I know this is long but give it a chance, you may even enjoy it.
Who was Georg Cantor?
I did some reading the other day on Russell’s Paradox. All to do with
sets and whether they are members of themselves. Got me thinking
about someone who I took a great interest in a few years back. Georg
Cantor. Cantor had done work on sets. I decided to look him up again
and as I read I then remembered why I had been so interested in
him. As I say, I had to look up many of his more personal details but
I thought perhaps one or two might find him interesting, as I had
done a few years ago.
Cantor was born in Russia, sometime in 1845. In 1862, he started off
at the University of Zurich but switched to the University of Berlin. It is
known that while at Berlin he studied mathematics, philosophy and
physics, under the guidance of some of the greatest mathematicians
around at that time (including Kronecker and Weierstrass). After
completion of his studies our main man was unable to find a tidy job.
So he did what most people would have done. He was forced to accept
a position as an unpaid lecturer in the hope that it got him a foot on
the ladder. He later became an assistant professor at the University of
Halle…a university it should be noted which had little or no reputation.
In 1874 he got married, and, fair play to him, had 6 children.
It was in 1874 that Cantor published his first paper on set theory. He
investigated sets and particularly infinite sets. He kept digging and
was surprised at what he saw. For Cantor, at this stage, seeing was not
believing. “I see it but I don’t believe it” he wrote to a friend of his.
Over the next 23 years he proved many things. He showed that the
set of integers (ie the “group” of all the whole numbers) had the
same number of members as the set of even numbers. It even had
the same number of members as squares and cubes etc! How so?
Consider for a second what happens if we line the integers above their
respective squares.
1, 2, 3, 4, 5, … , n, …
1, 4, 9, 16, 25, … , n^2 , … and so on.
Every integer can be mapped up with it’s square. So even though
there would appear to be far more integers than there would be
squares, in fact there are an equal number. What Cantor did was to
prove this.
But now check this out. Consider the number of “points” on this line
segment: ____ and now the number of “points” on a line of infinite
length. There must be more points on the line of infinite length right?
Wrong. Cantor showed that there would be an equal number of points
on a line segment as there are on a line of infinite length. (He did
some other much more complex stuff involving transcendental
numbers such as Pi and e and algebraic equations).
Something I didn’t know but found when I looked Cantor up was the
Jesuits used Cantor’s work to prove the existence of God and the Holy
Trinity. Imagine that!! The work that you do being used to “prove” the
fundamental existence of the Almighty. Cantor, a top class theologian
himself, put some distance between himself and the claims of the
Jesuits.
What he wasn’t able to do though was put distance between himself
and those that attacked him, his work, his ideas, which ultimately
proved to be his downfall and subsequently, tragically, his death.
Why the big deal? Why did people have a go at Cantor? All Cantor
had done was mess around with concepts of the infinite when
discussing set theory.
Little bit of rumour for y’all. It's relevance will become apparant. A
pupil of Pythagoras asked his teacher/leader what was the length of a
diagonal in a unit square. Pythagoras said root 2 of course. The pupil
wondered what *number* root 2 was. The story goes that Pythagoras
had the pupil tied up in a bag weighted down with stones and threw
him in a nearby river, in order to send out a message that numbers
like root 2 weren’t worth bothering about (sad fact was that Pythagoras
just didn’t know what to do, irrational numbers had been discovered
but it was hoped that they weren't important). This shows there is a
history of mathematicians being worried about things they can’t
control. They try and cover up these new ideas.
So it was with Cantor. One mathematician (generally regarded as the
last all round great of mathematics) called Poincare reckoned that
Cantor’s set theory would be considered by future generations as a
disease from which one has recovered from. So one minute people
are using Cantor’s work to prove the existence of God, the next he
has the big Daddy of maths at the time calling it a disease that
people will eventually get over! Whereas Poincare hurt Cantor with
words, his former teacher Kronecker went after Cantor far more
sadistically. Kronecker was a real old school nut, I believe he once
said “God created the integers, all the rest is down to man”, implying
that the integers were all that truly mattered. Thus conveniently
ignoring fractions, imaginary numbers and his pet hate, irrational
numbers (a la Pythagoras some two millennia before him).
At the time in the world of mathematics people were obviously aware
of a concept of infinity. But that’s all it would be; a concept. A concept
no less that should be avoided at all costs. What Cantor was
suggesting was actual infinity. Kronecker was a professer at the Uni of
Berlin remember, he was a powerful man. Among other things, he
either delayed or totally suppressed Cantor’s publications. He went
after Cantor with written and verbal attacks. He belittled Cantor in front
of his students. What really crushed Cantor was that Cantor’s lifelong
goal was to become a professor at the famous University of Berlin. It
was what Cantor’s father would have wanted before he (his father)
died from ill health many years previously. This was Cantor’s dream
and Kronecker prevented this for no other reason than spite and fear
at what Cantor was doing to mathematics.
There were those that backed Cantor, Weierstrass and a chap called
Dedekind supported Cantor’s ideas and slagged off Kronecker for all
they were worth, but it was not enough. I’ll quote something I read
earlier:-
“Stuck in a third-rate institution, stripped of well-deserved recognition
for his work and under constant attack by Kronecker, he suffered the
first of many nervous breakdowns in 1884. The rest of his life was
spent in and out of mental institutions and his work nearly ceased
completely”.
Cantor died in a mental institution in 1918 at the age of 56. By this
time he had lost the plot completely. His mind had been ravaged by
the personal attacks on him and his work.
Around about 1900 people started realising that there was some truth
in what Cantor was saying. The trickle became a stream which became
a river. It was too late for him to enjoy it or understand. In 1904,
Cantor was awarded a medal by the Royal Society of London and was
made a member of both the London Mathematical Society and the
Society of Sciences in Gottingen. It was too little to late. He had
wanted the post at the Uni of Berlin above anything.
Today, in 2002, what Cantor did, his work, his ideas, his vision, is
widely accepted by the world of mathematics. His theory on infinite
sets shook the foundations of mathematics, and, with work done in
later years reset the foundation of nearly every mathematical field
and brought mathematics to its modern form. Zeno’s paradoxes which
had baffled and perplexed for 2,500 years were finally within grasp of
total understanding.
Cantor, like most geniuses in history, had the grave misfortune of
being born before the world was ready for him.
Mark
The Squirrel Lover
I know this is long but give it a chance, you may even enjoy it.
Who was Georg Cantor?
I did some reading the other day on Russell's Paradox. All to do with
sets and whether they are members of themselves. Got me thinking
about someone who I took a great interest in a few years back. Georg
Cantor. Cantor had done work on sets. I decided to look him up again
and as I read I then remembered why I had been so interested in
him. As I say, I had to look up many of his more personal details but
I thought perhaps one or two of you out there who would be willing to
put in the effort and read this might find him interesting, as I had
done a few years ago.
Cantor was born in Russia, sometime in 1845. In 1862, he started off
at the University of Zurich but switched to the University of Berlin. It is
known that while at Berlin he studied mathematics, philosophy and
physics, under the guidance of some of the greatest mathematicians
around at that time (including Kronecker and Weierstrass).
After completion of his studies our main man was unable to find a tidy
job. So he did what most people would have done. He was forced to
accept a position as an unpaid lecturer in the hope that it got him a
foot on the ladder. He later became an assistant professor at the
University of Halle...a university it should be noted which had little or
no reputation. In 1874 he got married, and, fair play to him, had 6
children.
It was in 1874 that Cantor published his first paper on set theory. He
investigated sets and particularly infinite sets. He kept digging and
was surprised at what he saw. For Cantor, at this stage, seeing was not
believing. "I see it but I don't believe it" he wrote to a friend of his.
Over the next 23 years he proved many things. He showed that the
set of integers (ie the "group" of all the whole numbers) had the
same number of members as the set of even numbers. It even had
the same number of members as squares and cubes etc! How so?
Consider for a second what happens if we line the integers above their
respective squares.
1, 2, 3, 4, 5, ... , n, ...
1, 4, 9, 16, 25, ... , n^2 , ... and so on.
Every integer can be mapped up with it’s square. So even though
there would appear to be far more integers than there would be
squares, in fact there are an equal number. What Cantor did was to
prove this.
But now check this out. Consider the number of "points" on this line
segment: ____ and now the number of "points" on a line of infinite
length. There must be more points on the line of infinite length right?
Wrong. Cantor showed that there would be an equal number of points
on a line segment as there are on a line of infinite length. (He did
some other much more complex stuff involving transcendental
numbers such as Pi and e and algebraic equations).
Something I didn't know but found when I looked Cantor up was the
Jesuits used Cantor's work to prove the existence of God and the Holy
Trinity. Imagine that!! The work that you do being used to "prove" the
fundamental existence of the Almighty. Cantor, a top class theologian
himself, put some distance between himself and the claims of the
Jesuits.
What he wasn't able to do though was put distance between himself
and those that attacked him, his work, his ideas, which ultimately
proved to be his downfall and subsequently, tragically, his death.
Why the big deal? Why did people have a go at Cantor? All Cantor
had done was mess around with concepts of the infinite when
discussing set theory.
Little bit of rumour for y'all. It's relevance will become apparant. A
pupil of Pythagoras asked his teacher/leader what was the length of a
diagonal in a unit square. Pythagoras said root 2 of course. The pupil
wondered what *number* root 2 was. The story goes that Pythagoras
had the pupil tied up in a bag weighted down with stones and threw
him in a nearby river, in order to send out a message that numbers
like root 2 weren't worth bothering about (sad fact was that Pythagoras
just didn't know what to do, irrational numbers had been discovered
but it was hoped that they weren't important). This shows there is a
history of mathematicians being worried about things they can't
control. They try and cover up these new ideas. So it was with Cantor.
One mathematician (generally regarded as the last all round great of
mathematics) called Poincare reckoned that Cantor's set theory would
be considered by future generations as a disease from which one has
recovered from. So one minute people are using Cantor's work to
prove the existence of God, the next he has the big Daddy of maths
at the time calling it a disease that people will eventually get over!
Whereas Poincare hurt Cantor with words, his former teacher Kronecker
went after Cantor far more sadistically. Kronecker was a real old school
nut, I believe he once said "God created the integers, all the rest is
down to man", implying that the integers were all that truly mattered.
Thus conveniently ignoring fractions, imaginary numbers and his pet
hate, irrational numbers (a la Pythagoras some two millennia before
him).
At the time in the world of mathematics people were obviously aware
of a concept of infinity. But that's all it would be; a concept. A concept
no less that should be avoided at all costs. What Cantor was
suggesting was actual infinity. Kronecker was a professer at the Uni of
Berlin remember, he was a powerful man. Among other things, he
either delayed or totally suppressed Cantor's publications. He went
after Cantor with written and verbal attacks. He belittled Cantor in front
of his students. What really crushed Cantor was that Cantor's lifelong
goal was to become a professor at the famous University of Berlin. It
was what Cantor's father would have wanted before he (his father)
died from ill health many years previously. This was Cantor's dream
and Kronecker prevented this for no other reason than spite and fear
at what Cantor was doing to mathematics.
There were those that backed Cantor, Weierstrass and a chap called
Dedekind supported Cantor's ideas and slagged off Kronecker for all
they were worth, but it was not enough. I'll quote something I read
earlier:- "Stuck in a third-rate institution, stripped of well-deserved
recognition for his work and under constant attack by Kronecker, he
suffered the first of many nervous breakdowns in 1884. The rest of his
life was spent in and out of mental institutions and his work nearly
ceased completely".
Cantor died in a mental institution in 1918 at the age of 56. By this
time he had lost the plot completely. His mind had been ravaged by
the personal attacks on him and his work.
Around about 1900 people started realising that there was some truth
in what Cantor was saying. The trickle became a stream which became
a river. It was too late for him to enjoy it or understand. In 1904,
Cantor was awarded a medal by the Royal Society of London and was
made a member of both the London Mathematical Society and the
Society of Sciences in Gottingen. It was too little to late. He had
wanted the post at the Uni of Berlin above anything.
Today, in 2002, what Cantor did, his work, his ideas, his vision, is
widely accepted by the world of mathematics. His theory on infinite
sets shook the foundations of mathematics, and, with work done in
later years reset the foundation of nearly every mathematical field
and brought mathematics to its modern form. Zeno's paradoxes which
had baffled and perplexed for 2,500 years were finally within grasp of
total understanding.
Cantor, like most geniuses in history, had the grave misfortune of
being born before the world was ready for him.
Mark
The Squirrel Lover
that was very inteesting, Mark.
Just a comment on Russel's paradox. You know I like 'intuitive' stuff. Here is an 'intuitive' version, you probably have heard
before. It may not be 100% correct, but illustrates the paradox:
you have libraries. assume all libraries maintain a catalog of books. sometimes the catalog is considered as a book itself, and,
logically, mentions itself in the catalog. Sometimes it isn't.
Now, if you were to make a library of catalogs, but only of the catalogs that do NOT mention themselves. And you make the
catalog of that library. Should it contain itself?
That's the paradox: if it doesn't, it should. If it does, it shouldn't.
for what it's worth.
Gilbert
You're right Gibert, that's pretty much it spot on. The librarian and the
catalogues is the clearest way that I have seen it described. Slightly
more formal (though still a long way off general mathematical
notation) is as follows. One thing though, it should be noted that
Russell's Paradox absolutely rocked the very foundations that
mathematics was built on.
Cats. Take the set of all the cats in the world. We ask ourselves, is
this set of all cats actually a member of itself? To be a member of
itself we would need the set of all cats to be a cat. Clearly the answer
is no. The set of all cats is an abstract grouping, all the cats in the
world don't make up one big cat. Conclusion, the set of all cats is not
a cat, and therefore NOT a member of itself. So far, so good.
Not cats. Consider trees, chees pieces, Beethoven's 7th and all the
sets which are non-cats. Taking all the sets which are non-cats we now
ask is this big set of non-cats (everything in the world that isn't a cat)
a member of itself? To be a member of itself we would need the set
of all non-cats to be a non-cat. Obviously, the set of all non-cats isn't
actually a cat. Conclusion, the set of all non-cats is not a cat, and
therefore IS a member of itself. Luvverly jubberly.
Define a "normal set" to be a set which does NOT include itself as a
member of the set (in other words the set of cats is a normal set,
since this set isn't a cat - it's just an abstract grouping remember).
Russell's Paradox is whether or not *the set of all normal sets should
include itself or not*.
Let's firstly say YEP, the set of all normal sets *should* include itself.
To be included in itself then we have to say that the set of all normal
sets is itself in fact a normal set (since only normal sets can be
included in the set of all normal sets, like to be a member of the set
of all cats, the set would have to be a cat if it was to be included). But
if it is in fact a normal set, then by the definition of what a normal set
is, it CAN'T actually include itself. For if it does include itself then it
isn't a normal set! In summary, to be included it must be a normal
set. If it is a normal set then by definition it can't be included in itself.
Contradiction. If it is, then it isn't.
Let's now say NOPE, the set of all normal sets should NOT include
itself. If it is not included in the set of all normal sets then it must be
a non-normal set (for if something isn't in the set of all cats then it
simply can't be a cat, for if it were a cat it would be included in the set
of cats). BUT if it is a non-normal set then it must include itself in the
set (since if it didn't include itself in the set it is a normal set by
definition). Again, we have another contradiction. In this case if we
don't include it we arrive at the conclusion of it having to be included.
If it isn't, it is.
So what can be done? It seems like nothing can be done. It seems
that Mathematics and Logic are felled by a bunch of cats and a bunch
of non-cats. Or, as Gilbert said, a catalogue of catalogues that don't
mention themselves.
If anyone is interested in all this and has something to add, however
trifling or small, then please write something. If anyone is interested I
will post later (unless someone else does it) why Russell's Paradox
had such far reaching effects, why it destroyed the life of one man in
particular, and perhaps some proposed solutions to the Paradox.
Kurt Godel was an Austrian logician who did some crazy stuff in the
1930s. This all kinda builds up to Kurt Godel. Anyone know anything
about him?
Mark
The Squirrel Lover
I've enjoyed your posts Mark, and wish I had the time to chime in
more. I'm suprised when you were talking about Cantor you didn't
bring up his diagonal proofs, perhaps the weirdest of which is the one
that shows that the set of real numbers is larger than any
denumerable infinite set (that there are orders of infinity, and some
are larger than others). Working through this extremel.y elegant
proof shook my metaphysical foundations. I've been forced to a
procedural interpretation of the infinite, because I can't shake the
feeling that any result that shows one infinite set to be larger than
another is just a reductio of the concept of the infinite. By the way,
did you hear the David Foster Wallace has been hired to write the
biography of Cantor? I'm actually working through a lot of Godel right
now, concerning the computability of functions. Building turing
machines and the like. I came to all this through philosophy
generally, and logic in particular. You presumably came to this through
mathematics.
Bennett
Thanks Bennett! Marvellous post.
I did start typing up Cantor's diagonal dabblings (it really is
astonishing isn't it!) in the middle of the bit about points on a line and
the number of squares matching the number of integers...but felt that
it was getting long-winded enough as it was and ultimately wanted
many people to read it, rather than those with specialist knowledge.
Plus also, the other reason for not expounding upon the nitty gritty of
Cantor is that, quite honestly, I don't know it. Most of this stuff comes
from when I started doing some reading when I was 16/17 so a lot of
the hardcore maths was out of my reach. Am just getting back into it
again now having graduated this year (and having studied a load of
statistics at university, heaven help me) and wanting to get back to
why Maths got me excited 5 or so years ago.
I do know though that what Cantor did, especially regarding differing
sizes of infinity, and what can be counted and what can't, and so on
really, REALLY did shake up a lot of people (one of the reasons
Kronecker gave him such a hard time no doubt). Like you say, you
were forced to think about something that doesn't seem to sit
comfortably in a mathematical or philosophical construct.
A lot of people are still deeply scared and thrown by it. Very, (VERY)
simplistically for people who don't know, (as far as I understand it
anyway) Cantor started off by assuming actual infinity existed. He
then went on to prove that there must be different "kinds" of infinity,
infinity of different sizes. People still argue with the premise, but
Cantor's proofs would suggest that the existence of other infinities is
unarguable.
Just off the top of my head (and perhaps not related to Cantor at all)
I remember reading about how it was possible to create a box with
finite dimensions but yielding an infinite volume. And (I think, I may
be wrong), more incredibly, a box with infinite dimensions but with a
finite volume. No wonder a lot of mathematicians and logicians end up
a little doolally! The amazing thing about this is that, mathematically,
they appear to be watertight conclusions.
To answer your question though, I came to this kind of thing after
having stumbled across the story of Archimedes's death when I was
about 16. It captivated me. I read more about Archimedes and then
on about other mathematicians. At that stage it was more the
personal, humanastic (is that a word?!) point of view that I was
interested in...what made these great minds tick? How were they
viewed by others? What is it like to see something no other person
has ever seen before? I loved the stories.
It was around this time that Andrew Wiles was finally laying Fermat's
Last Theorem to rest. It was in the newspapers and so on. I bought a
book called Fermat's Last Theorem by Simon Singh. Chrismo
mentioned it a while back. It is very accessible for the non-maths bod
and contains lots of stories, as well giving inklings of many different
things in mathematics. Gilbert's post on the library catalogues was in
there, so too were mentions of Godel's work on undecidability and
inconsistency and so on.
The logic side of it really grabbed me, as well as the fact that this stuff
was done by people, just the same as you and I but with a gift. These
people are fascinating just from how they lived their lives, regardless
of any kind of maths thang going on.
It was a real challenge trying to keep up with the maths; I have no
real outright ability for it. I sort of did the opposite to Bennett (albeit
not at such an advanced level). As I read about people like Descartes
and Leibniz so too did I see that they had philosphical ideas, as well
as mathematical (although the two are linked in ways that is almost
uncanny).
I was really interested in the philosophy side. On the bookshelf at
home was a copy of Descartes "Discourse de la Méthode et des
Méditations" (in English mercifully). I was embarrassed to be liking
this kind of thing though, and so took it off the shelf without telling
my Mother, hoping she wouldn't notice and would read it secretively on
the bus to school away from my parents and my friends.
I get the piss ripped now from people who know. They tell me that
most adolescent angst kinda people kept a copy of Playboy under
their beds, and there was me sneaking around with Descartes.
Repressed is a word they use a lot! I wanted to study philosophy at
university but didn't have the guts to pursue it as people around me
regarded it as a wishy washy useless thing to do.
Once again, a very lengthy post (I need to get out a bit(!)...I'm going
for a run now I think) but this time with a point at the end.
Being allowed and encouraged to pursue one's interests, and having
the confidence to do so, is the greatest gift that any parent can give
to their child.
Mark
The Squirrel Lover
PS Bennett, I didnae even know there was going to be a biography of
Cantor coming out! Thankyou! I love biographies...I'll keep an eye
out for it.
Bingo.
A neat book on the subject is an old one. Nagel and Newmann's "Godel's Proof".
Accessible to the layperson (aka ME). Old, written in the late 1950s. Does a
good job of explaining the history of the problem, it's importance in
mathematics, the brilliancy of Godel's approach, and quite a bit of the
details of the proof, all without being too painfully mathematical. Of course,
it is a book designed to accurately explain the proof, so if you don't like
mathematics, it's probably a bit dry.
And of course, Douglas Hofstadter's Godel, Escher, Bach is a fabulous read
too. Godel and much much more. A rambling wonder of a book and great fun as
well.
This is my version of Cantor's proof, its form is that of an indirect
proof, or a reductio of the claim that the set of real numbers is the
same size as the set of natural numbers.
1. For any set X and any set Y, X and Y are of the same size if and
only if the elements of X bear a one-to-one correspondance with the
elements of Y. (This is assumed by Cantor, but seems pretty
intuitive. If I want to know whether the number of students in my
room equals the number of chairs, I tell everyone to sit down and see
what's left over. That is, I try to establish a one-to-one
correspondance between students and chairs.)
2. For any set X and any set Y, the elements of X and the elements
of Y bear a one-to-one correspondance with each other if and only if
both X and Y are denumerable.
3. For any set X, X is denumerable if and only if (1) the elements of
N can be arranged into a list with a first member, second
member,...,nth member and (2) each element of X is quaranteed a
definite unique position on the list.
4. The set of natural numbers N is denumerable (The natural
numbers are 0, 1, 2, etc. no negative numbers, no fractions with
denominators other than 1.)
5. Suppose the set R of real numbers in the interval from 0 to 1 is
denumerable. [Real numbers are those that start with a decimal and
go on forever to the right of the decimal. Among the real numbers
are (1) the rational numbers (that repeat some digit or string of digits
infinitely) and (2) the irrational numbers (that have no infinitely
repeating pattern). .499999....is rational, in fact, it's equal to .5. The
decimal expansion of pi is irrational.]
6. If so, then the elements of R can be arranged in a list with a first
member, second member,..., nth member, and each element of R will
be guaranteed a definite unique position on the list. (This follows
from our 3 and 5).
7. Let L be the list of the elements of R. Let each element of R be a
decimal point followed by an infinite string of digits. We can represent
each element of R in the following fashion: Where B is some digit in
some element E of R, let x be the ordinal position of B in the string of
digits that constitute E; let y be the oridinal postiion of E in L. Then,
the ordered pair (x,y) specifies a definite unique location within L for
any particular B. Thus, B(x,y) picks out a definite unigue digit within L.
8. Suppose we apply the following procedure: Beginning at B(1,1),
followed by B(2,2)....B(n,n), for any B(x,y) if B(x,y) is the digit '1',
change it to '2'. If B(x,y) is any digit other than '1', change it to '1'.
The resultant string of digits, B(1,), B(2,2)....B(n,n), subsequent to
our procedure, is the Diagonal of L.
9. The Diagonal of L will itself be a real number within the interval
from 0 to 1, and thus will be an element of R. But the procedure
above guarantees that the Diagonal of L will not itself appear as an
entry of L, because for any entry (e.g., the nth entry) in L the
Diagonal of L will have some digit (the nthe digit) that differs from the
digit found in the entry in L.
10. Thus, if R is denumerable, then any list of the elements of R will
necessarily by incomplete (it will not contain its diagonal). But this
contradicts 6, so R must not be denumerable. (Any assumption that
leads to a contradiction is rejected and the assumption's negation is
adopted)
11. But if R is not denumerable, then the elements of R cannot be
put in a one-to-one correspondence with the the elements of N.
12. Thus the sets R and N are not of the same size.
So Cantor shows us that not all infinite sets are the same size, but he
assumes that all denumeralbe sets are of the same size. This
assumption already may do violence to your metaphysical intuitions,
because you may think that the set of even numbers is roughly half
as big as the set of natural numbers. I admit to finding this view
persuasive, but imagine you have two perfect random number
generators. The first generator is set up so that it only produces an
even number when operated. The second can produce any of the
natural numbers when operated. Each operation yields just one
number from each machine. If the set of evens is half as big as the
set of naturals, then it ought to be twice as likely getting a '2' from
the first machine than from the second. But each machine picks their
numbers completely at random and both have an infinite number of
choices. So the probability of getting a '2' on either should be
1/infinity. At this point my intuitions become silent and I begin to
drink beer.
Thanks for the great topic Mark.
Bennett
My response to Cantor, FWIW:
Every square is an integer, so the set of all squares is included in the
set of all integers.
There exist integers that are not squares.
Accordingly, the number of integers is larger than the number of
squares.
Am I missing something? How did Cantor address this point?
Another way of saying what Cantor showed was that there is ONLY one infinity;
there are no "bigger" or "smaller" infinities. (OK you mathematicians,I'm
lying, I'll leave the discussion of countably infinite vs uncountably infinite
to you).
Intuition is that there "must" be more integers than squares of integers, just
as you state. Yes it WOULD seem there are more integers than squares, since
not all integers are squares, but all squares are integers. My understanding
of what Cantor showed was that this intuition is actually wrong - there are
just as many squares as there are integers! Infinite is infinite, ain't no
bigger or smaller infinities.
My weak understanding of what Cantor did was say, essentially "ok, you think
there are more integers than squares. So start listing the integers, in some
order. For each unique integer YOU name, I can name a unique square. No matter
how far YOU count along your infinite list of integers, I can rattle off a
unique square from MY infinite list of squares. You will never be able to come
up with an integer for which I can't give you a square number."
Since I can match every member of you set with a member from my set, the
"size" of the two infinities, if you will, are the "same."
Doing this, he showed there were the same number of integers as there are
rational fractions. Freaky freaky.
Cantor was also the grandfather of fractals, most of a century before
Mandelbrot. Love those Cantor dusts.
The integers to squares thing isn't completely obvious, since, for example 1^2 = (-1)^2. But
if you go from integers to naturals, turning negative numbers into odd numbers and positive
ones into evens, the list of squares you could use is clear.
"Doing this, he showed there were the same number of integers as there are rational
fractions. Freaky freaky." This one is more disturbing, because between any two rationals
there are infinitely many more rationals! How can you possibly put all them in a list? I did
this once, but I've forgotten exactly what I did. Maybe this should go in the Posers and
Puzzles forum.