Name: Plober
Who is asking: Student
Level: Secondary
Question:
How can I explain to a friend (in a bar, using as a pen and a paper napkin)
that the integer's infinity is 'smaller' than the irrationals's one? The
demo I tried was that you couldn't match the integers with the real
numbers between 0 and 1 (that 0.xxxxx replacing the Nth number from a
different one... that demo), but she used my argument >:| saying that
you can add one to the integer's infinite, and the number I was creating
was only one more...
I can't think of any other way, and I KNOW the real's cardinality is greater than the integer's one
Hi Plober,
Galileo knew that squaring produced a bijection between
the integers and their squares, hence the saying "the whole is greater than the
part" did not apply to infinite sets; his conclusion was that infinite
sets were totaly incomprehensible and it was pointless to try to analyse
them logically. Cantor later classified infinite sets in terms of bijections,
but his arguments were not fully accepted in his age. In that respect,
your bar room conversation is part of an long historical debate on the
nature of infinity. Perhaps it is good that many people can express their
opinion rather than having "the right answer" taught
to everyone.
Your real insight on the subject is that you stick to a formal definition of why a cardinality is larger than another one, rather than avoid definitions or make them up along the way as suits you best. We often get questions of the type "How many sides does a circle have?" Depending on what is meant by a side, the answer can be 0, 1, 2 or infinity; students picking one of them are sometimes marked wrong and told that the answer is another one of them, without further justification.
We presented your argument to Carlos who asked earlier about the size of infinite sets. The point is that Cantor's condition is "If every attempt to construct a one-to-one correspondence between A and B, leaves B with elements that are not matched with elements of A, then we say that B is larger than A." In your argument you showed precisely this.
Claude and Penny