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mathematics faq

THE CONTINUUM
HYPOTHESIS FAQ

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By Nancy McGough (nm@noadsplease.ii.com)

 

This is a draft of an article that will become part of the sci.math FAQ, which is regularly posted to the sci.math news group.

 
The continuum hypothesis was proposed by Georg Cantor in 1877 after he showed that the real numbers cannot be put into one-to-one correspondence with the natural numbers. Cantor hypothesized that the number of real numbers is the next level of infinity above the number of natural numbers. He used the Hebrew letter aleph to name the different levels of infinity: aleph_0 is the number of (or cardinality of) the natural numbers or any countably infinite set, and the next levels of infinity are aleph_1, aleph_2, aleph_3, et cetera. Since the reals form the quintessential continuum, Cantor named the cardinality of the reals c, for continuum. Cantor's original formulation of the continuum hypothesis, or CH, can be stated as either:

  1. card(R)=aleph_1
  2. c=aleph_1
where `card(R)' means `the cardinality of the reals.' An amazing fact that Cantor also proved is that the cardinality of the set of all subsets of the natural numbers -- the power set of N or P(N) -- is equal to the cardinality of the reals. So, another way to state CH is:
  1. card(P(N))=aleph_1
Since the cardinality of the power set of any set, X, is 2^card(X), card(P(N)) is 2^aleph_0 and CH can also be stated as:
  1. 2^aleph_0=aleph_1
Even though these different versions of CH are equivalent in ZFC they have different flavors and help us to think about the problem, and its possible solution, in different ways. The first two lead us to think about sizes of sets of reals, the third leads us to think about subsets, sequences, and trees of natural numbers, and the last leads us to think about cardinal exponentiation.

Cantor and many other great mathematicians spent years trying to prove CH or its negation, ~CH. This problem was so important that Hilbert put it first in his list of 23 problems that he thought were the most important for twentieth century mathematics. In 1938 significant progress was made when Goedel proved that CH is consistent with ZFC by constructing a model of ZFC+CH. At this same time Goedel proved his famous incompleteness theorems and showed that ZFC is an example of an incomplete system. This means that there are statements in the language of set theory -- called undecidable statements -- that can neither be proved nor disproved. Mathematicians suspected that CH was undecidable in ZFC but it took until 1963 until this was proved. Paul Cohen constructed a model of ZFC+~CH and this, along with Goedel's model of ZFC+CH, showed that CH is undecidable in ZFC. So this means that either CH or ~CH could be added as an axiom of ZFC but since neither of these seem axiomatic or `self evident,' mathematicians have instead tried to find another axiom that will tell us the cardinality of the continuum. The pursuit of the continuum problem has motivated a lot of work in set theory and in mathematics in general.

 
need to write more...   brief discussion of descriptive set theory, martin's axiom, singular cardinal hypothesis, large cardinals, philosophy.

 
Lots more about the continuum hypothesis is at www.ii.com/math/ch/.

 
 

 

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Copyright © 1992-1998 Infinite Ink and Nancy McGough
Last significant update July 20, 1997
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