Revenge of the Countables

By Kelly Craig Walling

REVENGE OF THE COUNTABLES

Beginner: The Mass Noun Conspiracy!

Who wants to be a rice counter?

Don't count rice! They invented a machine for that. They want to control

the universe with this machine!

It affects grammar, for example: a cup of rice, a grain of rice. A lot of rice. Not very much rice. ~~Many rice, one rice.~~
I eat a meat every Friday.
Adjectives are not countable:
The important thing is to study. ~~The important is to...~~
He is the responsible one. ~~He is the responsible.~~
Is your car red? No, mine is the blue one. ~~Mine is the blue.~~

Intermediate: Matter over Mind

Modern English hates counting. In fact, it counts things only when absolutely necessary.
Since necessity is the mother of invention, I have invented this list of exceptions that are getting longer and longer. English loves exceptions. Remember: Learn the rule before the exceptions.

You can only defeat it with the Time/Space Continuum!

This was Steve Queen's first film in 1958. They couldn't defeat this ooze until they knew it was called a blob. Suddenly it was countable!

States of Matter.

Anything expressed as a plasma, gas, liquid or even solid is generally uncountable. Consumables: food, household products, textiles, sources of energy, pharmaceuticals,
natural resources such as clay, wood, etc. Be careful with the gerund. You have to determine by the context whether it is a verb, noun or adjective. Words with -ing are generally uncountable (hiking, swimming) as an action but sometimes countable as a thing (a painting, a hanging).

Advanced: Mind over Matter

General references to abstract concepts are amorphous and intangible, meaning they do not take up determined physical space and are therefore treated as even less countable than vapor. Art and science, for example, are not counted when used in a general sense, as in modern science. Notice the uncountable forms should not receive an article of any kind. I bought a furniture. I study ~~the~~ philosophy. But you can count the different kinds of arts and sciences (painting is an art, geology is a science). Modern English frequently turns nouns into verbs (I will friend you on Facebook) but turning verbs into nouns is another can of worms.

States of Mind:

           Emotional states (anger, happiness, lust, jealousy, love, confusion, obsession), attitudes (respect, indifference, bigotry, oppression, sycophancy), philosophies (religion,
democracy, capitalism, karma, freedom), virtues and vices (patience, forgiveness, specific crimes such as bribery and extortion), knowledge, service, forgetfulness, peace, entertainment, politics (slavery, sycophancy, eugenics, euthanasia, etc.) are all uncountable.
             You can say “some” with almost all of these (some time) or “a bit of time”. You can also say things like "a long time", “lots of time” or “a lot of time” etc. If you are referring to a particular manifestation of something you can say: A kind of (vegetation), a sort of (vegetation), a type of (vegetation), a variety of, an instance of, an example of, a specimen of a version of, and of course the individual components of the various taxonomies.
        Some of the best cinematographic antagonists have been uncountable, for example, The Fog, the Blob, the Haunting, the Borg, Communism, Deception, Eugenics, Ebola. A virus is countable but: he has an Ebola. An outbreak is countable so watch the film. The plot thickens:

We are (the) uncountable. Resistance is uncountable. You will be uncounted.

Uncountable Form Possible Countable Forms

Freedom isn't free.                      3 basic freedoms: life, liberty and the pursuit of happiness.
Life is like a box of chocolates. A cat has nine lives. Run for your lives!
Never give up hope.                    There was never a hope.
Let's talk about painting.       Let's talk about your painting.
My travel agent likes to travel. A trip. A voyage (on a boat).
Bribery, to bribe.                         A bribe.

Countable Food:
    Some foods go with “a piece of” or “a slice of”, for example:
A slice of cheese, bread, pie, cake, ham, meat, pizza etc.

Units of measurement: time, currency, length, weight width, depth, electricity, pressure money, elements (atoms) compounds (molecules) information bytes, radiation. All chemicals.

Meteorological and geological manifestations:
A drop of rain. A rainstorm. A snowflake.
A bolt of lightning. A clap or bolt of thunder. A gust of wind. Colder waters.
Natural disasters are countable: A hurricane, a volcano, an earthquake. A flood.

It is Noam Chomsky's contention that the mind cannot begin to fathom something
until it has been labelled. Therefore, I have assembled an armament of labels indicating
countable forms to help you defend the universe from the onslaught of the uncountable!

Uncountable Possible Countable

Advice*      “Do you have any advice for me?” “Yes, some. In fact, two pieces of advice.”
Art*         A piece of art is fine, but a work of art may be a masterpiece. Martial Arts. You can study the Arts or the Sciences. I've got this down to an art. See work. See science.
Baggage      Suitcase. See luggage.
Clothing*    An article of clothing. An accessory.
Education    He had a good education
Equipment*   A tool. A device. An apparatus. A gadget. A rig. See technology
Evidence*    Exhibit 'A'. Proof is not countable, either.
Experience   I had a bad experience. I have 12 years of experience teaching English
Fiction*     A work of fiction. See work, literature.
Frustration He is always taking his frustrations out on her.
Furniture*   A set of furniture, but more specific would be a dining room set, for example.
Glass*       A drinking glass. A wine glass. He wears glasses. A pane of glass. A windowpane. A shard of glass.
Gossip       A rumor
Grass        Blade of grass, a lawn
Ground       The ground wire or grounds for divorce. Coffee grounds may sound plural but they're hard to count. Soil is not countable.
Gum*         A stick of gum (chewing or bubble). Your gums are the fleshy pink tissue that holds your teeth in.
History      He has a history. An event. A historical moment or occasion. See time.
Homework     an assignment
Information* Bytes, etc.
Life         He leads a life of luxury.
Literature* Book, magazine, an article etc. See paper, fiction
Luck         A bit of luck. Some luck
Luggage*     See baggage
Memory       I have a few good memories of high school. He has photographic memory.
Military     Usually an adjective. “The military” is the armed forces.
News*        A report, an article. A recent event. A current affair
Paper*       sheet of paper. I wrote two papers last year means two academic documents.
Parking      Parking space, parking lot, parking garage, parking meter.
Personnel    Usually an adjective. See Staff.
Philosophy   Tenet. Credo. Ethos. A thought. A concept. A precept. A dogma. I studied the philosophy of existentialism in Copenhagen.
Police       A policeman, policewoman, police officer. Consider it a verb first and then an adjective: They are always policing the main plaza near the police department. If you say “the police are here” that implies more than one.
Progress           An advance in progress. A progression. An advancement. See science and technology.
Propaganda         Campaign. Slogan. Chant. Sycophancy. Operation. Project. A plot. A conspiracy.
Religion           Pastafarianism is a new religion
Research           A trial. An investigation. A project. See technology.
Science            An experiment. A hypothesis. A theory. Theo rum. A project. Alchemy was once considered a science. See art. See technology.
Space              A parking space. A gap. A space. A place. The Time/Space Continuum. See time.
Staff              Staff member. Staff can be an adjective, verb or noun but cannot be counted.
Surgery            An operation. A procedure. A surgical procedure is often heard but redundant.
Technology         An invention. A technique. A treatment. An experiment. A procedure. A component. A widget. An advance. A breakthrough. See equipment. See measurements. See science.
Time               There was a time when medicine was more brutal. I've been to the Algarve 4 times. A while. A long time/while. A short time/while. See measurements. See History. See space.
Sex                All mammals have 2 sexes (genders). “The genitals” takes an article but “genitalia” is never countable.
Speed              He was travelling at an incredible speed. As a verb, it can mean to exceed the speed limit.
Work*              A work of art or public works. A job. A position. See literature. See art.
         *A piece of...

Single items referred to as pairs:

      a pair of pants/trousers
   •
      a pair of underwear
   •
      a pair of swimming trunks
   •
      a pair of shorts
   •
      a pair of jeans
   •
      a pair of glasses
   •
      a pair of sunglasses
   •
      a pair of binoculars
   •
      a pair of goggles
   •
      a pair of scissors
   •
      a pair of pliers
   •
      a pair of clippers
   •
      a pair of tweezers
   •
      a pair of tongs
   •
      a pair of headphones/earphones
   •
      a pair of earplugs
   •
      a pair of clippers
   •
      a pair of handcuffs
   •
Things that come in pairs
      a pair of shoes
   •
      a pair of socks
   •
      a pair of slippers
   •
      a pair of boots
   •
        a pair of shoelaces
     •
        a pair of gloves
     •
        a pair of cuff links
     •
        a pair of earrings
     •
        a pair of contacts
     •
        a pair of eyes
     •
        a pair of lungs
     •
        a pair of kidneys
     •
        a pair of wings
     •
        a pair of dice
     •
        a pair of skis
     •
        a pair of skates
     •
        a pair of crutches
     •
        a pair of stilts
     •
        a pair of chopsticks
     •
        a pair of speakers
     •
        a pair of headlights
     •
        a pair of windshield wipers
     •
        a pair of knitting needles
     •
You can also count pairs of things: two pairs of pants, three pairs of socks, four pairs of
shoes, etc.

Theoretical:

1. Should Spanish tortilla be countable? What about paella? Give reasons why or why not.
2. Beyond is a preposition but The Beyond is the supernatural. Even though it
can take a definite article it cannot take an indefinite article. Why not?

Natives Only:

Let me give you a piece of my mind (I have a bone to pick with you).
That guy is a real piece of work.
A slice of life. the great outdoors
Everybody wants a piece of the action!
The Good, the Bad and the Ugly (if you are an advanced English Speaker you will understand how this exception could exist)
An ounce of prevention is worth a pound of cure.

For Mathematics: I leave that to the professionals: www.math.brown.edu/~res/MFS/handout8.pdf

Countable and Uncountable Sets

Rich Schwartz

November 12, 2007

    The purpose of this handout is to explain the notions of countable and
uncountable sets.
1     Basic Definitions
A map f between sets S1 and S2 is called a bijection if f is one-to-one and
onto. In other words
    • If f (a) = f (b) then a = b. This holds for all a, b ∈ S1 .
    • For each b ∈ S2 , there is some a in S1 such that f (a) = b.
We write S1 ∼ S2 if there is a bijection f : S1 → S2 . We say that S1 and
S2 are equivalent or have the same cardinality if S1 ∼ S2 . This notion of
equivalence has several basic properties:
   1. S ∼ S for any set S. The identity map serves as a bijection from S to
      itself.
   2. If S1 ∼ S2 then S2 ∼ S1 . If f : S1 → S2 is a bijection then the inverse
      map f −1 is a bijection from S2 to S1 .
   3. If S1 ∼ S2 and S2 ∼ S3 then S1 ∼ S3 . This boils down to the fact that
      the composition of two bijections is also a bijection.
These three properties make ∼ into an equivalence relation.
    Let N = {1, 2, 3...} denote the natural numbers. A set S is called count-
able is S ∼ T for some T ⊂ N . Here is a basic result about countable
sets.
                                        1
Lemma 1.1 If S is both countable and infinite, then there is a bijection
between S and N itself.
Proof: For any s ∈ S, we let f (s) denote the value of k such that s is the
kth smallest element of S. This map is well defined for any s, because there
are only finitely many natural numbers between 1 and s. It is impossible for
two different elements of S to both be the kth smallest element of S. Hence
f is one-to-one. Also, since S is infinite, f is onto. ♠
Lemma 1.2 If S is countable and S ′ ⊂ S, then S is also countable.
Proof: Since S is countable, there is a bijection f : S → N . But then
f (S ′ ) = N ′ is a subset of N , and f is a bijection between S ′ and N ′ . ♠
    A set is called uncountable if it is not countable. One of the things I will
do below is show the existence of uncountable sets.
Lemma 1.3 If S ′ ⊂ S and S ′ is uncountable, then so is S.
Proof: This is an immediate consequence of the previous result. If S is
countable, then so is S ′ . But S ′ is uncountable. So, S is uncountable as well.
♠
2        Examples of Countable Sets
Finite sets are countable sets. In this section, I’ll concentrate on examples
of countably infinite sets.
2.1       The Integers
The integers Z form a countable set. A bijection from Z to N is given by
f (k) + 2k if k ≥ 0 and f (k) = 2(−k) + 1 if k < 0. So, f maps 0, 1, 2, 3... to
0, 2, 4, 6... and f maps −1, −2, −3, −4... to 1, 3, 5, 7....
                                          2
2.2       The Rational Numbers
I’ll give a different argument than the one I gave in class. Let Lq denote the
finite list of all rational numbers between −q and q that have denominator
at most q. There are at most q(2q + 1) elements of Lq . We can make the list
L1 , L2 , L3 , ... and throw out repeaters. This makes a list of all the rational
numbers. As above, we define f (p/q) to be the value of k such that p/q is
the kth fraction on our list.
2.3       The Algebraic Numbers
A real number x is called algebraic if x is the root of a polynomial √   equation
c0 + c1 x + ... + cn xn where all the c’s are integers. For instance, 2 is an
algebraic integer because it is a root of the equation x2 −2 = 0. To show that
the set of algebraic numbers is countable, let Lk denote the set of algebraic
numbers that satisfy polynomials of the form c0 +c1 x+...+cn xn where n < k
and max(|cj |) < k. Note that there are at most k k polynomials of this form,
and each one has at most k roots. Hence Lk is a finite set having at most
k k+1 elements. As above, we make our list L1 , L2 , L3 of all algebraic numbers
and weed out repeaters.
2.4       Countable Unions of Countable Sets
Lemma 2.1 Suppose that S1 , S2 , ... ⊂ T are disjoint countable sets. Then
S = i Si is a countable set.
Proof: There are bijections fi : Si → N for each i. Let Lk denote the set of
elements s ∈ S such that s lies in some Si for i < k, and fi (s) < k. Note that
Lk is a finite set. It has at most k 2 members. The list L1 , L2 , L3 ... contains
every element of S. Weeding out repeaters, as above, we see that we have
listed all the elements of S. Hence S is countable. ♠
     The same result holds even if the sets Si are not disjoint. In the general
case, we would define
                                            k−1
                                   ′
                                                Si ,
                                  Sk = Sk −
                                            i=1
and apply the above argument to the sets S1 , S2 .... The point is that Si′ is
                                                 ′   ′
countable, the various S sets are disjoint, and i Si = i Si′ .
                             ′
                                          3
3     Examples of Uncountable Sets
3.1     The Set of Binary Sequences
Let S denote the set of infinite binary sequences. Here is Cantor’s famous
proof that S is an uncountable set. Suppose that f : S → N is a bijection.
We form a new binary sequence A by declaring that the nth digit of A is
the opposite of the nth digit of f −1 (n). The idea here is that f −1 (n) is some
binary sequence and we can look at its nth digit and reverse it.
    Supposedly, there is some N such that f (A) = N. But then the Nth
digit of A = f −1 (N) is the opposite of the Nth digit of A, and this is a
contradiction.
3.2     The Real Numbers
Let R denote the reals. Let R′ denote the set of real numbers, between 0 and
1, having decimal expansions that only involve 3s and 7s. (This set R′ is an
example of what is called a Cantor set.) There is a bijection between R′ and
the set S of infinite binary sequences. For instance, the sequence 0101001...
is mapped to .3737337.... Hence R′ is uncountable. But then Lemma 1.3
says that R is uncountable as well.
3.3     The Transcendental Numbers
A real number x is called transcendental if x is not an algebraic number.
Let A denote the set of algebraic numbers and let T denote the set of tran-
scendental numbers. Note that R = A ∪ T and A is countable. If T were
countable then R would be the union of two countable sets. Since R is un-
countable, R is not the union of two countable sets. Hence T is uncountable.
    The upshot of this argument is that there are many more transcendental
numbers than algebraic numbers.
3.4     Tail Ends of Binary Sequences
Let T denote the set of binary sequences. We say that two binary sequences
A1 and A2 are equivalent if they have the same tail end. For instance
1001111... and 111111... are equivalent.
                                        4
Lemma 3.1 For any binary sequence A, there are only countably many bi-
nary sequences equivalent to A.
Proof: Let Ln denote the set of sequences that differ from A only in the
first n digits. Then Ln is a finite set with at most 2n elements. Now we list
L1 , L2 , L3 .... This gives a list of all the binary sequences equivalent to A.
The rest of the proof is as above. ♠
     Say that a tail end is the collection of sequences all equivalent to a given
one. Note that T is the union of tail ends. Each tail end it a countable
set, and T is uncountable. Hence, there are uncountably many tail ends, by
Lemma 2.1.
3.5       The Penrose Tiles
To each Penrose tiling P we can associate a tail end τ (P ). Recall that there
is an infinite sequence P = P0 , P1 , P2 , ... where Pn is the parent of Pn−1 . In
other words, Pn is obtained from Pn−1 by the grouping process discussed in
class.
     We say that the nth digit of τ (P ) is a 0 if x is contained in a kite of Pn
and a 1 if x is contained in a dart of Pn . We might need to move x slightly
to avoid choosing a point that lies right on a crack. If we replace x by x′ ,
then only the initial part of the sequence changes. So, τ (P ) is well defined.
     By using the subdivision operation, we can produce a Penrose tiling P
that has any τ (P ) we like. Hence, there are uncountably many different
Penrose tilings.
4      A Heirarchy of Infinite Sets
For any set S let 2S denote the set of subsets of S.
Lemma 4.1 There is no bijection between S and 2S .
Proof: This is really a generalization of Cantor’s proof, given above. Sup-
pose that there really is a bijection f : S → 2S . We create a new set A as
follows. We say that A contains the element s ∈ S if and only if s is not a
member of f (s). This makes sense, because f (s) is a subset of S.
                                           5
    Since A is a subset of S, we have A = f (a) for some a ∈ S. If a ∈ A then
a ∈ f (a). But then, by definition, a is not a member of A. On the other
hand, if a ∈ A, then a ∈ f (a). But, again, this is a contradiction. The only
way out of the contradiction is to realize that there can be no bijection f . ♠
    We can start with S0 = N , and recursively define Sn = 2Sn−1 . That is,
Sn is the set of subsets of Sn−1 . Then, the sets S0 , S1 , S2 , ... form an infinite
heirarchy of sets, each one so much larger than the previous one that there
is no bijection between it and the previous one.
    The fun doesn’t stop there. We can define
                                        ∞
                                           Sn .
                                  Σ0 =
                                       n=0
Then, there is no bijection between Σ0 and Sn for any n. The set Σ0 is larger
than all of the sets previously defined. One can now define Σn = 2Σn−1 . And
so on.
                                       6

Revenge of the Countables

Monday, June 25, 2012