> 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.
This idea you reject, or my applications of it, and the result of such a rejection yields another counter-intuitive result, that 0.999... is exactly in and of itself equal to 1.