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Topic: Why do we need those real non-constructible numbers?
Replies: 68   Last Post: Dec 11, 2017 1:42 AM

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 Jim Burns Posts: 831 Registered: 9/26/15
Re: Why do we need those real non-constructible 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 UTC-7,
> bassam king karzeddin wrote:

>> Why do we really need those real non-constructible 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

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,
non-empty, with one set completely to the right of the other,
there is at at least one point between them.
-- For every upper-bounded non-empty 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 non-constructible numbers.

This is essentially the reason that there are non-constructible
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.)

Date Subject Author
11/6/17 bassam king karzeddin
11/6/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 bassam king karzeddin
11/8/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 bursejan@gmail.com
11/8/17 bursejan@gmail.com
11/8/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 bursejan@gmail.com
11/8/17 bursejan@gmail.com
11/9/17 bassam king karzeddin
11/9/17 zelos.malum@gmail.com
11/9/17 bassam king karzeddin
11/9/17 bassam king karzeddin
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/9/17 bassam king karzeddin
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/13/17 bassam king karzeddin
11/14/17 wolfgang.mueckenheim@hs-augsburg.de
11/14/17 genmailus@gmail.com
11/14/17 bursejan@gmail.com
11/14/17 bursejan@gmail.com
11/6/17 Stephen G. Giannoni
11/6/17 zelos.malum@gmail.com
11/7/17 bassam king karzeddin
11/9/17 zelos.malum@gmail.com
11/8/17 jsavard@ecn.ab.ca
11/8/17 bassam king karzeddin
11/8/17 Jim Burns
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 Jan Burse
11/8/17 Dan Christensen
11/8/17 Dan Christensen
11/8/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 bursejan@gmail.com
11/8/17 bursejan@gmail.com
11/8/17 Jan Burse
11/8/17 Jan Burse
11/11/17 bassam king karzeddin
11/11/17 zelos.malum@gmail.com
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/9/17 Tucsondrew@me.com
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 Dan Christensen
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/9/17 Tucsondrew@me.com
11/9/17 Dan Christensen
11/9/17 wolfgang.mueckenheim@hs-augsburg.de
11/9/17 Dan Christensen
11/10/17 wolfgang.mueckenheim@hs-augsburg.de
11/10/17 Dan Christensen
11/11/17 wolfgang.mueckenheim@hs-augsburg.de
11/11/17 zelos.malum@gmail.com
11/11/17 wolfgang.mueckenheim@hs-augsburg.de
11/8/17 Peter Percival
11/11/17 bassam king karzeddin
11/11/17 zelos.malum@gmail.com
11/15/17 bassam king karzeddin
11/15/17 Dan Christensen
11/15/17 zelos.malum@gmail.com
11/27/17 bassam king karzeddin
11/29/17 rebatchelor2@gmail.com
11/30/17 bassam king karzeddin
12/1/17 zelos.malum@gmail.com
12/2/17 bassam king karzeddin
12/4/17 zelos.malum@gmail.com
12/6/17 bassam king karzeddin
12/9/17 bassam king karzeddin
12/11/17 zelos.malum@gmail.com