numbers has a cardinality that is less than some even number.
That's an unwarranted extrapolation.
So there are legitimate proofs, and there are bogus proofs. You have to actually learn logic to be able to tell the difference.
