

Trapping A Transcendental - Finding numbers that aren't solutions to polynomials - ColinWright
http://www.penzba.co.uk/Writings/TrappingATranscendental.html?HN2

======
moomin
For extra craziness: you can extend the list argument to list all (finite)
programs that produce increasing but bounded sequences. We'll call this the
computable numbers. The transcendental number described in the article is
computable.

But as I've just mentioned, the computable numbers are still countable. This
really hurts my head. There's an uncountable set of numbers that aren't
expressible in any closed form whatsoever.

Continuous numbers get weirder the harder you look at them.

~~~
ColinWright
Problem there is you have to list all finite _terminating_ programs that
produce increasing bu bounded sequences. Then you start to have to worry about
being really, really careful in your definitions. See here:

[https://news.ycombinator.com/item?id=6967954](https://news.ycombinator.com/item?id=6967954)

That's a classic "paradox" being created by having insufficient care in the
definitions.

But yes, most (in a technical sense) reals can't be described or computed, and
yes, the reals get more and more weird the harder you look at them. In some
sense you never really understand them, you just get used to them.

Mostly.

~~~
poizan42
I don't see the problem here. It shouldn't be a surprise that listing the
computable numbers is itself uncomputable as it is equivalent to the halting
problem - to determine whether an algorithm will either keep on producing new
decimals or halt at some point is clearly undecidable.

It's when you introduce things as constructible/defineable numbers that things
gets hairy.

------
Tloewald
Um, what?

There are two well-known arguments from Cantor:

[http://en.wikipedia.org/wiki/Cantor's_first_uncountability_p...](http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof)

[http://en.wikipedia.org/wiki/Cantor's_diagonal_argument](http://en.wikipedia.org/wiki/Cantor's_diagonal_argument)

Describing the sequence at the end as something that seems like it should
converge is a bit unconvincing -- it's a sequence explicitly defined not to
land on an algebraic number. So the argument is that if you assume you can
always pick an interval that doesn't contain a specified algebraic you won't
land on an algebraic. This is hardly surprising and kind of presupposes the
result.

And why divide the interval into thirds? This seems to be a result of
confusing a different Cantor proof (that the measure of the
rationals/algebraics in [0,1] is zero, which involves dividing things into
thirds) with the first proof. If we accept the argument at all I don't see how
it requires three intervals at each step. At most one interval can contain the
posited nth algebraic, so you've always got the other interval.

So it's like a mixture of two badly remembered proofs :-)

~~~
ColinWright
To take your points:

    
    
      > There are two well-known arguments from Cantor:
      > http://en.wikipedia.org/wiki/Cantor's_first_uncountability_p...
      > http://en.wikipedia.org/wiki/Cantor's_diagonal_argument
    

Yes. And in fact the first uncountability proof comes in two flavours. This is
explicitly a re-casting of those two versions of Cantor's first proof.

More explicitly, one version of Cantor's first proof has you list the
algebraics, take an interval, find the first two algebraics _a_ and _b_ in the
interval, reduce the interval to [a,b], and continue. This has various
technical issues. What if there aren't two algebraics in your interval
(although there always will be, because they're dense, but the argument goes
through for any list, not just algebraics, so we need to consider this
possible eventuality if we're using the full generality.) What if the interval
doesn't converge to be of length zero? (Which again, for the algebraics, it
always will. And again, if we use the argument in general we need to allow for
this.) And so on.

The second version of Cantor's first proof avoids some of this by saying: find
just the first algebraic in the interval, then choose a smaller sub-interval
that avoid it, any carry on. Again, for algebraics the interval will go to
length zero, but for a general list that won't necessarily be the case.

    
    
      > Describing the sequence at the end as something that seems like
      > it should converge is a bit unconvincing -- it's a sequence
      > explicitly defined not to land on an algebraic number. So the
      > argument is that if you assume you can always pick an interval
      > that doesn't contain a specified algebraic you won't land on an
      > algebraic. This is hardly surprising and kind of presupposes
      > the result.
    

I think you've mis-read the argument. We produce a succession of nested
intervals. For a given algebraic, eventually every interval will not include
it. That means that no algebraic is in more than finitely many intervals.

The we consider the left end-points. That's an increasing sequence that's
bounded above. The reals are complete, so that sequence must converge. But by
construction the limit can't be an algebraic.

The comment that the sequence "seems like it should converge" is pointing out
that completeness is one of the things we expect of the number line, and it's
a way into giving that concept a technical meaning.

    
    
      > And why divide the interval into thirds? This seems to be
      > a result of confusing a different Cantor proof (that the
      > measure of the rationals/algebraics in [0,1] is zero, which
      > involves dividing things into thirds) with the first proof.
    

No, it's a way of being explicit in the choice of the sub-interval. I
sometimes use fifths. In that way I can discard the end sub-intervals, discard
any sub-interval that touches or contains the given algebraic, and thus the
end-points are _strictly_ increasing, and not just increasing. I didn't do
that here, and you make me wonder if it's worth changing it, so people don't
(incorrectly) make the connection with Cantor dust.

But no, this isn't confusing the construction of Cantor dust.

    
    
      > If we accept the argument at all I don't see how it requires
      > three intervals at each step. At most one interval can contain
      > the posited nth algebraic, so you've always got the other interval.
    

The algebraic may be on the boundary, rather than in an interior.

    
    
      > So it's like a mixture of two badly remembered proofs :-)
    

With respect, your comments seem like they are generated from an
insufficiently careful reading, filtered through an understanding of the
existing explanations. It feels like you know about this material, and were
assuming this is a simple re-hash.

But I could be wrong. Your comments might have substance that I haven't
recognised. If that's the case I'd be delighted if you could expand more on
why you think the article is wrong or misleading.

Thanks.

~~~
Tloewald
> We produce a succession of nested intervals. For a given algebraic,
> eventually every interval will not include it. That means that no algebraic
> is in more than finitely many intervals

Going back over the post -- this number only "exists" because of the least
upper bound _axiom_.

You're right that I probably didn't read it carefully enough but there's no
particular reason why the intervals need share any elements (e.g. [0,0.5)
[0.5,1]) or contain their boundary points -- we got division and < from the
rationals without violating anyone's intuition.

Personally, I've always found these arguments unsatisfying. The proof is of
the "existence" of something but in the end we have no idea what it looks like
-- is it >0.5? No clue. And the proof relies on an existence axiom that seems
contrived. (I guess at heart I'm a constructivist.) The desire for
"completeness" flows from a false intuition of the nature of the world -- the
surface of this table is a "continuum" right? But it turns out not to be. In
contrast, I really liked the discussion of the Reals linked yesterday which
sidetracked into a robust description of the Reals with infinitesimals (and
infinites) and no, I repeat no, least upper bound axiom.

~~~
gjm11
> in the end we have no idea what it looks like -- is it >0.5? No clue.

No, that isn't true. The proof Colin gives is quite explicit. There's an
explicit listing of the algebraic numbers (except that he leaves the order of
various things unstated; for present purposes I'll just work with the order of
appearance in his writeup). Then there's a procedure that considers these one
by one and chooses intervals of exponentially decreasing size to concentrate
on. You can get as many decimal places as you like of the transcendental
number Colin's constructing, just by doing enough steps.

Let's be more explicit. The first algebraic numbers in Colin's list are 0, 0,
-1, 1, 0. (I assume we aren't counting repeated roots of a single polynomial,
but also aren't eliminating duplicates between different polynomials in the
list.)

So, first we take the intervals [0,1/3], [1/3,2/3], [2/3,1] and the algebraic
number 0. We take the leftmost interval not containing 0, namely [1/3,2/3].

Now we subdivide into three again: [1/3,4/9], [4/9,5/9], [5/9,2/3]. And again
our algebraic number is 0. The leftmost interval not containing 0 is
[1/3,4/9].

(So we've already established that the number is < 0.5.)

In the next three steps we never see an algebraic number inside our interval
so we'll just take the leftmost third each time; so we get [1/3,1/3+1/243].

We've now found the first two decimal places of the transcendental number
we're constructing, namely 0.33. It would (at least in principle) be easy to
computerize the process and find many more decimal places.

------
ColinWright
I've found at least one typo in this, there may be more. Before I fix it,
what's the easiest way (other than BitCoin) to offer a reward to people who
find errors?

~~~
spodek
Normally I wouldn't look for such things, but since you mention it, there is a
period missing at the end of the bullet point "Then we need to know what we
mean by "a sequence that looks like it should converge""

Also, at the end of "Later: Build a sequence"

In later bullets too: "We can list them", for example.

~~~
ColinWright
All fixed - thank you.

I wouldn't normally worry about every last detail of punctuation, especially
in lists and/or bullet points, but your comment is perfectly valid, and of
value.

So, how can I buy you a coffee?

------
laurentoget
Maybe i am nitpicking but the existence of transcendentals was proven by
Liouville 30 years before Cantor brought up the considerations about
countability. (disclaimer liouville was the advisor of the advisor of the
advisor of the advisor of the advisor of the advisor of my advisor)

Also if you like that sort of things you would like
[http://amzn.com/0201038129](http://amzn.com/0201038129)

Nearly 30 years ago, John Horton Conway introduced a new way to construct
numbers. Donald E. Knuth, in appreciation of this revolutionary system, took a
week off from work on The Art of Computer Programming to write an introduction
to Conway's method. Never content with the ordinary, Knuth wrote this
introduction as a work of fiction--a novelette. If not a steamy romance, the
book nonetheless shows how a young couple turned on to pure mathematics and
found total happiness.

~~~
ColinWright
Indeed, Liouville proved the existence of transcendentals, exhibiting specific
numbers as examples. That was around the 1840s. Cantor showed in 1874 that
pretty much _every_ real number is transcendental, and the linked article is a
re-casting/adaptation of one version of that paper. Although it shows how to
construct a transcendental, it really shows that every list of numbers is
incomplete. Taking the algebraics as the list then gives the result.

And I actually have a copy of "The Surreal Numbers". Still working on how
it/they can be presented to a non-technical audience.

~~~
stephencanon
Both "Surreal Numbers" and "On Numbers and Games" do a not-terrible job of
presenting surreal numbers to a (relatively) non-technical audience.

~~~
ColinWright
Not convinced. I've tried to follow their presentations when talking about
these topics with different audiences and found that people really just don't
follow. There's a difference between teaching material and exposing people to
it - both of those works try to teach the topic to people, which isn't really
appropriate in a single lecture format, or a math club, or a masterclass.

~~~
stephencanon
Well, they _are_ full-length books (Ok, SN is shortish). There aren't many
books that can be presented as a single lecture as is. The surreal numbers (as
you know, I'm sure) are fairly subtle, and people don't have much intuition
for how they should behave.

Even non-technical people have lots of intuition for real numbers (even if
some of it is wrong); they know that they can add them, multiply them, compare
them, etc, and they even know algorithms to do so. When confronted with the
surreal numbers, it's not at all obvious that you _can_ perform those
operations, much less how to do so, so there's an enormous weight of basics to
build up before you can really do anything interesting (much like how the
first week or two of an introductory linear algebra course can be pretty
dull).

Games are perhaps easier to motivate and make interesting (in my experience),
as you can get to the point of analyzing and reducing simple positions and
establishing the addition operation pretty quickly, without getting bogged
down in "wait, why would I use this horribly complex approach to what seems at
first to be not terribly unlike the real numbers except maybe I can't even do
all the operations I know and love". Unfortunately, multiplication of games
doesn't really make terribly much sense, so you need to restrict to numbers
reasonably quickly to continue developing "conventional" mathematical
structures.

TL;DR: for a single lecture format, motivating and defining games and showing
that there's a reasonable embedding the natural (or dyadic rational) numbers
isn't a bad introduction in my experience.

------
mnx
The biggest problem I had with this, was convincing myself this method does
list ALL the algebraics. Otherwise a great read.

~~~
Tloewald
All orderings of the rationals / algebraics / etc. are counterintuitive.

~~~
ColinWright
Actually, the Calkin-Wilf tree[0] for enumerating the positive rationals is
really very good.

[0]
[http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree](http://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree)

~~~
Tloewald
Cute yes. Clever yes. Intuitive?

------
CurtMonash
If all we want to do is prove the existence of transcendental numbers, rather
than construct them, that follows as soon as we know:

1\. Algebraic numbers are countable. 2\. Real numbers aren't countable.

#1 is easy -- the number of polynomials with rational coefficients is
obviously countable, and they have finitely many real roots each. So pick your
favorite proof of #2 and you're done.

~~~
ColinWright
True. And in truth, the linked article is my favorite proof of #2.

And that's the point. This proof is really showing the uncountability of the
reals - nowhere does it use any property of the algebraics other the fact that
they're countable.

------
CurtMonash
Spivak's Calculus contains a 3-page proof that e is transcendental. I didn't
work through it, but it's unsurprisingly based on the famous series expansion
of e.

------
graycat
So, he likes what is known as the _completeness_ property of the reals. Good.
Glad he likes that! One remark is "Calculus is the elementary consequences of
the completeness property of the reals.". And, yes, calculus is nice stuff.
Newton did nice things with calculus but may not have understood the
completeness property, i.e., just took it for granted that his derivatives and
integrals did yield _numbers_.

Some of what is true about the reals is bizarre, nearly beyond belief, e.g.,
as in Oxtoby, _Measure and Category_.

The main reason for being careful with fine details going from the natural
numbers to the integers, rationals, algebraics, and reals is a theme that
started near 1900 of trying to clean up the logical basement of mathematics
and, there, build everything on just _sets_ so that, if believe in sets, then
could believe in everything that could be based on sets, that is, hopefully
all of mathematics. This goal seemed okay for about 10 minutes until B.
Russell thought of the set

    
    
         A = {x|x is not an element of x}
    

so that A is an element of A if and only it is not. Bummer. Seventy years
later the gods of axiomatic set theory came out with a lot of work that made
set theory look more solid. Okay, that's curious work, and on a good day with
four quarters will cover a $1 cup of coffee.

~~~
Tloewald
There was an article linked yesterday (sorry, don't have the link) which
discussed the Reals in considerably more detail and sidetracked into a
reformulation of the Reals that rigorously allows for Newton's infinitesimals
without being "complete" in the "least upper bound" sense (and indeed,
depending on the classes of functions you care about, doesn't even require an
uncountable set).

~~~
graycat
My view is that _infinitesimals_ were a mistake, a long, festering sore in
math and some of its applications. Can think intuitively about infinitesimals
if you want, but I see no need to mess up the now very clean situation on the
foundations of the reals.

Indeed, as a grad student, I had some results and went to see a world famous
prof to show him what I had and ask him if my work seemed new to him. At one
point he suggested an improvement, and, curiously, as a high end pure
mathematician in _analysis_ did his first cut, intuitive reasoning in terms of
infinitesimals! Fine. But Dedekind cuts, Cauchy sequences, etc. make a rock
solid foundation for the reals, and now can we just get on with, say, Fourier
theory and the rest of _analysis_ and their applications?

