Download Section 2.6 Cantor`s Theorem and the ZFC Axioms

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Section 2.6
Cantor’s Theorem
Section 2.6 Cantor’s Theorem and the ZFC Axioms
Purpose of Section: To state and prove Cantor theorem, which guarantees the
existence of infinities larger than the cardinality of the continuum. We also
state and discuss and Zermelo-Frankel axioms and the Axiom of Choice.
Introduction
Cantor must have felt he was on a great adventure. He had found sets
of infinite size, some countable with cardinality ℵ0 and some uncountable with
cardinality c . He then wondered if there were larger sets. That is a larger
infinities than the continuum.
Cantor’s Discovery of Larger Sets
Often little ideas lead to big ideas. We have seen that for finite sets the
power set of a set is always larger than the set itself. For example, the set
A = {a, b, c} has three elements, whereas the power set
P ( A ) = {φ , {a} , {b} , {c} , {a, b} , {a, c} , {b, c} , {a, b, c}}
has 23 = 8 elements. In other words, for finite sets A we have A < P ( A ) .
This prompted Cantor to ask if the same principle was at work for infinite
sets. This question was answered in the affirmative by the following theorem.
Theorem 1 (Cantor’s Theorem) The power set P ( A ) of any set A has a
strictly larger cardinality than the set A .
Proof:
As for many of Cantor’s theorems, the proof proceeds by contradiction.
The assumption that there exists a one-to-one correspondence between A
and P ( A ) leads to a contradiction. We first prove the result when A is
countable, which allows one to get a good visual grasp of the proof. We then
prove the result uncountable sets.
If
A is countable, then without any loss of generality we let
A = = {1, 2,3,...} , our favorite infinite countable set, and assume there exists
a one-to-one correspondence ≈ P ( ) . To get a good feel for the proof, we
list a few elements of P ( ) :
P ( ) = {∅, {3 } , {6 } , {3,5 } , {7 } , {5,9,13 } , {9,11,15,...} ,...}
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Section 2.6
Cantor’s Theorem
and attempt to pair off elements of with elements of P ( ) as shown in
Table 1:
↔
c ()
1
↔
2
↔
{3,5}
{1, 2, 7}
3
↔
4
↔
5
↔
{9,13, 20}
{1,5, 6}
{2,5,11, 23}
Trial Pairing of the Natural Numbers and Its Power Set
Table 1
Note that the numbers 1,3,4 are not members of the subset which they are
paired with, while 2 and 5 do belong to the sets in which they are paired. We
call the numbers 1,3,4 “uncaptured” numbers, and the numbers 2 and 5
“captured” numbers. We now create a “rogue” subset of the natural numbers
(a member of P ( ) ) that is not matched with any natural number, thus
contradicting the fact there is a one-to-one correspondence between and
its power set P ( ) . This “rogue” set is the set of all “unmatched” numbers.
That is
U = {n ∈ : n does not belong to the set it is matched to}
In our correspondences in Table 1, we have U = {1,3, 4,...} .
But U is a subset
of natural numbers so it must be paired with some natural number, say, let’s
say n ↔ U as we have illustrated in Table 2.
↔
c ()
1
↔
2
↔
{3,5}
{1, 2, 7}
3
↔
{9,13, 20}
n
↔
U = {1,3, 4,..., n,...}
n ∈ U or n ∉ U ?
Table 2
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Section 2.6
Cantor’s Theorem
We now ask the question? Is n ∈ U = {uncaptured numbers} ?
If you say n ∈ U
that means n is an uncaptured number, but n ↔ U = {1,3, 4,..., n,...} that means
n is captured or n ∉ U , which contradicts n ∈ U . On the other hand if you
say n ∉ U that means n is an captured number, but n ↔ U = {1,3, 4,...} which
means n is uncaptured or n ∈ U , which contradicts n ∉ U . In either case a
contradiction. Hence, we conclude that ≠ P ( ) . ≠ P ( ) . But we can’t
have P ( ) < since the power set P ( ) contains the singletons
{m }
of
members m ∈ , and these singletons form a “copy” inside P ( ) of elements
of , and so the only remaining possibility is < P ( ) , which proves
Cantor’s theorem for countable sets.
▌
Uncountable Sets:
The proof for uncountable sets is the basically same as
for countable sets; we just don’t have the nice pictures to help with our
understanding. Let f be any function from a set A to its power set P ( A ) .
We show that f is not onto the power set by exhibiting a subset of A not in
the range of f . The subset we find is our rogue set discussed earlier:
U = { x ∈ A : x ∉ f ( x)} ⊆ P ( A ) .
To show U is not in the range of f , suppose U is in the range of f . This
means that for some y ∈ A , we have f ( y ) ∈ U . Now ask whether y ∈ U or
y ∉ U . If y ∈ U then y ∈ f ( y ) , but that implies, by definition of U , that y ∉ U .
On the other hand if y ∉ U , then y ∉ f ( y ) and therefore y ∈ U . In both cases
whether y ∈ U or y ∉ U we arrive at a contradiction. Hence, we cannot
assume that f maps A onto its power set. Hence, Cantor’s theorem is
proven for an arbitrary infinite sets1.
▌
Margin Note: Cantor presented the proof of what we now call Cantor’s
Theorem in 1891 Űber eine elemenare Frage der Mannigfaltigkeitslehre,
which was the first appearance of Cantor’s Diagonal argument, used in the
proof that the real numbers are uncountable.
Summary: Since the power set of a set, finite or infinite, always has more members
than the set itself, it is always possible to find a set with larger cardinality by simply
taking the power set of a given set. Thus, one can obtain a sequence of sets having larger
and larger infinities ad infinitum
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We set is interpreted using the Zermelo-Frankel axioms of set theory. We have seen that logical
paradoxes arise when sets are chosen without regard to any conditions.
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Section 2.6
Cantor’s Theorem
.
(
)
ℵ0 = < P ( N ) < P ( P ( ) ) < P P ( P ( ) ) < all uncountable except for smallest infinity ℵ0 .
Cantor called these infinite numbers
transfinite numbers,
numbers which means numbers larger than the finite numbers. Since
Cantor’s theorem implies that for every set there is a greater set, from which it
follows there is not set of all sets. That is, there is no set of everything2.
Now that we know the power set of the natural numbers is larger than
the natural numbers itself, just how big is it? The following theorem answers
that question.
Theorem 2 ( c = P ( ) ) The cardinality of the power set of natural numbers is equal
to the cardinality of the real numbers. That is c = P ( ) .
Proof
Since the real numbers and the open interval ( 0,1) both have the same
cardinality ( c ), we find a one-to-one correspondence P ( ) ≈ ( 0,1) . We first make the
observation that any real number x ∈ ( 0,1) can be expressed in binary decimal form
x = 0. b1b2b3 where each b j is 0 or 1. For example
x = 0.1101 =
1 1 0 1
13
+ 2 + 3 + 4 + =
2 2 2 2
16
We now associate to each real number x = 0. b1b2 the subset of natural numbers
consisting of the indices j for which b j = 1 . For example, we associate
x = 0.1101 ↔ {1,3, 4,...}
x = 0.011011 ↔ {2,3,5, 6,...}
x = 0.11101 ↔ {1, 2,3,5,...}
.
which gives a one-to-one correspondence ( 0,1) ≈ P ( ) .
▌
We denote 2ℵ0 to stand for the cardinality of the power set of the natural
numbers, which from Example 1 we have seen is the cardinality of the continuum, i.e.
c = 2ℵ0 .
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The great German mathematician David Hilbert said in 1910 that "No one shall drive us from the
paradise which Cantor created. Later in 1926 Hilbert is quoted as saying "It appears to me to be
the most admirable flower of the mathematical intellect and one of the highest achievements of
purely rational human activity."
Section 2.6
5
Cantor’s Theorem
If we now call ℵ1 the next larger infinite number after ℵ0 , an interesting
question arises. We know ℵ0 is the smallest infinity and that the numbers have a larger
cardinality c , but are we sure there isn’t an infinite number between the two. In other
words, is there some mystery set (or sets) M that have cardinality larger than ℵ0
but less than c . In other words ℵ0 < M < c .
Stated another way, is c = ℵ1 ?
Cantor believed the assertion was true but was never able to prove or disprove
it. Cantor called the hypothesis c = ℵ1 the continuum hypothesis and went to his
death not resolving the question.
A proof of the continuum hypothesis would confirm that the continuum c
is the smallest uncountable set and would bridge the gap between the countable
and uncountable. After Cantor’s death, due to the paradoxes of Bertrand Russell
and others, various logicians, such as Ernst Zermelo and Abraham Fraenkel,
placed the study of sets on a firm foundation with the introduction of a set of
axioms. In 1938 under the framework of these axioms, the Austrian logician
Kurt Godel (1906-1978) proved that the continuum hypothesis can safely be
assumed true since he proved it can’t be proven false3 . This means that it can be
added as an axiom to the existing axioms of set theory without introducing any
inconsistencies that were not already present. Although this discovery was
significant, it was not the last word since he didn’t prove the hypothesis was
true, nor did he prove it was independent of the axioms of Zermelo-Fraenkel
(ZF). This later question was resolved in 1963 by the American logician Paul J.
Cohen who proved that the continuum hypothesis is independent of the axioms
of set theory, the net result being that Cantor’s continuum hypothesis c = ℵ1 is
undecidable,
undecidable meaning that anyone can accept it as true or accept it as false. If
taken as a true axiom (along with the Zermelo-Fraenkel axioms), the resulting
theory is called Cantorian set theory,
theory while if assumed false, the theory is called
nonnon-Cantorian set theory).
theory In either case, one has a valid set of axioms, albeit
much different. Very strange indeed4!
The generalized continuum hypothesis,
hypothesis also proposed by Cantor, has
piqued the interest of set theorists and logicians for 100 years. It states that
for any infinite set A there is no infinite set whose cardinality is between the
cardinality of A and the cardinality of its power set P ( A ) . Symbolically, it
states
ℵα +1 = 2ℵα , α = 1, 2,...
Margin Note: Cantor’s theory does test one’s intuition, knowing his theory is
able to prove that a line segment a thousand light years in length contains the
3
4
In other words consistent with the Zermelo Fraenkel axioms.
Most logicians accept Cantorian set theory. .
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Section 2.6
Cantor’s Theorem
same number of points, not one more or one less, than a line the width of an
electron.
Hmmmmmm Since there are a finite number of letters in the alphabet, one can
show the number of possible names for things is denumerable. But there are
sets that are uncountable, which means there are sets which cannot be named.
Example 2 (Cardinality of Sequences)
Show that the set of all sequences of 0’s and 1’s has cardinality c .
Solution
Consider two the typical sequences of 0’s and 1’s
Sequence 1:
Sequence 2:
(100101000...)
( 001011001000...)
where the dots at the end of the sequences mean all remaining members of
the sequences are zero. In other words Sequence 1 has 1’s in the 1st, 4th ,
and 6th position and zeros elsewhere. Sequence 2 has 1’s in the 3rd , 5th, 6th,
and 9th positions and zeros elsewhere. Taking note of the positions of the 1’s,
we can match any sequence of 0’s and 1’s subsets of the natural numbers.
For example, Sequence 1 can be matched with the set {1, 4, 6} and we can
match Sequence 2 with {3,5, 6,9} . Hence, the family of sequences of 0’s and
1’s can be matched in a one-to-one manner with the power set of , i.e.
family of all subsets of natural numbers.
See Figure 1. We will see in
Section 2.6 that the family of subsets of has cardinality of the continuum c .
Sequence of 0’s and 1’s
(1001001000...)
(11100001000...)
( 001000...)
( 000...)
↔
↔
Subset of ↔
{1, 4, 7}
{1, 2,3,8}
{3}
↔
∅
↔
One-to-One Correspondence
Figure 1
Everyday Sets Larger
The question now occurs, are
Larger than the Continuum
there sets familiar to the ordinary mathematician whose cardinality larger than
Section 2.6
7
Cantor’s Theorem
the cardinality of the continuum? One example is given by Theorem 3, which
we state and hint at its proof.
Theorem 3:
The set of all real-valued functions defined on
( 0,1)
has
cardinality larger than c . The cardinality of this set is ℵ2
Proof
Again, the proof is by contradiction, similar to the proof of Cantor’s
theorem. We assume we can match every real number in (0,1) with a realvalued function on ( 0,1) . We then construct a “rogue” function not on the list,
which contradicts the our assumption that such a correspondence exists.
Need for Axioms in Set Theory
The reader should not entertain the belief that Cantor was the first
mathematician to think about sets and their operations, such as union,
intersection and compliment. The concept of a set has been known for
centuries. What Cantor did was to introduce the formal study of sets and in
particular a deep analysis of infinite sets and their many unexpected
properties. Although Cantor produced many deep ideas, his interpretation of a
set was intuitive. That is, to him a set was simply a collection of objects. It
was Bertrand Russell who upset that view of sets with his 1902 discovery,
called Russell’s Paradox,
Paradox which showed that the naïve view of a set as “any
collection of objects” leads to contradictions. Thus, it became clear in order
to have a consistent theory of sets (no contradictions), one must formalize the
study of sets with “rules of the game.” That is, axioms. So for the first time
in the history of humankind, the most basic of all mathematical disciplines was
formulated into a set of “guidelines” for operation.
Russell’s Paradox
Russell’s paradox is the most famous of all set-theoretic paradoxes
which arises in naïve set theory and motivates the need for an axiomatic
foundation for set theory. The paradox was constructed by English logician
Bertrand Russell (1872-1970), who proposed a family of set R consisting of
all sets who do not contain themselves. He then asked whether the family R
was itself a member of R . If R ∈ R , i.e. R contains R , then we have a
contradiction since R is made up of sets that do not contain themselves. On
the other hand if R ∉ R , this is also a contradiction since R contains all sets
that do not contain themselves which means R ∈ R . Hence, we are left with
the contradictory statement
R∈R ⇔ R∉R
Hence, we must “restrict” the meaning of a set from the naïve point of view
which says “a set is an arbitrary collection of elements.”
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Section 2.6
Cantor’s Theorem
ZermeloZermelo-Fraenkel Axioms
The most commonly accepted axioms of set theory are the ZermeloFraenkel5 axioms (ZFC) axioms, developed by German logicians Ernst Zermelo
(1871-1956) and Abraham Fraenkel (1891-1965). Zermelo6 conceived of
providing a consistent set of axioms, like those of plane geometry, which
restricts the wide latitude of Cantor’s interpretation of a set, thus avoiding the
“bad” things, like Russell’s paradox, but allows enough “objects” to be taken
as sets for ordinary use in mathematics. The “C” in “ZFC” refers to the
Axiom of Choice, the most controversial and debated of the ten ZFC axioms.
Although there is no agreed upon names for each of the 10 axioms, and they
are often written in varying notation, we have settled on the following list.
ZermeloZermelo-Fraenkel Axioms7
1. Existence of a Set (sets exist)
There exists a set (albeit one with no elements). That is, there is a set
A such that
( ∀x )( x ∉ A)
Comment: This set is called the empty set and denoted by ∅ . This axiom
says that the whole theory defined by these 10 axioms is not vacuous by
stating at least one set exists.
2. Axiom of Equality (equality defined)
Two sets are equal if and only if they have the same members. That is
for all sets A, B
( A = B ) ⇔ ( ∀x )( x ∈ A ⇔ x ∈ B )
Comment: The Axiom of Equality (often called the Axiom of Extension)
defines what it means for two sets to be equal. The first two axioms say that
the empty set (guaranteed by Axiom 1) is unique.
5
There are other axioms of set theory than the ZFC axioms, such as the Von Neumann-Bernays-Godel
axioms (which are logically equivalent to ZFC) and the Morse-Kelly axioms, which are “stronger” than ZF.
6
Zermelo published his original axioms in 1908 and were modified in 1922 by Fraenkel and Skolem, and
thus today are called the Zermelo-Fraenkel (ZF) axioms.
7
Zermelo’s original axioms were stated in the language of second-order logic: i.e. sets were quantified (like
( ∀A) , ( ∃A) ) in addition to variables.
There are versions of the ZFC axioms that use only first-order
logic, but for convenience we have quantified sets in a few instances.
9
Section 2.6
Cantor’s Theorem
3. Axiom of Sets of Size One (singletons exist) For any a and b there is a
set, denoted by {a, b} , that contains exactly a and b . That is
( ∀a )( ∀b )( ∃A)( ∀z )  z ∈ {a, b} ⇔ ( z = x ) ∨ ( z = y ) 
Comment: If we pick a = b then the axiom implies there exists sets with one
element {a} . Note that the existential quantifier ∃A quantifies a set, which is
not allowed in first-order predicate logic.
4. Axiom of Union of Sets (unions of sets exist) We have unions of sets. For
any collection of sets F = { Aα }α ∈Λ , there exists a set Y = α ∈Λ Aα , called the
∪
union of all sets in F . That is
( ∀F )( ∃A)( ∀x ) ( x ∈ A ) ⇔ ( ∃C ∈ F )( x ∈ C ) 
Comment: From Axioms 3 and 4, we can construct finite sets. (We are making
progress.)
5. Power Set Axiom (power sets exist)
set B (think power set) such that
That is for any set A there exists a
X ⊆ A⇔ X ∈B
6. Axiom of Infinity (infinite sets exists) There exists a set A (think infinite)
such that
( ∅ ∈ A ) ∧ ( ∀x )( x ∈ A )  ⇒ ( x ∪ { x} ) ∈ A
Comment:
This axiom “creates” numbers A = {0,1, 2,3,...} according to the
correspondence: 0 = ∅ , 1 = {0 } , 2 = {1 } , 3 = {2 } , and so on.
From this we
conclude that the set of natural numbers exist.
7.
Correct Sets Axiom (axiom that avoids Russell’s Paradox) If p ( x ) is a
predicate and A any set, then
{ x ∈ A : P ( x )}
defines a set, i.e. the set of elements for which the predicate P ( x ) is true.
The key point is that the set
{ x ∈ A : P ( x )}
is a subset of a previously defined
set A , which rules out sets of the form
{ x : x ∉ x}
which is the Russell
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Section 2.6
Cantor’s Theorem
paradoxical set. Axiom 7 is what is called an axiom schema, which means it is
really an infinite number of axioms, one for each predicate P ( x ) .
8. Image of Sets are Sets The image of a set under a function is again a set.
That is, if A, B are sets and f : A → B is a function with domain A and
{
}
codomain B , then the image f ( A ) = y ∈ B : y = f ( x ) is a set. (A function can
be defined in terms of sets as the set of ordered pairs
{( a, b ) ∈ A × B : f ( a ) = b}
9. Axiom of Regularity (no set can be an element of itself) Every non-empty
set has an element that is disjoint from the set. That is for any set
( ∀A ≠ ∅ ) ⇒ ( ∃x ) ( x ∈ A ) ∧ ( x ∩ A = ∅ ) 
Comment: As an example if A = {a, b, c} then we can pick our element to be
" a " where we observe a ∩ {a, b, c} = ∅ .
a ∩ {a, b, c} = ∅ .
Note
{a} ∩ {a, b, c} = {a} ≠ ∅
but
What this axiom accomplishes is to “keep out” sets that
contain themselves, like A = {1, 2,3, A}
10. Axiom of Choice
Given any collection of non-overlapping, nonempty
sets, it is possible to choose one element from each set.
Comments: This is an existence axiom that claims the existence of a set
formed from picking one element from a collection of non-empty sets. The
objection to this axiom lies in the fact that the axiom provides no rule for how
the items are selected from the sets, simply that it can be done.
Comments on the
the Axiom of Choice
A group of mathematicians are attending a buffet dinner and while they
pass through the dessert line containing plates full of different kinds of
cookies, one mathematician says she will appeal to the Axiom of Choice and
selects one cookie from each plate. That’s it, that’s the Axiom of Choice.
The Axiom of Choice says given a family of nonempty sets (plates of cookies)
it is possible to select one member from each set. So what’s the big fuss over
AC? It seems so trivial.
But what happens if there are an infinite number of plates, maybe
uncountable? Of course if each plate has a “biggest” cookie, there is an
obvious way to pick one cookie from each plate. Pick the biggest cookie. In
Section 2.6
11
Cantor’s Theorem
that case one doesn’t have to apply the Axiom of Choice. But what if the
cookies on each plate look exactly alike, then how do you go about selecting a
cookie from each plate? What is your rule of selection? Can you write it on a
piece of paper? Of course not. However, there is a rule and we know this
since the Axiom of Choice says so. The Axiom of Choice is a pure existence
axiom and for that reason it is a bone of contention for many mathematicians
who prefer a constructive method for selecting the desired set.
There are good consequences and bad consequences proven when
adopting the Axiom of Choice (AC).
A “good” result is the proof of
Tychonoff’s theorem in topology, which states that the “product” of an
arbitrary family of compact topological spaces is compact. On the negative
side, a few “bad” results can be proven if one adopts AC as an axiom (along
with the other axioms of ZF) and one is the existence of a nonnon-measurable set
in measure theory. And last but not least is the rather disconcerting paradox
known as the Banach
Banach Tarski Paradox,
Paradox which states roughly that any sphere
can be decomposed into a finite number of disjoint pieces and then
reassembled as two spheres identical in size to the original. Of course, don’t
try that at home, since the procedure appeals to AC, which means it is a pure
existence result with no constructive way to show how its done. That’s the
nature of results that apply to AC, one arrives somewhere without knowing
how one got there.
Recently, logicians have tried to find a compromise for AC and seek for
new axioms8 which produces “good” theorems but does not lead to unwanted
consequences9.
Bertrand Russell’s Shoe Model for AC
Bertrand Russell once gave an intuitive reason why sometimes the Axiom
of Choice is required and sometimes when it is not. Suppose you are supplied
with an infinite number of pairs of shoes and asked to find a “selection” rule
for picking one shoe from each pair. In this case no Axiom of Choice is
required, since you can simply the left shoe (or right) from each pair.
On the other, suppose you are supplied with an infinite pair of socks and
asked the same question. The socks look alike so you are unable to provide a
“rule of selection” for choosing a sock from each pair. But the Axiom of
Choice comes to the rescue, which says there is such a rule, although not
explicitly stated, which specifies how to select the socks. It is the non
constructive aspect of the axiom that causes angst to some mathematicians
and logicians. To some pure intuitionists, the word “exists” belongs more to
religion and not mathematics.
8
See Hamilton, A.G. 1982. Numbers, Sets and Axioms, Cambridge University Press, Cambridge.
As the American geometer Oswald Veblen once said, “The test of any axiom system lies in the theorems
it produces.”
9
Section 2.6
12
Cantor’s Theorem
Axiom of Choice ⇔ Well-Ordering Principle
When Zermelo introduced the Axiom of Choice, most mathematicians
accepted it and never gave it a second thought10. However, in 1905 Zermelo
proved a theorem that gave everyone second thoughts. He proved that the
Axiom of Choice was equivalent to what came to be known as the WellWellOrdered Principle.
Principle We are getting ahead of ourselves since we don’t introduce
order relations until Chapter 3, but roughly a set is wellwell-ordered if there is an
“ordering” of elements of the set in such a way that every non-empty set has
a smallest element. For example the natural numbers = {1, 2,3,...} are wellordered by the usual 1 < 2 < 3 < The Well-Ordered Principle says that any
non-empty set can be well-ordered by some order relation. As an example
the principle says a well-ordering exists for the real numbers, although no
one has ever exhibited one, which leads one to suspect the axiom as false.
Although intuitively the axiom seems hard to believe, Zermelo proved it is
equivalent to the Axiom of Choice, which seems obvious. It is a rather
strange situation and has caused logicians a great deal of thought.
10
However, many of the foremost mathematicians of the day did object to the axiom, including measure
theory analysts Henri Lebesgue and Emile Borel.
Section 2.6
13
Cantor’s Theorem
Problems
1.
Prove that if we add a member to a denumerable set, we still have a
denumerable set.
2.
Show that the set of all sequences of 0’s and 1’s has cardinality c . Hint:
Relate each sequence of 0’s and 1’s to a subset of natural numbers and
then use Cantor’s theorem.
3. Show that the set of all one-to-one correspondences between and is c .