The author of my book stated
The Continuum Hypothesis: There exists no set such that
He then went on raising a question: Is there as set such that ?
Following the question, he proved that , and at the end of the proof, he said the proof shows that there is no largest set. In particular, there is a set S with .
Remark: I have not a clue what he meant in blue texts.
Question: Is there or is there not that ?
Open interval (0,1)
Theorem: The set (0,1) and are numerically equivalent.
I know that if I write the open interval as
where , and that , I also can conclude that and are numerically equivalent.
Question: What is the purpose of the theorem? Does it mean to say that any uncountable subset of is numerically equivalent with ?
Originally Posted by novice
The statement in blue simply means: given any set its power set has greater cardinality so there is always a 'larger' set.
There are two ways of comparing the number of elements in a set. You either count them, and see which one is bigger, which is fine for finite sets, or you can try and set up a one to one correspondence between the elements.
Kolmogrov gives the example of comparing the number of students is a classroom to the number of chairs: if there are still chairs left over when eveyone is sat down, then there are more chairs than students, and if there are still students standing up when everyone is sat down, then there are more students than chairs. And, in fact, this works well for infinite sets.
The fact that means that we can match every element of to an element of like students to chairs, with some elements in to spare, which intuitively means that the set is 'bigger' than . The fact that there is no biggest set means that there is no set which cannot be 'fit inside' another set in this way: no matter how many students we have, we can always find a classroom to sit them all with some chairs left over.
In particular, suppose that is the biggest possible set, then is bigger, a contradiction. To answer the question, , so yes there is such an .
And as for your second question, the fact that and are numerically equivalent doesn't mean that any infinite subset of is numerically equivalent to because yet is not countable. The purpose of the theorem is anybody's guess, you might as well ask what the meaning of life is!
That might be misleading. In general
Originally Posted by nimon
|S| < |T| if and only if there is an injection from S into T but there is no bijection from S onto T.
That there is an injection from S into a proper subset of T (the fact you've mentioned) does not in itself entail that |S| < |T|.