

The Hilbert Hotel - appfactories
http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel

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btilly
I am amused that only one interpretation of the facts is considered, and not
its bizarre consequences.

At the end it is mentioned that the real numbers from 0 to 1 are not
enumerable. Therefore there exist real numbers that are not in any particular
enumerable set. Now consider the set of all potential finite definitions of a
number using a finite set of symbols. This set turns out to be perfectly
enumerable. Not all definitions in it will turn out to be well-formed, numbers
have multiple definitions (for instance 5/10 is the same as 1/2) and many that
are defined are not in the interval (0, 1). But this is no barrier. According
to the classical view of mathematics, the set of definable numbers in (0, 1)
is an enumerable set. Therefore almost all real numbers in that interval do
not have any possible definition.

In what sense, I ask you, does a number with no possible definition actually
exist?

Is there another way to look at this problem? It turns out that there is. If
you take the constructivist view of things, things only exist if you can lay
your hand on a concrete construction. "Constructions" that classical
mathematics does not blink an eye at may not qualify. In particular this
"enumeration" over possible definitions, and being able to sort through to
figure out which definitions are valid, and which are equivalent, and which
fall in the interval, is an example of something that is beyond the pale. Why?
Well many of the possible definitions are computer programs. Figuring out
which of them actually define numbers means doing things like solving the
Halting problem. Which we can't actually do.

From the point of view of constructivism, "uncountable" does not mean "really
big infinite set". It means "really convoluted set structure".

Yes, I know that constructivism lost. Nobody wants to consider these odd
questions. But if somewhere in the middle of a proof you ask what it all
means, you may find that classical mathematics doesn't necessarily make that
much sense.

~~~
gruseom
Do you mean: for each finite symbol set S, the set of all finite definitions
using S is enumerable? That's clearly true. But then you're not really saying
that there are numbers with no possible definition, just that there is no
particular finite symbol set S that can define them all.

Or do you mean: the set of all finite definitions in _any_ finite symbol set
is enumerable? That's not clear. Are there countably many such symbol sets?

~~~
btilly
What I mean is this.

Given a finite symbol set, the set of all finite definitions is enumerable.
And the enumeration is simple. We enumerate all possible definitions of length
1, 2, 3, 4, etc. We order them first by length, and then by alphabetical
order. We strike out all definitions that are not definitions, or define a
number already defined. This leaves us with an enumeration of all numbers
definable with that set of symbols. (Note, a constructivist will insist
strenuously that this is not actually an enumeration.)

Given that there are uncountably many real numbers, uncountably many of them
are not in that enumerable set, and therefore almost all real numbers cannot
be defined using that finite set of symbols.

The possibility of uncountably many possible symbol sets with a similar
plethora of possible associated definitions is irrelevant, because any
potential symbol set that cannot ultimately be described in the finite
alphabet that we use with the language English is irrelevant to us.

~~~
phaedrus
As a programmer I find constructivism is my default position if I think of
math intuitively; perhaps that's why I butted heads so much with many of my
math teachers. But, your argument actually makes me less sure of my agreement
with it: it seems clear that at bottom constructivism must lead to the
rejection of the concept of continuously variable quantities. It is the horns
of a dilemma: if you reject Cantor you reject the existence of "analog" as
even an admissible _concept_ , on the other hand if you accept continuous
variability then you must also accept the shocking conclusion that the
overwhelming majority of the set you assert you believe exists (the reals)
cannot even be described or calculated.

~~~
btilly
But constructivism does not give up continuity. All of the traditional
definitions carry over.

What you do have to give up, though, are discontinuous functions on continuous
sets. They are not well defined for numbers that are the result of an infinite
sequence converging near the border of the discontinuity. You can still give
them a formal treatment, of course, but they aren't really functions.

However if you're attached to your step functions, never fear. It would not be
the first function-like thing that isn't a function. For another example
consider the well-known derivative of the step function, the Dirac delta. Not
actually a function in classical mathematics, though it is certainly useful.

------
carbocation
On a tangentially related topic, the human genome is essentially a Hilbert
curve in real life. It's behind a paywall, but in the very bottom thumbnail
you can kind of get the idea:
<https://www.sciencemag.org/content/326/5950/289.figures-only>

