On 1 Jun., 03:28, chaja...@mail.com wrote: > > No, by "are they real" I mean much more. As well as an order you also > > need > > the field operations (addition, subtraction, multiplcation and > > division) > > and you need them to preserve order. > > Makes sense to me and I agree. > > > No. If we use the standard definition of real numbers and > > the standard way of talking about decimals (infinite decimals > > are defined in terms of limits), then > > decimals can express any real number (including > > sqrt(2) and pi). > > I find this quite simply impossible to accept. For any decimal that > tries to express a number beyond zero, a whole infinity of real > numbers lie in between that and zero. Decimal can never point out a > value beyond zero that has no intervening infinity. To me the > consequence is that is cannot point out them all, for to point one out > only shows there are others. > > > It was to avoid nonsense like > > "would be able to comply in infinite completion" > > that the concept of limits was developed. > > > - William Hughes > > You reject the idea of infinite completion seemingly because it is > counter-intuitive to you, and indeed to many people, as infinity is > unbounded it our most general concepts. What about this: if you take > the sets of all real numbers in the intervals [0, 1] and [-1, 0] they > can be paired in cancelling values in infinite completion. The sets of > all real number in the intervals of [0, 1) and [-1, 0] could not be > paired in cancelling values in infinite completion, we could only > traverse infinity and yield pairs, refusing to complete the operation > and acknowledge that one value will not pair.
Take any element x of [0,1). If x is the reciprocal of a natural number, map x to -x/(1-x). Otherwise, map x to -x.