Curiously, set theory arose in the context of sets of real numbers related to
the convergence of Fourier series...
In 1829, Dirichlet had shown that
a function always had a Fourier series
converging to itself, under certain sufficient
conditions (he considered periodic functions with finitely many extrema in
every period and equal to the half-sum of their lower and upper limits at every point).
Georg Cantor (1845-1918) got his doctorate in 1867
under Karl Weierstrass (1815-1897).
In 1870, the young Cantor proved that the only trigonometric series which converges
to zero everywhere is the zero series.
In 1871, he introduced the concept of a U-set (or set of uniqueness) with
the characteristic property that the zero series is the only trigonometric series
converging to zero outside of any given U-set.
He proved that any finite set is a U-set.
In 1872, Cantor introduced the concept of
limit point and he properly defined real numbers
as equivalence classes of rational Cauchy sequences.
Then, he introduced the derived set E' of a set E as the
set of its limit points. He established that a set whose derived set is a U-set is
also a U-set.
Cantor went on to prove that every set with countably many limit
points is a U-set.
Ernst Zermelo (1871-1953)
considered that analytical result to mark the actual birth of modern set theory,
as introduced on this page.
(2014-01-21)
Aristotelian logic
For over two millenia, the framework of logic was rigid and frozen.
Aristotle (384-322 BC)
showed how to deduce
categorical propositions
from each other. In a typical syllogism,
a conclusion is derived from two premises. For example :
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
The pattern seems foolproof, and it is. Yet, difficulties may arise:
Rare things are not cheap.
A cheap horse is a rare thing.
Therefore, a cheap horse is not cheap.
As discussed below, the silly conclusion is not necessarily
invalid (as it merely implies that there's no such thing as a cheap horse,
which isn't so objectionable in this context).
Nevertheless, clarifications are needed...
Such threats to logic from circularities in the premises bothered
Gottlob Frege (1848-1925) as he seeked to put modern logic
on solid foundations.
As he thought he had done so, Frege was proofreading his work, just before
going to press, when he received a letter from his younger
colleague Bertrand Russell (1872-1970)
who pointed out to Frege what's now called Russell's paradox.
To his credit, Frege saw immediately that his work could not be fixed
and he did what he could to apologize to his readership.
Russell's objection was a fundamental one.
The consequences for the very foundations of mathematics are still with us. Read on...
(2014-02-11) Formal Logic
The basis for reasoning
(violated in the realm of quantum physics).
A system of axioms is consistent if, and only if, at least one statement is not provable in it.
Although natural language can be used for reasoning, it's notoriously
unreliable in its rigor, or lack thereof
(pay close attention to any debate involving a religious apologist to illustrate this point).
As with the rest of mathematics, a formal language is needed
for logical statements.
That language should be unforgiving enough to make mistakes detectable.
Definitions are equivalences involving newly named concepts.
The formal counterparts of Aristotelian categories are called sets.
A set can be envisioned as the collection of all things that belong to it
(something that belongs to a set is said to be a member of that set).
Two sets that have the same members cannot be distinguished and are
thus considered identical
(that's the principle of extensionality).
Surely, you want to consider sets themselves as legitimate objects of study
and allow sets whose members are sets.
However, the freedom to do so cannot be completely unrestricted,
as Bertrand Russel first discovered.
There's a need for a theory of sets
which spells out what is required to avoid subtle or not-so-subtle contradictions
in our discourse.
(Ashlee of Braithwaite, LA. 2000-05-03twice)
There is a barber who lives in a small town.
The barber shaves all those men and only those men who do not shave themselves.
Does the barber shave himself?
Stay away from the many "cute" answers that do not really address the question:
The barber can't be
a woman (otherwise the term "himself" used in the question would be improper).
It would also be cheating to consider that the barber is a boy (and therefore not a "man")
or any other kind of non-human male creature for that matter...
The dilemma remains (it's not a paradadox, as we shall see):
If the barber doesn't shave himself then he shaves himself.
If he shaves himself, then he doesn't shave himself.
The answer to this classic "problem" is simple:
There cannot possibly be any such barber!
The fact that we are wrongly told at the outset that such a barber exists
just does not make it appear out of thin air in spite of the logical contradictions
his very existence would imply.
This is no more paradoxical than a silly question like:
There's a number whose square is 4 and whose cube is 24. What is it?
Both this and the barber's "problem" are just plain nonsense.
If the existence of something leads to a contradiction,
the thing is thereby shown not to exist and any statements about it are to be considered
either nonsensical or vacuously true :
Any such barber does shave himself.
Any such barber does not shave himself.
Any such barber has purple hair.
The name of any such barber is Isaac Newton.
All of this is true.
Vacuously so
(ex falso quodlibet).
Russell's Paradox:
More seriously, it is worth pointing out that a fundamental logical
dilemma had to be resolved the same way by the pioneers of set theory:
In what's known as Russell's paradox, we are asked to consider
the "set" of all sets that are not members of themselves.
Is it a member of itself or not?
Neither answer is acceptable...
Therefore, such a thing can't possibly be called a set!
This paradox was instrumental
in clarifying the (Zermelo-Fraenkel) axioms of modern set theory:
A "set" cannot be just anything we like.
In particular, the axioms imply that no set can be a member of itself.
Therefore, there is no "set of all sets" (it would be a member of itself).
The collection of sets that are not members of themselves thus includes all
sets and it is not a set itself.
That's the way out of Russell's paradox :
"such a set" simply does not exist, exactly like "such a barber"
cannot possibly exist in the barber's dilemma.
Meaningless Statements in Natural Languages:
A related issue about "natural" languages is that there may be
inherently meaningless statements or descriptions in a language
that's allowed to refer to itself.
For example,
"the integers that can be specified in twenty English words or less"
cannot possibly describe any unambiguous set of integers.
Otherwise, there would be such a thing as
"the smallest integer that cannot be specified in less than twenty English words",
which would thus be specified in only 13 English words...
This goes to show that natural or formal languages may include self-contradictory sentences,
not only when they refer to themselves (e.g. "This sentence is false.")
but also when they include other references to the language being used.
Bertrand Russell coined the term
Berry paradox for sentences of this latter type,
because of a letter he received from G.G. Berry (in 1906)
introducing the earliest such paradox by discussing, among transfinite ordinals:
The first ordinal that cannot be named in a finite number of words.
Even if you don't know what an ordinal is,
you can tell that the above phrase must be meaningless.
Its meaning, if it had any, would be an ordinal.
However, by assuming that such a meaning exists, we see that another
ordinal must be described instead.
Thus, the above can't possibly describe any specific ordinal.
There's actually no paradox here; just a lack of meaning.
(Melissa of Denver, CO.
2000-11-06)
What is infinity and what does it mean?
(Nagaraj of Anamoose, ND. 2000-11-18)
What is the definition of infinity? Please explain to me in details.
Albert Einstein once said that
we can't solve problems by using the same kind of thinking we used
when we created them.Large
numbers begot the concept,
but they are of no help in the definition of infinity.
Let's try common sense first:
A finite set has n elements, for some nonnegative integer n.
Of course, any set which isn't finite in this sense may be called infinite.
Although this common sense approach is perfectly sound, it would seem that
the basic distinction between finite and infinite sets ought not
to depend explicitely on the properties of something
as complicated as the integers.
A more fundamental approach was first suggested by
Dedekind,
who turned a perceived paradox about infinite sets
into a natural definition of infinity:
Only with a finite set are all injections
from the set to itself surjective.
An infinite set is equipollent to one of its proper subsets.
A subset
is a set contained in the set being considered
(i.e., all elements of one are elements of the other).
A proper subset is a subset
not equal to the entire set.
The empty set is a proper subset of any set but itself.
Two sets are said to be equipollent
when there's a one-to-one correspondence between the elements of one and the
elements of the other.
At the most fundamental level (which may not be the most intuitive one)
the existence of infinite sets is built into the
axioms of modern set theory.
Consider the set N of natural integers
{0,1,2,3,4,5,6,...}. One of its proper subsets is the set of
all counting numbersN* = {1,2,3,4,5,6,7,...}.
Yet, there's a trivial one-to-one correspondence betweeen those two
(the function y = x+1).
There's also a one-to-one correspondence between N and the even numbers,
between N and the squares, etc. (This basic observation was made by
Thabit ibn Qurra before 900 AD.)
N is thus an example of an infinite set.
The same thing cannot be done with any finite set:
{1,2,3,4,5,6,7} cannot be put in a one-to-one
correspondence with any set of fewer than 7 elements.
Nowadays, a set equipollent to a proper subset is dubbed Dedekind-infinite.
Tarski's definition of infinity :
The problem with Dedekind's definition of infinity is that
a proof of its equivalence with the previous definition
(involving integers) requires the
axiom of choice
(or, at least, the weaker Axiom
of countable choice )
which is normally reserved for advanced nonconstructive proofs.
It's undesirable in a context where natural integers
themselves are considered too elaborate.
There are no such problems with the following definition, proposed in the doctoral dissertation
(1924)
of Alfred Tarski (1902-1983).
It's based on the notion of a minimal element in a family of subsets.
(A minimal element is a set
not containing another set from that family. See below.)
Tarski's Definition of Finite Sets (1924)
A set E is finite if, and only if, every nonempty family
F of subsets has a minimal element. Formally:
"F Í 2E ,
( F ¹ Æ )
Þ
( $xÎF ,
"yÎF ,
y Ë x )
That clever definition is simply perfect from a theoretical standpoint
(although it certainly doesn't qualify as an intuitive one).
The above goes to show how fundamental the concept of an extremal
point (i.e., minimal or maximal) really is.
It applies to all partial ordering relations,
whether or not elements exist which are (absolute) extrema
(a minimum or a maximum).
This cannot be overstated. Let's clarify the concept we just used
for the fundamental inclusion ordering of set theory:
Minimal and maximal points in a partially ordered set (poset) F :
Considering a set F endowed with a partial ordering relation:
An element is minimal when no other element of F is below it.
An element is maximal when no other element of F is above it.
Infinity as a Limit of Large Numbers
If you allow mentally for the existence of such infinite sets (as you should)
you realize that there's no such thing as the "largest" integer.
The process of building larger integers is never ending.
That's what infinity is all about.
This is like the discovery game children play when trying to name the
"largest" number.
Sooner of later, one of them realizes that the game cannot possibly
be won by either player:
Regardless of what number is named, it's always possible to say something
like "twice that" (or "this plus one") and not concede.
Mathematicians routinely study things whose infinite versions turn out to be
much simpler than the finite ones.
One example is the following sum (properly called a power series):
This sum is equal to 1-2-n when carried out only to its n-th term.
It's simply equal to 1 if all of the infinitely many terms are added up.
When the ancient Greeks were still wrestling with the concept of infinity,
the above sum was underlying one of
Zeno's paradoxes :
Before an arrow reaches its target, it must first travel
half of the distance to it (1/2), then half of what's left (1/4),
half of what's left after that (1/8), and so forth.
Although there are infinitely many such "steps", the arrow does reach its target. (Try!)
(2012-06-15) The Peano axioms (1889)
Defining the set of the natural integers.
The five Peano Axioms are:
Zero is a natural integer.
Every natural integer has a successor in the natural integers.
Zero is not the successor of any natural integer.
Two different natural integers have different successors.
A set which contains zero and the successor of every element in it,
contains all natural integers.
Hermann Grassmann (1809-1877)
was the first person who discussed in print how operations on natural integers could be
defined inductively using only the successor function (1861).
Formal systems of axioms for natural numbers were proposed by
Charles S. Peirce (1839-1914) in 1881 and by
Richard Dedekind (1831-1916) in 1888.
In 1889, Giuseppe Peano (1858-1932)
published his own version of the axioms in a book he wrote in Latin
(Arithmetices Principia). Presumably, he was thus stressing the
timeless nature of the work, at a time when Latin was rarely used in Science.
(2005-07-09)
Ordered Classes of Numbers
From integers to rational, real and surreal numbers.
The most rudimentary numbers are the counting numbers
(1, 2, 3, 4 ...) but it's better to let
the story begin with the natural numbers (0, 1, 2, 3 ...)
although zero
is a sophisticated concept of relatively recent origin.
Alas, the tradition in US education
[1,
2, etc.]
differs from the international conventions used here, following
Bourbaki and others.
In US high-schools, "natural numbers" are often understood to exclude
zero, which makes the term synonymous with "counting numbers".
The dubious convention within the US educational system that makes
"whole numbers" include zero but exclude negative integers
is best forgotten by anybody who wants to be understood outside of the US.
You may talk about "positive integers" or "nonnegative integers" instead.
Negative integers originated after the fractions and
constructible numbers which result from the application
of classical rules (compass and straightedge)
to Euclidean geometry.
Algebraic numbers were also considered before negative numbers were accepted
(along with complex numbers, during the Renaissance).
Real numbers were first formally defined
(as cuts over the rationals)
by Richard Dedekind (1858)
to close all possible "gaps" in the number line.
This is to say that whenever the real numbers are divided into
two sets L and R such that no element of L exceeds
any element of R, there's one element at their common boundary.
(This need not be the case with rational numbers. For example,
there is no rational number at the
boundary between the sets L and R
of rational numbers whose squares are respectively less than 2
and more than 2.)
Archimedes attributes to
Eudoxus
what's now called the Archimedean
property of ordinary numbers:
There's always an integral multiple of any given positive quantity
that exceeds any other prescribed quantity.
The surreal numbers, mentioned above,
are not Archimedean, as they include infinitesimal positive
quantities whose product by any integer can't exceed a
number like 1
(likewise, no integer exceeds an infinite surreal quantity).
Totally-Ordered Number Classes
(each one contains those below it)
ciderspider
(Mark Barnes, UK.
2000-10-11) The Number Line
How do you prove that there are more real numbers than rational ones?
Any infinite set which may be put in a one-to-one correspondence with the
integers is said to be countable (or to have cardinality
Ào).
Infinite sets that are not countable are said to be uncountable.
As shown below, real numbers are not countable,
while rational numbers are.
There is some disagreement about the meaning of the adjective "countable"
when applied to sets which are not known to be infinite.
As the term is clearly a mathematical one, it should have the
inclusive meaning normally implied by
mathematical discourse.
In other words, finite sets are countable too.
However, some authors simply won't apply
the term "countable" (or its synonyms:
"denumerable" and "enumerable") to finite sets...
To avoid misunterstandings, you should either clarify things at the outset
(in Numericana, I consider finite sets to be countable)
or use unambiguous locutions, like "countably infinite" or
"countable or finite" (although the latter could be construed as pleonastic).
The uncountability of real numbers :
The diagonal argument of
Georg Cantor (1845-1918) is the remark that,
with any infinite "list" of real
numbers between 0 and 1 (such a "list" is in one-to-one correspondence with the integers),
an unlisted number may be constructed by choosing for its N-th decimal
a digit that's
different from the N-th decimal of the N-th number listed.
To avoid the problem presented by numbers having
two decimal representations
(ending with infinitely many zeroes or infinitely many nines)
never choose a 0 or a 9. This still leaves 7 valid choices.
The number so constructed is not in the list,
since it has only one
decimal expansion and differs from any listed number in at least one decimal digit.
Thus, any list of reals between 0 and 1 fails to contain all
the reals between 0 and 1.
Therefore, the real numbers between 0 and 1 are uncountable.
This is one of the simplest and most beautiful proofs of all times !
Its elegance is often spoiled
by presenting it as a proof by contradiction,
starting with something like: "Suppose there was such a list..."
It's related to the superb proof of Cantor's theorem which
is even simpler and purer, albeit far more abstract.
The countability of rational numbers :
Now, to prove that there are more reals than rationals, it suffices
to show that rationals may be put in a list.
There are neater ways to do so, but we'll describe only a
simple-minded scheme here:
At coordinates (p,q) of a whole planar grid, put the value p/q
[use zero as placeholder, if q is zero].
Starting at the center,
run through the grid along a square spiral like the one pictured at right.
Put in a growing list each ratio you encounter for the first time as you travel
this way through all possible fractions...
Any given rational will eventually appear in that list,
which is another way to say that all of them belong to the list.
(2000-10-11)
Cantor's Ternary Set
Some vanishing sets of reals have as many points as the whole line.
The probability is zero that a randomly chosen real number is rational.
This statement (which could be made precise in the proper context
of Lebesgue measure theory)
holds for the set of rational numbers or
any other countable set.
However, not all sets of numbers with zero probability
(or, rather, zero measure in the sense of
Lebesgue) are countable.
It's easy to describe a set of numbers which has zero probability but yet contains
just as many elements as the whole set of real numbers
(which isn't countable).
One such example is the ternary Cantor set
obtained by removing from a segment
(of numbers from 0 to 1, say) its "middle third" and similarly removing the middle third
of any smaller segment that shows up in this (infinite) process.
Cantor's ternary set, introduced by Cantor in 1883, was first considered by
Henry Smith
in 1875. It's thus the earliest mathematical example of
a fractal endowed with self-similarity,
since it actually consists of two copies of itself one third as big.
Its fractional dimension
d (in the sense of Hausdorff) is less than 1:
Equivalently, you may consider the set of all the numbers
(between zero and one) whose decimal expansions contain
only zeroes and nines.
Its probability is zero, in spite of the fact that this set contains
as many elements as the whole number line
(as sequences of zeroes and nines are in one-to-one correspondence with
sequences of zeroes and ones, which are
at least as numerous as the uncountable
set of numbers between 0 and 1).
The fractal dimension of that decimal version of the Cantor set is a
familiar number
(smaller than the above dimension of the ternary set):
d = log 2 / log 10 = 0.3010299956639811952...
A self-similar countable set of reals
must have dimension 0. Example:
For { 1, 1/2, 1/4, 1/8, ... 1/2n ... }
we have (1/2)d = 1 so, d = 0.
(2015-02-14)
The formal language of Set Theory :
Symbols and idioms which can be used in the rest of mathematics.
The rest of this page may involved advanced topics of Set Theory
(the biggest hurdle being the fundamental axioms) which can be skipped
by most readers. However, the language introduced in this section
will be of interest to everybody. It allows many mathematical
statements to be expressed quickly and accurately and its mastery
will definitely help structure the thought process of aspiring mathematicians
at all levels of expertise.
Some of it used to be taught systematically in primary and secondary education
(to avoid highbrow fundamental queezes, what was taught was actually
mostly the "algebra of the parts of a given set E").
Membership : x Î A .
x belongs to set A (it's an element of A).
Inclusion :A Í B .
Every element of A belongs to B.
Equality :A = B .
A and B are included in each other.
Union :A È B
is the set whose elements belong to
Aand/orB.
Intersection :A Ç B
(A inter B). All elements in both
AandB.
Complement :A .
All elements (of E ) not in A.
Powerset : 2A or P(A) .
The set of all subsets of A.
The original symbol for inclusion
was Ì
(so that A Ì A
may serve to express that A is a subset of itself).
To express that A is a proper
subset of B, the unambiguous notation is
A ÌÌ B.
We introduced unions and intersections for two operands, but those
are defined for several operands, possibly infinitely many of them...
Closely related to the above are the standard symbols of modern logic :
Universal quantifier : " x
Whatever follows is true for any x.
Existential quantifier : $ x
Whatever follows is true for some x.
Defining quantifier : $! x
Whatever follows is true for just one x.
Implication : Þ
If the LHS is true, so is the right-hand side.
Equivalence : Û
Either both sides are true or both sides are false.
Disjunction : Ú
At least one side is true.
Conjunction : Ù
Both sides are true ("&" is also used for that).
Negation : Ø
Whatever follows is false.
Needless to say that we may also use comma separators, parentheses or various brackets whenever
the precedence of the above operators needs to be clarified...
Curly brackets are usually reserved to denote sets, in particular with the
very important set-builder notation.
The next order of business for a well-rounded basic vocabulary would be to consider
cartesian products,
relations and
functions, as introduced below.
(2002-06-04)
On the fundamental axioms of Set Theory :
Fraenkel's replacement schema (1922) was added to Zermelo's list.
As Russell's paradox once showed,
we cannot simply call a set anything we like.
The "membership" of a set has to be somewhat restricted;
a set simply cannot be a member of itself (or else we'd run into trouble real fast).
On the other hand, infinite sets must be part of the theory,
if it is to be of any use whatsoever.
So, it is postulated at the outset that
some sets exists which may be put in one-to-one correspondence with a proper part
of themselves.
Axioms of Set Theory
Existence of the Empty Set :
The empty set Æ has no elements.
( A set with at least one element is called nonempty.)
( $ Æ )
( "x )
( x Ï Æ )
Extensionality :
A set is characterized by its elements.
( In particular, there's only one empty set.)
( "x )
( x Î A
Û
x Î B )
Þ A = B
Subset Schema :
Elements of A with property
b form a subsetB.
( This is to justify the set builder
notation: B =
{ x Î A | b(x) }. )
( $ B )
( "x )
( x Î B
Û
( x Î A
& b(x) ) )
Pairing :
Two elements form a set.
( $ A )
( "z )
( z Î A
Û
(z = x) Ú (z = y) )
Union of a set of sets : C =
È
B
BÎA
( $ C )
( "x )
( x Î C
Û
( $ B Î A )
( x Î B ) )
Power Set :
The power set C = 2A is the set
of all subsets of A.
( $ C )
( " B )
( B Î C
Û
B Í A )
Regularity :
Set membership is neither reflexive
( A Ï A ) nor circular.
Every nonempty set has an element with which it has no common element.
( " A ¹ Æ )
( $ x Î A )
( " y Î x )
y Ï A
[Zermelo, 1930]
Infinite Sets : One example is
{ Æ, {Æ}, {{Æ}}, {{{Æ}}} ... }
( $ A ¹ Æ )
( " x Î A )
( {x} Î A )
Replacement Schema :
Any collection equipollent to a set is a set.
To obtain cardinals as large as
Àw ,
Fraenkel and/or Skolem
added this axiom in 1922.
In 1923, von Neumann would show
that the assumption is also indispensable to prove that every well-ordered set is equipollent to an
ordinal.
The axiom of replacement had first been proposed by
Dmitri Mirimanov (1861-1945)
from Switzerland, in 1917.
The above axioms constitute ZF theory whose major variant is ZFC
(with the addition of the axiom of choice).
(2002-06-04)
ZFC is ZF set theory with the Axiom of Choice (AC).
AC is a tricky addition to the Zermelo-Fraenkel (ZF) list of axioms.
More things are recognized as legitimate sets in ZFC than in ZF by itself.
The Axiom of Choice (AC) stated below
postulates that sets need not be explicitely constructible
in the various ways allowed by ZF
but may instead be abstract collections of virtual choices.
It has been shown that,
if Set Theory is consistent without the Axiom of Choice,
then it's also consistent with it.
AC is sometimes perceived as allowing more sets than strictly needed,
which may falsify interesting theorems valid within a lesser realm of sets.
To mitigate that, various alternate theories have been formulated
where AC is replaced by some weaker statement
(which would be a consequence of the full-fledged Axiom of Choice).
A proof using the Axiom of Choice
(or, possibly, one of its weaker counterparts) is usually called nonconstructive,
and is rejected by constructivists.
However, such a proof may serve to demonstrate [even to constructivists]
that the negation
of a nonconstructive theorem cannot possibly be shown to be true within the framework
of the rest of Set Theory (since the Axiom of Choice must be consistent with it)...
Axiom of Choice :
A set may be formed which contains a single element from every
nonempty set of a given family of pairwise disjoint sets.
( " A Í
2E)
( "MÎA "NÎA
MÇN=Æ )
Þ
( $ C Í E )
( " B Î
A-{Æ} )
( $ x Î E )
B Ç C = {x}
More brutally stated:
( "MÎA, "NÎA,
MÇN=Æ )
Þ
( $C,
"B Î
A-{Æ},
$x,
B Ç C = {x} )
Either of the above formulations of the axiom of choice
is known as the Multiplicative Axiom
(that name can be useful to distinguish that from several equivalent statements which
will be discussed shortly).
That original formulation of the Axiom of Choice
was introduced in 1904 by Zermelo to establish that every set can be well-ordered
(a statement which turns out to be equivalent to the axiom of choice).
In 1963, Paul Cohen would show that the Axiom of Choice
is essential to the proof of that "desirable" statement
(i.e., replacing the Axiom of Choice by any weaker
statement would describe a theory of sets in which some sets cannot
be well-ordered).
Incidentally, Zermelo showed that a set is well-ordered if and only if is equipollent
to its cartesian square.
(This allows a very short nontrivial formulation of the Axiom of Choice.)
In spite of its annoying consequences (see below) some
formulations of the Axiom of Choice make it look trivial and unavoidable
(e.g., "the cartesian product of any collection of nonempty sets is nonempty").
Even if one finds the nonconstructive proofs based on the Axiom of Choice
particularly repugnant, it does impose nice constraints on the definitions of key concepts.
For example, in general topology,
the Axiom of Choice is found to be equivalent to
Tychonoff's theorem
(the product of compact topological spaces is always compact,
when compactness and product topology are properly defined).
In that sense, paradoxically, the Axiom of Choice has a
beneficial influence even in the "constructivist" axiomatic framework
(called ZF or Zermelo-Fraenkel) where it is explicitely rejected.
The full set of axioms stated above is usually referred to as ZFC
(Zermelo-Fraenkel with Choice).
Note that ZF or ZFC assert only the existence of mathematical objects that
can be defined as sets.
Every mathematical object so defined, with the sole exception of the empty set,
has some kind of sructure, in the sense that some simpler
mathematical objects belong to it.
This remains true even when convenient names are assigned to basic
objects whose set-theoretical definition is rarely used, if ever, starting with
the natural integers:
As part of ZF or ZFC, the additional assertion is often understood that all
mathematical objects ultimately reduce to the empty set in that way.
That's all there is to the mathematical universe we may talk about.
This is what establishes the validity of the standard method of proof by
structural induction
(also called mathematical induction ).
Nevertheless, it's also possible to postulate the existence of
atoms (such fundamental elements
or urelements are objects,
different from the empty set, to which no other object "belongs").
In the simplest such model, called ZFA (Zermelo-Fraenkel with Atoms)
all atoms are alike but distinct (e.g., there's only one set
of two nonempty elements that are both atoms).
(2013-02-15) The Axiom of Restriction
Ruling out infinite descending membership chains.
In 1917, Dmitri Mirimanov
(1861-1945) discovered that certain models compatible with the axioms of ZF set theory
allow the existence of infinite descending membership chains:
xn+1Îxn
John von Neumann
found this repugnant and proposed his Axiom of Restriction
to rule that out.
Philosophically, the axiom of restriction disallows sets beyond those
created in finitely many applications of the other axioms of ZF.
Interestingly, if the axiom of dependent choice
is postulated instead of the full axiom of choice, then the axiom of restriction
implies regularity (so regularity no longer needs to be postulated as an axiom).
??? There's an injection or a surjection from one given set to another.
Weaker alternatives to the axiom of choice :
Here are a few interesting statements that are undecidable within ZF and
proven within ZFC, although their proofs does not require the full power of
the unrestricted axiom of choice (the corresponding theorem
may be true even if the axiom of choice isn't).
Statements equivalent to some of them have been proposed as weaker
alternatives to the full axiom of choice to complement the rest of the ZF axioms.
König's lemma :
In a infinite connected graph where every node has finite
degree, there is an infinite simple path.
This can be proved using the above
axiom of countable choice.
Also, it's true in ZF for any graph with a well-ordered set of nodes
(in particular, it's true for any countable graph).
Is the following lesser statement a theorem in ZF?
"In an infinite connected graph where every node has finite
degree, there is a simple path of any prescribed length n."
Hahn-Banach theorem :
A restricted linear form dominated by a sublinear form can be extended to the whole space.
Krein-Milman theorem :
A compact convex nonempty set in a locally convex space has an extreme point.
The Hahn-Banach theorem and the Krein-Milman theorem taken together
are equivalent to the full axiom of choice.
Likewise, the Boolean prime ideal theorem and the Krein-Milman theorem taken together
are equivalent to the full axiom of choice.
(2010-05-06)
Are there sets of real numbers that are not measurable?
The Axiom of Choice guarantees it ! (Giuseppe Vitali, 1905)
The Lebesgue theory of integration is based on the notion of
measurable sets of real numbers:
The Lebesgue measure is a nonnegative
real function
m defined on measurable
sets, having the following properties:
It's translation-invariant:
m({x}+A) = m(A)
The measure of an interval
is equal to its length :
m ( [x,y[ ) = y-x
It's countably additive :
The measure of a countable union of pairwise disjoint sets
is the (countable) sum of their measures.
Surprisingly enough, the Axiom of Choice
then implies that there are sets
of real numbers that are not measurable. Here's the proof:
Consider the equivalence relation among real numbers in the interval
[0,1[ whereby two numbers are dubbed equivalent when their
difference is a rational number (i.e., the quotient of two integers).
By the Axiom of Choice, we may form a set W
(called a Vitali set )
containing one and only one number from every equivalence class.
If W was measurable, then every set Wq =
{ (q+x) mod 1 | x Î W }
would be measurable of measure M, for any rational number q.
Well, M couldn't be positive because the
union of finitely many Wq
(they're all pairwise disjoint)
would then have a measure exceeding unity.
M couldn't be zero either, because that would make
the measure of [0,1[ vanish
(as the union of countably many disjoint sets of measure zero).
Therefore, W cannot be Lebesgue-measurable.
Note that the above relies on the
Archimedean property of real numbers.
If we were investigating surreal measures,
that would merely establish that some sets of reals must have
infinitesimal nonzero measures...
(2010-05-06) Should we drop the (full) Axiom of Choice ?
Robert M. Solovay (b. 1938)
has argued in favor of dropping the Axiom of Choice from the standard axioms
of set theory to ensure that every set of real numbers is Lebesgue-measurable.
This could make every linear map on a Banach space continuous
(a so-called dream space ).
H.G.
Garnir (1921-1985)
One possiblity, advocated by the late Belgian mathematician
Henri Garnir
(1921-1985)
is to replace the Axiom of Choice (AC) by the weaker
Axiom of Dependent Choice (DC)
and to postulate that every set of reals has the
Baire property
(which is incompatible with the full axiom of choice).
In that system, every set of reals is Lebesgue-measurable.
(2014-12-04)
Cartesian Products
Binary or multiple cartesian products and infinite ones.
The cartesian product of two sets A and B
is the set, denoted
A ´ B , of all ordered pairs
(a,b) whose first element (a) is
in A and whose second element (b) is in B.
Multiple cartesian products :
This definition is readily extended to the cartesian product of a finite sequence
of n sets, which consists of all the n-tuples formed
with an element from every set, in the stated order.
The cross symbol (´) used to denote cartesian
products doesn't have a fixed arity.
The lack of parentheses is meaningful
(it's not like an associativebinary operator, where parentheses can be omitted or not).
For example:
(1,2,3) is an ordered triple belonging to
´
´
=
3
(1,(2,3)) is an ordered pair belonging to
´
(
´
)
=
´
2
From the highbrow perspective of category theory where new constructions are
defined up to isomorphism, that distinction isn't made and the corresponding
categorial product is indeed a binary associative operator
(at least when all relevant products are well-defined, as would be the case in a cartesian-closed category).
Indexed and/or infinite cartesian products :
Far more generally, we may define the cartesian product of a family of sets
indexed by a set I
as a function which sends every element i of I to
some element of Ei .
When I is finite, this is equivalent to our previous description.
When at least one of the sets involved is empty, the cartesian product is trivially empty.
However, the converse is only true if we postulate the Axiom of Choice
which is equivalent to the following statement
(which could seem obvious to the uninitiated):
The cartesian product of nonempty sets is not empty.
Formal definitions of cartesian products :
The intuitive notion of ordered pairs ought to be
defined first. This allows binary cartesian products to be defined
as above. Building on that, relations and functions can be stated to be subsets
of binary cartesian products (as is done in the next section).
Only then, can the above definitions for multiple or infinite cartesian products be given
in terms of "functions".
Classically, an ordered pairs are awkwardly defined in terms
of the fundamental concepts of set theory:
The pair (x,y) is defined as the set {x,{x,y}}.
(2014-12-04)
Binary Relations
A binary relation between two sets is a subset of their cartesian product.
Technically, a [binary] relation between two sets X and Y is just a
subset of their cartesian product X×Y.
When (a,b) is
in that subset, we say that "a is related to b"
(it's not the same thing as "b is related to a"
since we are talking about ordered pairs here).
Two elements a and b are said to be unrelated
(with respect to R)
when neither (a,b) nor (b,a) belong to R.
The converse relation
(RC ) or transpose (RT ) of a relation R
is also called its opposite or dual (Ro ).
Avoid calling it the inverse of R
(R-1 ) unless there's no risk of confusion with
ubiquitous group inverses which are entirely differerent.
Using the set-builder notation, wa have:
RC = RT =
{ (y,x) Î X×Y : (x.y) Î R }
A subrelation is a subset of a relation. For example,
among positive integers, the relation "divides"
("is a divisor of") is a subrelation of "is less than or equal to".
Likewise, "is a power of" is a subrelation of "divides".
For a relation R, the infix notation a R b
simply means (a,b) Î R.
A relation R from A to B is said to be
entire (or serial)
when any element of the domainA is related to some element of
the rangeB:
" a Î A ,
$ b Î B ,
a R b
Composition of Binary Relations :
If R is a relation from set A to B
and S is a relation from B to C,
then the composition T = S o R
is the relation from A to C defined by:
a T c
Û
{ $ b Î B ,
a R b , b S c }
Readers who have been exposed to this notion for functions
(which are just a special type of relations,
as introduced below) may remark
that the above reduces to the familiar composition of functions, when applicable.
If the relation R is a function,
a R b means the same as b = R ( a )
The peculiar order of the operands in the above definition
is meant to accommodate functional definitions. Indeed,
if R and S are functions, then the above definition translates into:
c =
S o R ( a ) = S ( R ( a ))
The composition of general binary relations (not necessarily functions)
is especially useful to define uniform spaces,
whose completeness can be determined even when
distances are not defined.
Binary Relations within a Set :
A relation R from a set A to itself could verify the following properties.
Reflexivity :
" x Î A , x R x
(thus, reflexive relations are entire).
Anti-reflexivity (or irreflexivity) :
" x, Ø ( x R x )
Symmetry :
x R y Þ y R x
Antisymmetry :
{ x R y & y R x } Þ x = y
Transitivity :
{ x R y & y R z } Þ x R z
Totality (or connexity) :
" x,
" y,
( x R y ) Ú ( y R x )
WARNINGS :
Under this last definition, a connex relation is always reflexive.
Some authors will instead call "connex" a relation for which two different
elements are related either way. Such a thing need not be reflexive.
We prefer to call it semi-connex.
Although a (partial) function is just a special kind of relation,
the idiom "total function" is used only for a function which
is an entire relation (which is more restrictive than
a "total relation" and makes sense even when the domain and range are different).
Those concepts are used to define several important types of relations:
A preorder relation (or quasiorder) is reflexive and transitive.
An equivalence relation is reflexive, symmetric and transitive.
An ordering relation is reflexive, antisymmetric and transitive.
A strict ordering is irreflexive, antisymmetric and transitive.
A total ordering relation is what the name implies, using the
above definitions (it's an ordering relation which is total).
The locution partial ordering denotes an ordering relation which need not be
total (although it may well be, per the usual
inclusive meaning of mathematical terms).
A poset (partially ordered set) is simply a set endowed with a
partial ordering in the above sense. It's usually denoted as a couple
like (E, R) or (P, ≤).
The notion is one of the most fundamental notions of mathematics.
For example, the most fundamental way to distinguish between a finite
or infinite set (due to Tarski, 1924) is expressed as a
property of the poset of the parts of that set with respect to the inclusion
relation: (2E, Í).
A directed set (more precisely, an upward-directed set)
is a nonempty set endowed with a preorder relation (i.e., a reflexive and transitive relation, not
required to be antisymmetric) which is such that two elements always have an upper-bound
(i.e., an element which is greater than or equal to both).
It's also called a directed preorder or a filtered set.
This structure is more general than a upper-semilattice
because the set of all the upper-bounds of two elements may not have a minimal element.
An ordering lattice
(French: treillis)
is a poset where any pair of elements has a unique join
and a unique meet (maximal lower bound and minimal upper bound, respectively).
One example of a lattice is the set of positive integers under the relation denoted
by the symbol "|" (pronounced "divides") whereby x | y means that
y is a multiple of x. The meet of two integers is
their greatest common divisor (GCD, wedge) while their join
is their lowest common multiple (LCM, vee).
(x Ù y)
(x Ú y) = x y
A similar relation holds in a lattice of subsets (under inclusion), where the meet
and join are respectively the intersection (Ç)
and union (È) of sets, if
implicit multiplication is understood
to denote the symmetric difference
D (a.k.a,, disjunctive union).
Well-founded binary relations :
A relation is well-founded in X
when every subset of X has a so-called "minimal element"
(to which no other element of that subset is related):
" A Í X ,
( A ¹ Æ ) Þ
{ $ m Î A ,
" a Î A ,
( a R m ) Þ ( a = m ) }
A set in whose transitive closure
the membership relation is well-founded is said to be a well-founded set.
The Axiom of Restriction
(which isn't part of ZF) postulates that all sets are well-founded.
A relation R whose converse is well-founded is said to be Noetherian :
" A Í X ,
( A ¹ Æ ) Þ
{ $ m Î A ,
" a Î A ,
( m R a ) Þ ( a = m ) }
" A Í X ,
( A ¹ Æ ) Þ
{ $ m Î A ,
" a Î A ,
( a ≤ m ) Þ ( a = m ) }
A well-ordering is said to satisfy the
descending chain condition (DCC).
A Noetherian ordering satisfies the
ascending chain condition (ACC).
(2014-12-04)
Quotient of a set modulo an equivalence relation.
It's the set formed by all the equivalent classes so defined.
When an equivalence relation
R is defined on the set A,
that set can be partitioned into disjoint equivalence classes
such that two elements are related iff
they belong to the same class. Any element of A belongs to
one and only one such class. The empty set isn't an equivalence class.
The set of all those equivalence classes is denoted A/R and is called
the quotient of A by R
(also dubbed Amodulo R).
That's often used to cleverly define a new set from a more elementary one:
For example the set of signed integers
can be construed as the cartesian square of the natural integers
( 2 )
modulo the following relation R :
(a,b) R (c,d)
Û
a + d = b + c
The motivation is to make subtraction a well-defined operation verifying:
(a , 0) - (c , 0) =
(a , c)
This construction is one key reason why it is best to include zero in the set of natural integers
(zero is excluded from
* )
in spite of the dubious tradition prevalent in US high-schools.
It is then a simple (albeit tedious) exercise to show that the set
so constructed is given a ring structure using the
following operations which respect the equivalence R
(i.e., the equivalence class of a result depends only on the equivalence classes of the operands).
(a,b) + (c,d) =
(a + c , b + d)
(a,b) (c,d) =
(ac + bd , ad + bc)
The same approach can be used to form the rationals
as the field of fractions of the
ring formed by the above group
endowed with the usual multiplication
(defined by induction through repeated addition).
Likewise, the field of real numbers
can be construed as the set of rational
Cauchy sequences modulo
Cauchy equivalence.
(2010-09-23)
Functions and applications are special
binary relations. Some
functions from A to B
can be injective, surjective, or both.
A function f from set A to set B is something
that associates one and only one element
f (x) of B with every element x of A.
Formally, functions are special relations between
the elements of A and B,
in the above sense
(namely subsets of A×B).
Thus, a function f is characterized by:
f Í
A ´ B
( " x Î A )
( $! y Î B )
( (x,y) Î f )
By convention, the above is expressed with the following notations:
f :
A
¾® B
x
¾® y = f (x)
Bourbaki calls the above an
application, Other authors may call it a total function,
A partial functionf from A to B is
merely an application from a subset of A
(called the definition domain of f ) to set B.
In numerical analysis, functions are normally understood to be partial ones.
Elsewhere, it's prudent to lift all ambiguities with the systematic use of
qualifiers (either "total" or "partial").
A function f from set A
to set B is said to be surjective
when every element of B is the image of some element
of A.
(The idiom "onto" is synonymous with surjective.)
A function is said to be injective when no element of
B is the image of more than one element of A.
("One-to-one" is synonymous with injective.)
Both terms were introduced by
Nicolas Bourbaki
(the collective pen name of a famous ongoing collaboration of industrious
mathematicians, mostly French, founded on January 14, 1935).
A function that's both surjective and injective is said to be
bijective, or "one-to-one onto".
It's also said to be a "one-to-one correspondence".
Functions that are surjective, injective, or bijective are respectively called
surjections, injections, or bijections.
A bijection may also be called a transformation;
(as in the common locution
linear transformation).
The idioms "one-to-one" and "onto" are often misunderstood by scientists whose first language
isn't english, so both idioms are best avoided in international communications.
Direct and reciprocal images of a set :
For a given function f from A to B,
the following definitions and notations are commonly used:
Direct image (or simply image)
of a subset A' of A :
f ( A' ) =
{ y Î B |
$ x Î A' , y = f (x) }
Inverse image (also called preimage)
of a subset B' of B :
f -1 ( B' ) =
{ x Î A |
$ y Î B' , y = f (x) }
It's useful to know that the [direct] image of an intersection is contained
in the intersection of the [direct] images, but is not necessarily equal to it
(unless the function is injective).
In particular, the images of two disjoint sets aren't necessarily disjoint
(unless the function is injective).
On the other hand, we do have equality for unions of direct images and for
both unions and intersections of inverse images. All told:
f ( X Ç Y )
Ì
f ( X ) Ç
f ( Y )
f ( X È Y ) =
f ( X ) È
f ( Y )
f -1 ( X Ç Y ) =
f -1 ( X ) Ç
f -1 ( Y )
f -1 ( X È Y ) =
f -1 ( X ) È
f -1 ( Y )
The proofs are left as an enlightening exercise for the reader.
One application of those relations pertains to the characterization
of the continuity of a function in terms of
the conservation of certain topological properties of sets for either
their direct or their inverse images...
(2005-07-26)
Cantor's Theorem:
Any set is smaller than its powerset. No
function from A to 2A
can possibly be surjective (Cantor, 1891).
A beautiful proof :
Consider anyfunctionf from set A to its powerset 2A
(by definition, 2A
is the set of all the subsets of A).
Let M be the subset of A consisting of all the
elements x of A such that
x Ï f (x).
Well, M is clearly an element of 2A
which is not the image of any
element x of A.
(HINT: Such an x could be neither in
M nor outside of it.)
Thus, our arbitrary function f is not surjective.
Therefore, there's no surjection from A
to its powerset.
This much seems obvious for finite sets,
but the above proof shows that it holds for all infinite sets as well.
The reader is encouraged to check the above wording for conformity with
the Axioms of Set Theory presented in the
above article,
assuming familiarity with the concept of a function.
The above result is known as Cantor's theorem and it provides
yet another argument (formerly known as the paradox of
Burali-Forti) against the existence of a set of all sets
because the powerset of such a thing could not possibly
be of greater cardinality than itself.
This argument was put forth by
Cesare Burali-Forti
(1861-1931)
in 1896 and it predates the formulation of Russell's paradox
(1901).
Cantor's Paradox (15:33)
by Gabe Perez-Giz (PBS Infinite Series, 2018-03-23).
(2003-11-10)
Ào
and the transfinite cardinals
On the different sizes of infinite sets.
Two different approaches make sense of actual
infinite numbers which were both investigated by
Georg Cantor (1845-1918).
The cardinal numbers are discussed in this section;
ordinal numbers will be considered later.
Equipollence, Cardinality and Cardinal Numbers
Two sets which can be put in one-to-one correspondence
are said to be equipollent.
This is a fundamental notion which does not require an ability to count
the elements in either set; the ability to pair them up suffices.
Loosely speaking, equipollence is an equivalence relation
and we can give a name to each equivalence class.
Such a name is called a cardinality.
Two sets are equipollent if and only if they
have the same cardinality.
WARNING : There's no such thing as the
"set of all cardinalities". This was known to
Cantor
even before Russel pointed out that
there is no such thing as the "set of all sets".
Of course, finite sets, have the same cardinality if and only
if they have the same number of elements.
If you have as many apples as you have oranges,
you may count either the apples or the oranges and you'll
come up with the same number (0 if you have no fruit).
One may thus consider natural integers (0,1,2,3,4 ...) to be
indicative of the
cardinality of either set (apples or oranges).
In that capacity, the natural integers are
called cardinal numbers.
The fancy vocabulary may be intimidating,
but the whole thing is really the most basic kind
of numbers,
only described in a philosophical way which can be extended
to infinite sets...
Can infinite sets be "counted" too?
Well, the finite integers are clearly not up to the task.
But can't we just add one more "number"
(using tentatively the "¥" symbol for it)
and say that "¥" is the number of elements
in any infinite set?
Cantor showed that this won't do,
because there are pairs of infinite sets which cannot
be put into one-to-one correspondence with each other.
To be consistent with the very concept of cardinal numbers presented
above, there has to be more than one "transfinite" cardinal number.
The symbol Ào (pronounced
"aleph zero", "aleph null", or "aleph nought") was introduced by Cantor to
denote the smallest of these;
it's the number of all the [finite] integers.
An infinite set with this many elements is said to be countable.
An uncountable set is an infinite set which is not countable.
Cantor's diagonal argument
shows that the continuum of the real numbers
is uncountable.
Its cardinality (called the power of the continuum)
is the same as that of the powerset of the integers (the set of all sets
of integers)
as can be established directly by observing that
Pierce expansions yield a one-to-one
correspondence between sets of positive integers and the reals between 0 and 1.
By Cantor's theorem,
the powerset of any set has a greater cardinality
than that of the set itself.
Therefore, there must be infinitely many different infinite cardinalities and
infinitely many transfinite cardinals, including, at least, the following
sequence of so-called beth numbers :
Á0 =
À0
; Á1 = C =
2
Á0
; Á2 =
2
Á1
... Án+1 =
2
Án
...
Georg Cantor used the notation
C = 2Ào
for the power of the continuum. and
called Àn+1
the smallest transfinite cardinal beyond Àn
(that definition is valid only if the
axiom of choice is retained).
With that notation, in 1877, he conjectured the continuum hypothesis (CH)
which states that there's nothing between
Ào and C :
À1 = C
The generalized continuum hypothesis (GCH) is the deep generalization:
Cantor could not prove or disprove the [restricted] continuum hypothesis,
Neither could anyone else... because this can't be done,
as discussed next.
(2009-04-11)
The Continuum Hypothesis (CH)
Is every uncountable set of real numbers
equipollent to the whole line?
Cantor thought that any uncountable subset of a continuum
was equipollent to the whole continuum.
He formulated this statement, known as the continuum hypothesis,
in 1877 and it bedeviled him till the day he died.
The reason why Cantor failed to prove or disprove
the Continuum Hypothesis is that it cannot be proved or disproved.
It's undecidable...
The axioms of set theory are equally consistent
if we rule in or rule out the existence of transfinite cardinals strictly between
À0 and C.
This was established in 1963 by
Paul Cohen
(1934-2007) who was awarded the
Fields medal for that work, in 1966.
Some preconditioned versions of CH are true :
In 1884, Cantor observed that an uncountable
closed set of real numbers must have a subset
equipollent to the Cantor set and, therefore,
must be equipollent to the entire real line.
So, the continuum hypothesis holds if restricted to closed sets.
More generally, Felix Hausdorff (1868-1942)
proved, in 1916, that any uncountable
Borel set of real numbers is
equipollent to the real line.
(2020-05-04)
Martin's Axiom
Implied by CH, but consistent with the negation of CH.
(2003-11-10)
w
and Cantor's transfinite ordinals (1897)
Counting to infinity and beyond.
The second kind of transfinite numbers introduced by Cantor
are called transfinite ordinals.
They're more intricate than cardinals.
The key concept is that of order-type,
which only applies to well-ordered sets
(if we reject the Axiom of choice, not all sets can be well-ordered).
Two well-ordered sets are said to have the same order-type if there's an
order isomorphism between them, namely a monotonous bijection
whose inverse is also monotonous.
A function f is said to be monotonous if it respects the orders respectively defined on its domain and codomain:
x ≤ y Þ
f (x) ≤ f (y)
Although Bertrand Russell
dubiously did so in Principia Mathematica,
it wouldn't be acceptable to define an order-type as an equivalence-class,
because an equivalence relation can only be properly defined within a set.
Instead, we define special sets dubbed ordinals canonically
so that there is one and only one ordinal of each order-type.
The modern way to do so was devised by
John von Neumann in 1923
and it's now universally accepted:
Following von Neumann, we may start with the following remark:
A natural integer may be represented by
the set of all nonnegative integers before it,
starting with the empty set ({} or Æ)
for 0 (zero) because there are no nonnegative integers before 0.
Then, 1 corresponds to {0},
2 is {0,1}, 3 is {0,1,2}, etc.
That prompts the modern recursive definition of ordinals as
transitive sets:
Every ordinal is the well-ordered set of all smaller ordinals.
Thus, any member of an ordinal is also an ordinal and a union of ordinals is an ordinal.
So, any set of ordinals has a
supremum
which is just the ordinal obtained as union of all of them.
This implies that there is no set of all ordinals (since such a set would be a member
of itself, which is ruled out by Regularity):
The ordinals form a proper class.
The set of all finite ordinals {0,1,2,3...} is the first transfinite ordinal.
For this, we use the symbol w introduced by
Cantor.
{0,1,2,3...w} is the next transfinite ordinal, which is best
called w+1.
{0,1,2,3...w, w+1} is w+2.
{0,1,2,3...w, w+1, w+2} is w+3, etc.
Addition of Ordinals :
You end up doing two different
types of counting if you enumerate a finite set first and an infinite set second or
the other way around...
The addition of ordinals is associative
but not commutative:
1 + w
= w
¹ w + 1
Incidentally, the maximum number of different results
which can be obtained by changing the order of the terms in a Cantor-sum of n
ordinals may be tabulated as follows.
(The last pair of lines gives the general pattern,
which has exceptions only below n = 15, as indicated by dimmed headers.)
Number of different ways to add n ordinals
(A005348)
n
0
1
2
3
4
k = -3
1
1
2
5
13
n
5
6
7
8
9
k = -2
33
81
193
449
1089
n
10
11
12
13
14
k = -1
2673
6561
15633
37249
88209
n
5k + 15
5k + 16
5k + 17
5k + 18
5k + 19
k ≥ 0
33 ´ 81 k+2
81 k+3
193 ´ 81 k+2
193 2 ´ 81 k+1
193 3 ´ 81 k
Infinite Numbers among Surreal Numbers :
The addition of infinite numbers such as w
may also be defined in other ways which retain commutativity.
The best such approach was proposed by
John H. Conway (1937-2020)
who defined w as
above, but in the context of his surreal numbers,
a very satisfying generalization of the familar concept of number
(including integers, rationals and real numbers ).
Expressions like
w-1,
2w,
w2,
Öw,
ww
www, etc.
have consistent meanings in that superb
construction of John Horton Conway, as presented below.
(2003-11-10)
The Class of Surreal Numbers (John H. Conway, 1972)
What's the most general type of ordered numbers?
The most general type of number line
(as opposed to unordered things like complex numbers
or p-adic numbers) turns out to be
simpler to define than the familiar real numbers.
With the right approach,
it's actually easier to include infinitesimals and transfinite ordinals
than to rule them out...
Conway's approach
was inspired by the aforementioned view of Cantor's ordinals
(Von Neumann, 1923) and also by the "Dedekind cuts" introduced in 1858
by Richard
Dedekind (1831-1916, last doctoral student of
Gauss).
Dedekind defined real numbers as gaps "between" splits of the rational line
(this was introduced before Georg Cantor found it more
convenient to identify real numbers with
equivalence classes of Cauchy sequences).
A Dedekind cut consists of two disjoint sets
L and R whose union is the entire set of rationals.
L does not have a greatest element and contains all rationals to the left of any
member of L, whereas R may
have a lowest element and contains any element to the
right of any member of R.
The mnemonic notations L and R ("left" and "right") are due to
J.E. Littlewood (1885-1977).
The term surreal numbers was coined by
D.E. Knuth
in 1973 to describe the beautiful construct of
John H. Conway,
who gave simultaneously recursive definitions of the numbers
and the ordering among them...
Technical Presentation of Surreal Numbers :
Conway introduced the compact notation x = { L | R }
for what he called a game. Namely,
an ordered pair of two sets L and R, given either by name or
by listing their elements, all of which being (simpler) games.
Conway's notations (used here) involve the
shorthand introduced by Hermann Minkowski:
If L is a set, then L+y is the set of all elements
obtained by adding y to an element of L,
just like E+F is the set whose elements
are sums of an element of E and an element of F.
If pedantically desired, the rest of Conway's conventions could be cast into
the language of set theory by stating that a number may stand for a single-element set
within the first level of a "{...|...}" notation, where a
comma ( , ) is synonymous with the
union operator ( È ).
For a given game x,
typical elements of L or R are respectively denoted xL
or xR and are called left or right options of x.
Numbers are defined as special games whose options are other numbers,
and which lay between their own left and right options,
in the following sense, recursively defined:
If L = { xL } and R = { xR } are two sets of
known surreal numbers, such that xR ≤ xL
is never true, then one representation of another surreal number x
is obtained as:
x = { L | R } = { xL | xR }
x ≤ y means that we never have
yR ≤ x or y ≤ xL
Those two mutual recursions allow the introduction of a fundamentalequivalence relation:
x = y is defined to mean that
x ≤ y and y ≤ x.
The true statement x ≤ x (implying
x = x) requires a nontrivial proof!
So does the transitivity of the "≤" relation.
From that follows the fact that the defined notion of equality is indeed
an equivalence relation (technically, surreal numbers are thus equivalence
classes of their representations modulo that relation).
Finally, we should observe that the construction ensures the antisymmetry
of "≤" which makes it a proper ordering relation.
The following definitions are merely introduced for convenience:
x ≥ y means that y ≤ x.
x < y means that x ≤ y is true and y ≤ x is false.
x > y means that y < x.
Here are the single-line definitions of the basic arithmetic operations:
Sum: x + y =
{ x L + y , x + y L |
x R + y , x + y R }
Opposite of a number: -x =
{ -x R | -x L }
Product: xy =
{
xL y + xyL -
xL yL ,
xR y + xyR -
xR yR |
xL y + xyR -
xL yR ,
xR y + xyL -
xR yL }
Well, nobody said that this was easy to work with, but it sure is beautiful...
Conway himself said that it took him "several weeks of hard work" to find
the above genetic definition of multiplication
(two years after defining addition c.1970) and another year to define division!
Constructing numbers using simpler numbers :
A representation of a surreal number is canonical
when it's formed only with previously constructed simpler numbers.
Let's now construct surreal numbers one generation at a time, using such canonical representations.
The simplest number is the game which has no left option and no right option
(that game is indeed a number because no left option is to the right of a right option).
For a reason which will soon be clear, we call it 0 (zero).
0 = { | }
We can show that this particular number must, indeed, be the
neutral element for addition using
structural induction
on surreal numbers which have a canonical representation (we haven't yet proved that
all of them do).
Indeed, if we apply the above definition of addition, we obtain:
x + 0
=
{ x L + 0 , x + 0 L |
x R + 0 , x + 0 R }
=
{ x L + 0 | x R + 0 }
=
{ x L | x R } = x
Expressions with 0R and 0L
are discarded because there's no such thing as an element of the empty set.
The remaining expressions can be simplified using the induction hypothesis,
which tells us that 0 is neutral for addition with the simpler numbers available
in either set of options of x.
0 ≤ 0 holds (since we can't have
0R ≤ 0 or 0 ≤ 0L ) therefore 0 = 0.
That goes to show that the game { 0 | 0 } is not a number.
The next generation of surreal number includes only two numbers:
-1 = { | 0 }
and
1 = { 0 | }
Indeed, those two are clearly opposite of each other and
we could prove, by structural induction, that the latter is the neutral
element of the multiplication defined above.
(At first, Conway didn't have the luxury to do so!)
The 3 surreal numbers identified so far can form 8 sets
which allow the construction of 4 new numbers, with multiple representations:
The seven simplest surreal numbers (the first three generations) :
Number
Canonical Representations (simplest one first)
Day
2
{ 1 | } { 0, 1 | } { -1, 1 | } { -1, 0, 1 | }
2
1
{ 0 | }
1
½
{ 0 | 1 } { -1 , 0 | 1 }
2
0
{ | }
0
- ½
{ -1 | 0 } { -1 | 0, 1 }
2
-1
{ | 0 }
1
-2
{ | -1 } { | -1, 0 } { | -1, 1 } { | -1, 0, 1 }
2
Any given number also has many noncanonical representations
involving at least one number of the same generation or greater. For example:
2n new numbers are born on Day n,
including two extremities (identified with -n and n)
and other new surreal numbers halfway between the 2n-1
numbers of the previous generations.
In finitely many days we obtain surreal numbers corresponding to all fractions whose
denominators are powers of two, but only those.
Structural Recursion :
The great power of Conway's notations is that they make it easy to specify recursively
new options for a number (or any other game) in terms of old options
(i.e., previously born ones).
For example, the following equation defines x = w ={0,1,2,3,4...| }
which is the simplest infinite surreal:
x = { 0, 1+xL | }
More precisely, that equation means to say that 0 is a left option of x
and that whenever xL is a left option of x, so is
1+xL.
w and -w
are the only infinite numbers born on Day w.
However, that Birthday is shared by all irrational numbers and all rational numbers
not born on a finite Day. That includes 1/3.
(2005-07-09)
Multidimensional (or Hypercomplex) Numbers
Real and complex numbers. Quaternions, octonions, sedenions, etc.
A division algebra over a field
(here, we'll use only the field of reals)
is an algebra
in which every nonzero element has a multiplicative inverse
(this doesn't rule out divisors of zero because no associativity is postulated).
The Cayley-Dickson construction doubles
the dimension of a previously established set of numbers, endowed with an involutive
conjugation (the conjugate of z is denoted z*)
homomorphic for addition and anti-homomorphic for multiplication, which is to say:
x** = x
(x+y)* = x* + y*
(x y)* = y* x*
If multiplication is commutative, the trivial
conjugation (z* º z) is fine.
The construct endows the cartesian square of the base numbers
(i.e., the set of all ordered pairs of the given numbers)
by defining addition, multiplication and conjugation in the following way
(defining conjugation allows the construct to be iterated):
( a , b ) + ( c , d )
=
( a + c , b + d )
( a , b ) ( c , d )
=
( ac - db* ,
a*d + cb )
( a , b )*
=
( a* , -b )
The base numbers are isomorphic to the pairs whose second component vanish
and are thus considered immersed into the composite numbers.
Those two expansions are easily checked to be identical if multiplication is commutative.
Thus, base commutativity implies composite associativity.
Otherwise associativity can't occur, since
two base numbers which don't commute (xy ¹ yx, say)
yield a counterexample to associativity:
Thus, associativity is preserved iff the base numbers are commutative.
Likewise, commutativity is preserved iff
the base conjugation is trivial.
However, the composite conjugation doesn't remain trivial
(except when base numbers have characteristic 2)
and commutativity is therefore broken when the construction is performed again on the composite numbers.
By design, the construction allows the iteration of one quadratic form:
X X* = (x,y) (x*,-y)
= (x x* + y y* , 0) ¹ (x*x + y y* , 0)
= (x*,-y) (x,y) = X* X
Thus, the quadratic form X X* is standard
because it's composed of two like forms
on the base numbers (while the form X* X, isn't).
Of particular interest is the above construction iterated on the field of real numbers,
as tabulated below.
They are called hypercomplex numbers.
The product of any such number by its conjuguate is a
non-negative real number (as is easily proved by induction on the dimensionality)
whose square-root defines the canonical norm of the number.
Hypercomplex Numbers: Normed Division Algebras (multidimensional
numbers with real coordinates)
Arguably, the above process should stop with the octonions, as the
next step yields a strange realm (the sedenions) where two
nonzero elements may have a zero product.
It seems abusive to call numbers
things that include such zero-divisors,
except for recreation (e.g., decadic
integers).
Among sedenions, the divisors of zero of unit norm are
homeomorphic to G2,
the only exceptional Lie group
not generated by the octonions in the following celebrated (symmetrical) magic square:
The noncommutativity of the quaternions
is rather benign.
The resulting algebraic structure
is called a skew field, and it repays study.
Sir William Rowan
Hamilton
(1805-1865) invented the quaternions
on October 16, 1843.
The
next day, he told John T. Graves about it.
John T. Graves (1806-1870)
came up with octonions on December 26, 1843.
They were rediscovered by Arthur Cayley
in March 1845.
In July 1844, Hamilton
realized that octonions failed to have the crucial property
for which he coined the word associativity, now universally used:
"x
"y
"z
( x y ) z = x ( y z )
A nonassociative division algebra may
contain divisors of zero.
In an associative division algebra, a product of nonzero factors is never zero.
In 1886, Frobenius
proved that the reals, the complex numbers and the quaternions
are the only three associative division algebras over the reals.
Multiplication of octonions isn't associative,
it's only alternative :
"x
"y
(xx)y = x(xy) and y(xx) = (yx)x
Alternativity is less demanding than associativity.
Less demanding still is the
power-associativity discussed below.
The Cayley-Dickson construction transforms a power-associative
algebra into another power-associative algebra, so all the algebras discussed here are
power-associative.
Likewise, they're all flexible.
(2013-07-08)
Beyond the sedenions :
Colorful names or systematic nomenclature.
There has been at least one colorful attempt to name some
Cayley-Dickson algebras
beyond the 16-dimensional sedenions.
Tony Smithreports the
macaronic
tongue-in-cheek nomenclature proposed by Robert de Marais :
Pathions (32 dimensions). Symbol: P.
The 32 paths of wisdom
in the Kabbalistic tree of life.
Chingons (64 dimensions). Symbol: X (since
is for dimension 2).
The 64 Hexagrams
of the I Ching (Book of Changes).
With 0-plexions denoting real numbers,
elements of the 2n-dimensional real algebra
could be called n-plexions where the number
n can be expressed with standard numerical prefixes:
The reals are plexions (avoiding "noplexions" or "nilplexions").
The complex numbers are monoplexions (2 dimensions).
The quaternions are diplexions (4 dimensions).
The octonions are triplexions (8 dimensions).
The sedenions are tetraplexions (16 dimensions).
The list then continues naturally with pentaplexions (32 dimensions),
hexaplexions (64 dimensions),
heptaplexions (128 dimensions),
octaplexions (256 dimensions),
enneaplexions (512 dimensions),
decaplexions (1024 dimensions),
hendecaplexions (2048 dimensions) etc.
(2012-12-20)
Power-associativity is a very weak form of associativity.
Any "multiplication" needs that for exponentiation
to be well-defined.
For an "addition", that property allows a consistent definition of
divisors.
A multiplication is said to be
power-associative
when positive powers
can be recursively well-defined as follows,
for any element a :
a1 = a
and
" m≥1 ,
" n≥1 ,
an+m = anam
The innocent-looking requirement that the above be "well-defined" entails
an infinite sequence of independent conditions :
a3 is well-defined only if
aa2 = a2 a
a4 is then well-defined only if
a a3 =
a2 a2 =
a3 a
a5 is then well-defined only if
a a4 =
a2 a3 =
a3 a2 =
a4 a
Etc.
If a power-associative multiplication has a neutral element (1)
the zeroth power of any element,
invertible or not, is also well-defined:
" a ,
a0 = 1
Invertible
elements can also be raised to the power of a negative integer:
a-n = (a-1)n
For commutative nonassociative rings or algebras
whose characteristic is either 0 or coprime with 30,
A.A. Albert has shown (1947) that,
if fourth powers are well-defined, then the multiplication is power-associative.
When the characteristic is 3 or 5, the same holds whenever fifth and sixth
powers are well-defined (Louis A. Sokotoris, 1952).
Examples of power-associative operators :
Alternative multiplications are power-associative,
so is the multiplication in any set of hypercomplex
numbers (even beyond the sedenions)
or any commutative multiplication having the following
property, known as Jordan's identity (which is verified,
in particular, by the Jordan product
used in quantum mechanics).
The following property is called either power-alternativity
or third-power associativity (for short, I call it
cube-associativity or cubic associativity):
"x ,
x ( x x ) = ( x x ) x
This merely means that cubes are well-defined.
The ugly multiplication tabulated below for a set of 3 elements,
has this property (since any square equals A and everything
commutes with A). This makes all cubes well-defined:
A3 = A,
B3 = C,
C3 = B.
However, neither B nor C have well-defined
fourth powers :
A
B
C
A
A
C
B
B
C
A
B
C
B
C
A
B B3 = B C = B
B2 B2 = A A = A
B3 B = C B = C
C C3 = C B = C
C2 C2 = A A = A
C3 C = B C = B
Therefore, this operation isn't power-associative.
(2005-07-19)
Von Neumann - Bernays - Gödel set theory
(NBG) Sets belong to classes.
A proper class isn't a member of anything.
Any class is a subclass of the class of all sets.
Everything in NBG set theory is a class
(a concept undefined in ZFC set theory)
but only a class which is a member of another class is called a set.
Any collection of sets can be considered collectively as a class
but such classes cannot be considered collectively unless they are sets.
Thus, quantified variables in NBG statements can't be
proper classes.
In the 1990's, it was established that NBG is a
conservative
extension of classical set theory (ZFC)
in the sense that NBG introduces
new language but no new theorems expressible in the language of ZFC.
In view of the fact that they have essentially the same theorems,
it's surprising to observe that NBG can be axiomatized with
finitely many statements and that ZFC cannot
(the latter requires at least one
schema of axioms
which asserts a given pattern rather
than a definite sentence about objects within the theory).
There's no need for a "statement about statements" which could become nonsensical
if the object statements feature free variables whose names
are used in the main statement.
(In the LISP programming language, the so-called FUNARG problem/bug
may arise when functions are passed as arguments to another function.)
For example, within NBG, we may speak of the class of all sets,
which is called the Absolute (always spelled
with a capital A).
The Absolute is a proper class because of
Russel's argument.
That statement isn't a "new" theorem because it can't be expressed at all in the language
of ZFC. It may be more satisfying to say that "the class of all sets is proper" rather
than "there's no such thing as the set of all sets" but both statements are
equivalent in a deep sense...
Gödel marveled at the fact that something
can be known about the Absolute by pure reason, namely:
Anything which is true of the Absolute is also true of at least one set.
(2014-01-19)
Tarski - Grothendieck set theory (TG)
The nonconservative extension of ZFC
used in the Mizar system.
Around 1973, Andrzej Trybulec
chose TG set theory as the basis for his Mizarproof-checker
(now, the most popular system of its kind).
TG set theory is essentially ZFC with the addition of Tarski's axiom
(originally formulated by Tarski in 1939).
All objects are sets;
the notion of class isn't part of the language of TG.
Tarski's axiom states (in modern language)
that every set is a member of at least one Grothendieck universe,
where a Grothendieck universe
is, loosely speaking, a set within which all usual mathematical operations can be performed.
More precisely, a set U is a Grothendieck universe if and only if the following four
conditions are met:
If x belongs to U and y belongs to x, then y belongs to U.
If x and y belong to U, then the set {x,y} belongs to U.