
Re: Why do we need those real nonconstructible numbers?
Posted:
Nov 8, 2017 9:42 PM


On 11/8/2017 4:25 AM, Quadibloc wrote: > On Monday, November 6, 2017 at 12:21:29 AM UTC7, > bassam king karzeddin wrote:
>> Why do we really need those real nonconstructible numbers, >> if it is impossible to express them exactly except only by >> constructible numbers or as meaningless notation in mind only? > > The real number line is a nice, simple thing.
I snipped all your good reasons because I have nothing to add to them.
However, I would like to say that _also_ the real number line is a good description of a continuous line with numbers marked on it. The main difference between a line of real numbers and a line of rational numbers (or of algebraic numbers, or of constructible numbers) is that _there are_ _no holes in the real number line_ (It is Dedekind complete.)
We don't think our number lines should have holes in them. Our real number axioms express this concept in clear language which we can use to reason with.
There are many equivalent ways to express the same concept.  If two continuous curves cross in the real plane, the curves meet at some point.  For any two collections of points on the real line, nonempty, with one set completely to the right of the other, there is at at least one point between them.  For every upperbounded nonempty set of real numbers, there exists a least upper bound.  And many more.
One consequence of Dedekind completeness of the number line is that there are nonconstructible numbers.
This is essentially the reason that there are nonconstructible numbers: no one cares that there are numbers that are not constructible, but we do care that our continuous lines behave like continuous lines.
 By the way, there is nothing unusual about there being too many of something to describe or construct or count.
Suppose we have the set of natural numbers. Now, suppose we have the power set of the natural numbers, the set of all subsets of the set of natural numbers.
There are more subsets of the natural numbers (uncountably many) than there are definitions of _anything_ including subsets of the natural numbers (at most countably many). Therefore, there are undefinable subsets of the natural numbers.
What was that sound? Was it the foundations of the universe trembling? No, it wasn't. The universe is fine, it's just that there are some things we can't define. (Or, elsewhere, construct.)

