16 April 2022 | New Scientist | 39>natural numbers, are also countable.
Perhaps more surprisingly, so are the fractions.
There are clearly a lot of fractions, but Cantor
used a cunning trick to prove that you can
still perfectly pair them up with the natural
numbers: they, too, are countably infinite
(see “A rational pairing”, page 40).Into the uncountable
That doesn’t exhaust all the numbers you
might think to string along your number line,
however. Numbers like π and √2 can’t be
written as fractions, and when written in
decimal continue forever after the decimal
point without a repeating pattern. Add in these
irrational numbers to the set of all numbers
already discussed, and you have what is called
the set of real numbers, or the continuum.
Cantor wondered whether, like the fractions,
the real numbers were countably infinite too,
but it turns out they aren’t. However you pair
the real numbers with the countable numbers,
there will always be real numbers that you have
missed (see “Accountably uncountable”, page
42). “Cantor discovered that there are a lot
more real numbers than natural numbers,”
says Ralf Schindler at the University of Münster
in Germany. Cantor had discovered that
infinity comes in different sizes. The countable
infinity was the smallest one, but there was
also a “continuum” infinity larger than it.
That wasn’t the half of it. If you take any set
of numbers, there is another set that contains
every possible combination of the elements in
the original one. Cantor discovered that this
“power set” of an infinite set was infinitely
large, of a larger size. Because you can repeat
this process, taking the power set of a power
set ad infinitum, he had found a method
to produce an infinite ladder of infinities.
This shocker just raised more questions.
“Once you discover such a thing, you want to
draw a map of the different kinds of infinity,”
says Schindler. Cantor knew that the countable
sets were the smallest infinity, but was the
continuum infinity the next level up? This
infinity is known as Յ1, or aleph-one, and
Cantor believed that it and the continuum
infinity were one and the same. This assertion
became known as the continuum hypothesis,
but Cantor was never able to prove it.
Neither was anyone else. This all happened
around 1878, and 22 years later, in 1900,
mathematician David Hilbert put proving or
disproving the continuum hypothesis top of“ Infinity’s wobbling tower suggests the
foundations of mathematics are unstable”
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