
The field of “useful reals” between rational and real numbers (2019) - peanut_is_yum
https://chittur.dev/math/2019/12/05/secret-field.html
======
Sniffnoy
None of this is somehow secret. The standard name for this is "definable"[0].
Although, one has to be really careful with this sort of thing; there are
apparently a number of subtle logical issues[1] that come up when talking
about these...

(Note, by the way, that there's any number of other fields one could put
inbetween; such as the field of algebraic reals, or computable reals, or the
fraction field of the ring of periods...)

[0]
[https://en.wikipedia.org/wiki/Definable_real_number](https://en.wikipedia.org/wiki/Definable_real_number)

[1]
[https://mathoverflow.net/a/44129/5583](https://mathoverflow.net/a/44129/5583)

~~~
dwohnitmok
Well that Math Overflow post is excellent.

One of the logical issues is that there is a model of ZFC where all reals are
definable/useful. I'm guessing that's not what the author of this blog post is
going for...

If this seems impossible given that the number of definitions is countable,
note first that it is possible that a model of ZFC is itself countable (in a
larger ambient model), but it cannot witness the countability of sets within
itself. So when we say that a set is uncountable in ZFC, it is sometimes
useful to make the distinction that it is only uncountable in the implicit
model under discussion.

Then note that definability, unlike countability, cannot be itself defined in
the language of ZFC (due to Tarski's undefinability of truth result). Note
that this is different from saying it's independent of ZFC. It cannot even be
expressed in ZFC. Hence, unlike countability, there is no "relative" concept
of definability, at least not relative to first-order ZFC. Therefore the
statement "every element of this model is definable" is more absolute than
"every element of this model is countable" (but not absolutely absolute, we
still have an ambient model we're working in, just a richer theory for that
model).

The usual diagonalization argument within our entirely definable model of ZFC
to try to construct a definable real number not contained in any countable
enumeration of definable real numbers fails because we have no enumeration of
definable real numbers. This is not a failure of constructivism (it is ZF _C_
after all, we do have choice), but rather a consequence of the fact that
definability cannot be expressed in ZFC so we don't have a way of even talking
about the set of all definable real numbers within our model.

~~~
dwohnitmok
Upon re-reading my reply, it might be worthwhile to simply quote Hamkins'
conclusion in full.

> And therefore neither are you able to do this in general. The claims made in
> both in your question and the Wikipedia page [the Wikipedia page has now
> since been updated] on the existence of non-definable numbers and objects,
> are simply unwarranted. For all you know, our set-theoretic universe is
> pointwise definable, and every object is uniquely specified by a property.

------
emacdona
The author claims in the notes that "The useful reals are similar, but not
quite equivalent to other ideas in mathematics, such as [...] computable
numbers."

Is that correct? What is the complement of the Computable Numbers in the
Useful Reals? What is the complement of the Useful Reals in the Computable
Numbers?

I've always thought of Computable Numbers as all numbers able to be
represented by a finite string, ie: a computer program that would generate the
number to any desired precision. How does that differ from the set of numbers
with a finite symbolic representation?

Hmmmm... maybe by asking that question I've led myself to the answer.
Chaitin's Constant has symbolic representations, one of which being the
Wikipedia page that describes it:
[https://en.wikipedia.org/wiki/Chaitin%27s_constant](https://en.wikipedia.org/wiki/Chaitin%27s_constant).
Does that mean it's included in the complement of the Computable Numbers in
the Useful Reals? Are the Computable numbers a subset of the Useful Reals?

~~~
Sniffnoy
The standard term is "definable", not "useful":
[https://en.wikipedia.org/wiki/Definable_real_number](https://en.wikipedia.org/wiki/Definable_real_number)

But yes, Chaitin's constant is an example of a number that is definable but
not computable.

~~~
bloomer
Yeah I think this terminology is odd because I think that the computable
numbers are much more “useful” than the definable numbers.

------
D_Alex
If you look at the _integers_ between say Graham's number
([https://en.wikipedia.org/wiki/Graham%27s_number](https://en.wikipedia.org/wiki/Graham%27s_number))
and
TREE3([https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem))
you can observe that practically _all_ of these integers, while "computable",
cannot be defined within the known constraints of this universe.

Which raises an interesting question: In what meaningful sense do these
numbers _exist_? They are just out of reach as the non-definable real
numbers...

~~~
voxl
You're mixing up "defined" with "defined via a decimal numeral" we can define
these numbers without much difficulty via finite formula that compute them.
This is a completely valid definition, it is just not a decimal numeral.

An interesting idea might be "useful integers" which requires whatever
definition we have to allow approximation of any finite subsequence with error
converging to zero given more computational power.

~~~
pwdisswordfish2
GP did not mix up anything. Some of those finite formulas _also_ will be too
long to be written within the constraints of this universe. The pigeonhole
principle applies just as much to finite formulas as it does to finite strings
of decimal digits.

------
klodolph
Note that like the rational numbers, the field of “useful reals” is not
complete.

So if you have a sequence of “useful reals” that is Cauchy, it _will_ converge
to a real number but it may or may not converge to a “useful real”.

~~~
function_seven
Sorry if I misunderstand you. If I have a Cauchy sequence of "useful reals",
wouldn't the convergence be, by definition, a "useful real"? That is, I can
write down the Cauchy sequence, so it's now symbolically noted, right?

Or are you referring to a Cauchy sequence that exists, but can't be defined
using our symbology?

~~~
klodolph
There are uncountably many Cauchy sequences of useful reals. You can’t write
them all down. So now you have to also restrict yourself to “useful Cauchy
sequences of useful reals”.

This is a rabbit hole with no end.

~~~
wolfgke
> There are uncountably many Cauchy sequences of useful reals. You can’t write
> them all down.

This does not hold if you demand that, for example, the map

k -> a_k

that represents the Cauchy sequence, is a computable function.

~~~
klodolph
I am a bit shocked, because that seems like a really dishonest way of doing
things. “Surprise! Actually, I am not talking about Cauchy sequences, but only
computable Cauchy sequences.”

If you change the rules you had better be up front about it.

What you are describing is a _completely different definition_ for “complete
metric space” than what is commonly accepted by the mathematical community at
large. So do not be surprised that by using different definitions, you come to
different conclusions.

~~~
wolfgke
> I am a bit shocked, because that seems like a really dishonest way of doing
> things. “Surprise! Actually, I am not talking about Cauchy sequences, but
> only computable Cauchy sequences.”

Rather: If countability is important to you, you should change the rules so
that the property that the field is closed w.r.t limits of Cauchy sequences
does not make your set uncountable.

Redefining the rules if something does not work is how you do mathematics
works all the time:

\- A PDE does not have a solution in a classical sense and you hate this? No
problem: You invent the theory of weak solutions and distributions and simply
change the concept what is to be considered a solution of the PDE.

\- The concept of algebraic varieties turns out to be to limiting to obey the
rules that you would love them to have? No problem: You define the concept of
algebraic schemes and now talk about algebraic schemes instead of varieties
([https://en.wikipedia.org/w/index.php?title=Scheme_(mathemati...](https://en.wikipedia.org/w/index.php?title=Scheme_\(mathematics\)&oldid=942687770)).

TLDR: Mathematics is often the art of "defining your problems away".

~~~
pdonis
_> If countability is important to you, you should change the rules so that
the property that the field is closed w.r.t limits of Cauchy sequences does
not make your set uncountable._

Ok, fine: I hereby declare that the set Q of rationals is a closed field,
because I define "closed" to mean "closed under Cauchy sequences whose limit
points are rational numbers".

Does that seem OK to you?

------
doomrobo
A nit:

"reals are a field extension of ℚ. They could be considered an algebraic
number field..."

This is not an algebraic extension. Pi is a "useful real number" and it is not
algebraic over Q.

~~~
klodolph
Yes—and to elaborate, the reason why an algebraic field extension of ℚ cannot
contain π is because:

\- If it is a field, it contains π, π², π³, … which are linearly independent.

\- By definition, an algebraic field extension is finite dimensional.

~~~
arberavdullahu
You are wrong! The algebraic field extension ℚ[π] contains π.

~~~
jopolous
I guarantee that is not an algebraic extension.

It's not even a finite extension

~~~
lonelappde
That's not a valid critique, as other commenters explained

~~~
jopolous
You're totally right, I could have worded that differently. I actually posted
an infinite field extension that is algebraic elsewhere in this thread, but
usually those are tricky and I haven't seen them pop up as often as finite
algebraic extensions

------
jepler
Not "between" in the sense of having an intermediate cardinality between
rationals and reals, since they are exactly the numbers available from strings
in some symbolic system or other. Seems to be a slightly expanded case of
algebraic numbers, since additional forms (like infinite definite integrals)
are allowed.

~~~
xtacy
Yep, that's right. Its cardinality is the same as rationals, since it's
countable.

~~~
perl4ever
How disappointing.

Unlike most of the time, I read the article first and now that I'm here, that
was the question I had - clearly it's smaller than reals, but how and why is
this field larger than rational numbers? Guess it's not.

~~~
klodolph
It’s larger than the rational numbers in the sense that it is a strict
superset. Cardinality is what a lot of people reach for when they are talking
about “larger” or “smaller”, but there are lots of other useful concepts which
we can translate to “larger” and “smaller”.

So when someone says “larger” or “smaller”, your first step might be to try
and translate that relationship into a more precise mathematical concept, like
cardinality or measure.

Casual terminology also leads to weird discussions. Like when someone asks
whether some function is “close” to another, and these functions are defined
in terms of vector spaces. Unfortunately, “closeness” does not necessarily
exist in a vector space. So the answer may be that the question does not make
sense.

~~~
effie
> “closeness” does not necessarily exist in a vector space.

The asker will give a definition. For example, two vectors are close if sqrt
of dot product of difference of the two vectors is smaller than some number
delta.

------
OscarCunningham
> A “useful real” is just a real number that can be precisely described (not
> just approximated!) by some symbolic notation. Obviously, this definition is
> loose and depends greatly on your choice of symbols and their definitions.

In fact, the definition is necessarily loose. If you could make it precise
then you could carry out Cantor's diagonalisation procedure to produce a
precise description of a real which couldn't be precisely described, a
contradiction.

~~~
thaumasiotes
> the definition is necessarily loose. If you could make it precise then you
> could carry out Cantor's diagonalisation procedure to produce a precise
> description of a real which couldn't be precisely described

Is this true without the Axiom of Choice? Don't you need a choice function to
order the numbers before you can diagonalize them?

~~~
lonelappde
Finite descriptions are countable. Axiom of Countable Choice is not
counterintuitive like Axiom of (Uncountable) Choice.

You can order the set of all definitions, by prepending each definition with
its length and then using the ordering (numerical order, alphabetical order).

~~~
OscarCunningham
That doesn't even need Countable Choice. You only need any form of Choice when
you can't explicitly specify an order, which you did.

------
stephencanon
Aside from all the other issues people have raised, equality is not decidable
for the “useful reals”. While they form a field, they do not form a
computably-ordered field, which makes them quite a bit less useful than many
other number systems.

------
NelsonMinar
Another related topic of interest is constructivism in mathematics.
Unfortunately the wikipedia article is pretty abstruse, anyone have a more
down to earth one?
[https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...](https://en.wikipedia.org/wiki/Constructivism_\(philosophy_of_mathematics\))

(Note this is different from constructible numbers, which the author mentions.
That has to do with classical geometry.)

~~~
btilly
My favorite is the collection of essays touching on the topic in the book *The
Mathematical Experience".

All of the other essays in the same book are also good. :-)

------
dchyrdvh
The premise of this idea - that anything describable can be written in a
binary firm and is thus countable - seems wrong. It's wrong because we easily
invent new concepts and put them into a symbolic form. We could invent a new
concept, agree on a new symbol for it and add it to our alphabet. The set of
ideas isn't countable and so our alphabet isn't countable. This alphabet can't
be translated into some binary form either.

~~~
PureParadigm
Why is the alphabet not countable? If each time you think of a new idea and
make a symbol for it, I can also assign it to an integer (because there is
always a next integer like there is always a new symbol you can come up with).

When you come up with a new concept, it should also be possible to write out a
definition of it. If you can write down your definition (in English, math
notation, etc.), then it comes from a countable set, since there are countably
many things that you can write down.

~~~
dchyrdvh
We don't "come up" with ideas from other ideas using some closed form rules of
logic, like in Coq or some Turing machine. Instead, we discover new ideas.

There is a world of ideas and the real world. People live in both worlds. When
they discover a new idea, often by accident, they label it with a symbol and
use it in the real world. Other people can see the same idea and since they
can't fully describe it with words, they agree to use the new symbol.

We describe new concepts with words, but those definitions are underspecified:
they refer to things with vague or non existent descriptions, or just common
sense. What is "set" for example? The same words often mean different things
in different contexts. This extra meaning that's always attached to words is
what makes these definitions non countable.

~~~
PureParadigm
Even if ideas come from an uncountable set (not convinced yet), there are
still only countably many ideas people will ever have. Each time anyone comes
up with an idea, I can assign it a new integer.

~~~
dchyrdvh
Im merely trying to drag the concept of separating ideas and reality as two
different but very real worlds under the spotlight of everyone's attention.
This concept is fundamental and very old. I won't be able to defend this idea
with formal proofs.

------
leni536
Many sets have a "useful" subset this way. Even the class of all sets have a
"useful" subclass.

------
lonelappde
Such a weird perspective. The author thought they discovered something that
true and interesting and kind of fundamental but wasn't already published, but
didn't think it was worth publishing to the math community?

~~~
joppy
Thinking about maths is fun, and some people do it for leisure and write what
they find in innocuous places like blogs. Usually the things you come up with
are already well-known by a different name (as was the case here), so one
would usually not publish something like this.

Think of it just like a random blog post on someone’s thoughts. Just because
it contains maths doesn’t mean it needs to be published or not, it can be free
to live its own life.

------
superjan
this reminds me of unit testing, where the tests come up with arbitrarily
defined numbers, and the function you test tries to come up with a consistent
way to count them. If you can change your function each time a test is added,
the tester never wins. Isn’t this similar? It seems like cherrypicking to
include simple formulas with e and pi in your numbering system.

------
scarejunba
Does this field behave differently from Q in some 'useful' way?

~~~
H8crilA
It has sqrt(2), for starters? Not sure what do you mean by useful.

It is not "useful" in the sense that reals are most "famous" for: it is not
complete. Cauchy sequences can diverge in the useful reals field.

~~~
lonelappde
Completeness in the "full" reals is a useless feature, though. All is gives
you is an emotional crutch to pretend your cauchy sequences can be mapped to
regular numbers. But it doesn't give you anything you didn't already have in
the cauchy sequences and useful reals.

~~~
steerablesafe
You are of course right, reals are isomorphic to equivalence classes of Cauchy
sequences on Q. But once you are dealing with equivalence classes of Cauchy
sequences on Q you might as well give it a name. Maybe call it R.

~~~
H8crilA
His point is different. You cannot (by definition) ever write a "name", a
formula, a rule, a lim expression, anything really, for a real that is not in
the useful reals.

------
currymj
it's good to see the "real numbers are fake" crowd out in full force!

~~~
lonelappde
Ahem.

Repeat after me, the Creed of Numbers:

" The imaginary numbers aren't imaginary.

The real numbers aren't real. "

------
dfox
There is well defined name for "useful reals": Algebraic numbers. Of course
the well-definedness necessitates some limit on how the symbolic description
looks like (ie. algebraic numbers are roots of polynomials with rational
coefficients) because every real number can be described by some arbitrarily
complex symbolic notation.

Edit: I vaguely remember that there used to be some name for the intersection
of algebraic and real numbers, but I neither can remember it nor can find it
on wikipedia.

~~~
JoshuaDavid
> every real number can be described by some arbitrarily complex symbolic
> notation

This seems like it would have to be false, because otherwise the reals would
be countable (iterate through every possible 1-character string, then every
possible 2 character string, then 3 chars, etc and in a finite (but
potentially very very large) amount of time you would come across the
description of any real number that can be described).

------
arberavdullahu
The author claims that this set is countable but not sure if that is true. My
argument is based on Cantor's theorem [1], which states that the power set has
cardinality strictly greater than the set.

In order for the set of symbols to be finite field it must grow therefore
since rational is infinitely countable from Cantor it must hold that "useful
reals" is uncountable.

[1]
[https://en.wikipedia.org/wiki/Cantor%27s_theorem](https://en.wikipedia.org/wiki/Cantor%27s_theorem)

~~~
selectionbias
If you have a finite set of symbols, then the set of finite sequences of those
symbols is countable. The key here is 'finite sequences', if you were to allow
for infinite sequences then the set if uncountable.

------
PaulHoule
This is one of my favorite obscure math topics.

I think of the "useful reals" being the "reals that have names". Alan Turing
developed the Turing machine to get a handle on the "useful reals" since you
can make a Turing machine write them out one digit at a time.

Given that, I don't like the term "real numbers" at all because they are phony
compared to the "useful reals" \-- if you reject the axiom of choice then the
construction that Cantor does to construct a real isn't valid.

Despite calling for a rebuild of math and science based on computation, Steve
Wolfram has yet to take the critical step of rejecting the axiom of choice. I
wish he would man up.

~~~
OscarCunningham
> if you reject the axiom of choice then the construction that Cantor does to
> construct a real isn't valid

Are you talking about Cantor's argument that the reals are uncountable? That
doesn't need choice.

~~~
thaumasiotes
Elaborating, the hypothesis that Cantor disproves is "The real numbers are
countable -- that is to say, the real numbers can be put into one-to-one
correspondence with the natural numbers".

You never have to use the axiom of choice, because the hypothesis tells you
there is a one-to-one function between the reals and the naturals. You can
then order the reals in the order suggested by their image in the naturals:
f(0), f(1), f(2), ...

