Activity 1-17: Infinity

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www.carom-maths.co.uk
Activity 1-17 Infinity
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Of all the ideas you will meet in mathematics,
the most elusive of them all is likely to be
infinity.
In no area are the preconceptions that we bring
to thinking about the subject more likely to be
adrift.
Many of the errors that mathematicians have made
down the years have come from not paying enough
respect to infinity and its implications.
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Consider the following sequence
What happens as the number of terms increases?
The sequence clearly heads towards 0.
What happens if we add the terms of the sequence?
It seems clear to us that this heads towards 2.
Indeed, the sum to infinity we can say IS 2.
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So we have an infinite number of numbers that
add to a finite number. Mmm
Does this explain Zenos Paradox?
Achilles gives the tortoise a 100m head start in
their race. By the time Achilles has run
100m, the tortoise has moved on say 1m. By the
time Achilles has run this 1m, The tortoise has
moved on...
This argument can be repeated infinitely often.
So Achilles can never overtake the tortoise.
But we watch the race, and he does just that!
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The error is to say that the sum of an infinite
number of things must be infinite.
If the things become infinitely small, then they
can add to something finite.
But...
Just because they become infinitely small, they
dont have to add to something finite!
for example...
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Is infinity just infinity?
Or might there be different types, different
sizes of infinity?
The mathematician Cantor spent much of his life
thinking about infinity, and maybe he paid the
price his mental health was fragile. But he
came up with two marvellous arguments that still
shine a great light onto the idea of infinity
today.
Georg Cantor, German,
(1845-1918)
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What is the simplest idea of infinity that you
can have? Maybe
1, 2, 3, 4, 5
Is this the only infinity we can have? How about
the infinity given by all the integers?
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
The looks to be a bigger infinity but it turns
out that you can rearrange all the integers to
pair off perfectly with the positive integers.
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Task what is the rule for getting from n to m
here? Can we find a similar rule for getting
from m to n?
We say that there is a bijection between the set
of ns and the set of ms and if there is a
bijection between any two sets, then they are of
equal size.
What about the rational numbers?
Surely this is a bigger set than the counting
numbers!
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But actually, the set of rational numbers is no
bigger than the counting numbers.
So between every two rationals there is another
rational thats not true for the counting
numbers.
Suppose we arrange the rational numbers like this
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That means we can count them like this
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So once again, it seems that the infinity
represented by the counting numbers is the one
possessed by the rational numbers.
Maybe this infinity is the only one then?
What about the infinity given by all the
numbers between 0 and 1?
Cantor discovered a wonderful argument here.
Suppose that the numbers between 0 and 1 can be
written out in a list as decimals that is,
suppose the infinity we are dealing with is the
same as that of the counting numbers.
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Cantor then said, suppose I take the following
number
and then I change each digit for another, any
other.
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It cannot be at number n, say, because it
differs with the nth number in the list at the
nth digit.
This new number will be between 0 and 1, but
where will it be in the list?
The only conclusion we can come to is that the
number is not in the list, and so the list is
incomplete
which means that the infinity of the real numbers
is bigger than the infinity of the counting
numbers.
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For a long time a big question in mathematics was
Is the Continuum Hypothesis true? The CH says
there is an infinity between that of the counting
numbers and that of the numbers between 0 and 1.
Kurt Godel, Austrian-American (1906 1978)
Saying the CH is true is consistent with the
axioms of standard mathematics.
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Paul Cohen, American (1934 - 2007)
Saying the CH is false is consistent with the
axioms of standard mathematics.
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So Godel and Cohen showed mathematics works
perfectly well (in a non-contradictory
way) whether we assume this infinity between the
counting numbers and the infinity of numbers
between 0 and 1 exists, or whether we assume it
doesnt.
So we now have two different versions of
mathematics to work with with the Continuum
Hypothesis being true, and with it being false.
I did say infinity was tricky!
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With thanks toWikipedia, for another excellent
article.
Carom is written by Jonny Griffiths,
hello_at_jonny-griffiths.net