with a problem of their own – they didn’t have to explain what straight lines
were, but only how they should be dealt with.
In the case of set theory, this was the origin of the Zermelo–Fraenkel axioms
for set theory which prevented the appearance of sets in their system that were
too ‘big’. This effectively debarred such dangerous creatures as the set of all sets
from appearing.
Gödel’s theorem
Austrian Mathematician Kurt Gödel dealt a knockout punch to those who
wanted to escape from the paradoxes into formal axiomatic systems. In 1931,
Gödel proved that even for the simplest of formal systems there were statements
whose truth or falsity could not be deduced from within these systems.
Informally, there were statements which the axioms of the system could not
reach. They were undecidable statements. For this reason Gödel’s theorem is
paraphrased as ‘the incompleteness theorem’. This result applied to the Zermelo–
Fraenkel system as well as to other systems.
Cardinal numbers The number of elements of a finite set is easy to count,
for example A = {1, 2, 3, 4, 5} has 5 elements or we say its ‘cardinality’ is 5 and
write card(A) = 5. Loosely speaking, the cardinality measures the ‘size’ of a set.
According to Cantor’s theory of sets, the set of fractions Q and the real
numbers R are very different. The set Q can be put in a list but the set R cannot
(see page 31). Although both sets are infinite, the set R has a higher order of
infinity than Q. Mathematicians denote card(Q) by) , the Hebrew ‘aleph
nought’ and card(R) = c. So this means <c.
The continuum hypothesis
Brought to light by Cantor in 1878, the continuum hypothesis says that the
next level of infinity after the infinity of Q is the infinity of the real numbers c. Put
another way, the continuum hypothesis asserted there was no set whose
cardinality lay strictly between and c. Cantor struggled with it and though he
believed it to be true he could not prove it. To disprove it would amount to
finding a subset X of R with < card(X) < c but he could not do this either.