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mathematics

THE CONTINUUM
HYPOTHESIS
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
By Nancy McGough (nm@noadsplease.ii.com)

 

 

  1. Overview
    • 1.1 What is the Continuum Hypothesis?
    • 1.2 Current Status of CH
  2. Alternate Overview
  3. Assumptions, Style, and Terminology
    • 2.1 Assumptions
      • 2.1.1 Audience Assumptions
      • 2.1.2 Mathematical Assumptions
    • 2.2 Style
    • 2.3 Terminology
      • 2.3.1 The Word "continuum"
      • 2.3.2 Ordered Sets
      • 2.3.3 More Terms and Notation
  4. Mathematics of the Continuum and CH
    • 3.1 Sizes of Sets: Cardinal Numbers
      • 3.1.1 aleph0 < c = 2aleph0
      • 3.1.2 CH and GCH
      • 3.1.3 Sample Cardinalities
    • 3.2 Ordering Sets: Ordinal Numbers
    • 3.3 Analysis of the Continuum
      • 3.3.1 Decomposing the Reals
      • 3.3.2 Characterizing the Reals
      • 3.3.3 Characterizing Continuity
    • 3.4 What ZFC Does and Does Not Tell Us About c
  5. Metamathematics and CH
    • 4.1 Consistency, Completeness, and Compactness of ...
      • 4.1.1 a Logical System
      • 4.1.2 an Axiomatic Theory
    • 4.2 Models of ...
      • 4.2.1 Real Numbers
      • 4.2.2 Set Theory
        • 4.2.2.1 Inner Models
        • 4.2.2.2 Forcing and Outer Models
    • 4.3 Adding Axioms to Zermelo Fraenkel Set Theory
      • 4.3.1 Axioms that Imply CH or GCH
        • 4.3.1.1 Explicitly Adding CH or GCH
        • 4.3.1.2 V=L: Shrinking the Set Theoretic Universe
        • 4.3.1.3 Diamond
      • 4.3.2 Axioms that Tell Us About Cardinal Exponentiation
        • 4.3.2.1 Martin's Axiom: A Weak Version of CH
        • 4.3.2.3 The Singular Cardinal Hypothesis
      • 4.3.3 Large Cardinal Axioms: Exploding the Set Theoretic Universe
      • 4.3.4 ZF+AD: A Set Theoretic Universe in Which the Axiom of Choice is False
    • 4.4 Alternate Mathematical Frameworks
      • 4.4.1 Category Theory
      • 4.4.2 Intuitionist Mathematics
      • 4.4.3 Alternate Logics
  6. Philosophy of CH
    • 5.1 Mathematical Realism: CH is True or False
      • 5.1.1 CH and the Physical World
    • 5.2 Not Mathematical Realism
  7. Conclusion

  8. Appendix 1: Timeline
  9. Appendix 2: Confusion About CH in Popular Literature
  10. Glossaries
  11. References
  12. Acknowledgments
  13. Search the Net for "continuum hypothesis"
  14. Public Discussion


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1. Overview


People have tried to understand space, time, motion, and the notion of "continuum" for thousands of years. This pursuit lead to the Pythagoreans discovery of irrational numbers, Zeno's paradoxes, infinitesimal calculus, transfinite set theory, relativity theory, quantum physics, and many more intriguing ideas.

What do we mean when we say "continuum"? Here's a description Albert Einstein gave on p. 83 of his Relativity: The Special and the General Theory:

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a "neighboring" one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here by "neighbouring" and by "jumps" (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.

Hermann Weyl said "let us stick to time as the most fundamental continuum" and gave the following description on p. 92 of his The Continuum:

So we can gather the following concerning objectively presented time:
  1. an individual point in it is non-independent, i.e., is pure nothingness when taken by itself, and exists only as a "point of transition" (which, of course, can in no way be understood mathematically).

  2. it is due to the essence of time (and not to contingent imperfections of our medium) that a fixed time-point cannot be exhibited in any way, that always only an approximation, never an exact determination is possible.
Corresponding remarks apply to every intuitively given continuum; in particular, to the continuum of spatial extension.


1.1 What is the Continuum Hypothesis?

In 1874 Georg Cantor discovered that there is more than one level of infinity. The lowest level is called "countable infinity" and higher levels are called "uncountable infinities." The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set. In 1877 Cantor hypothesized that the number of real numbers is the next level of infinity above countable infinity. Since the real numbers are used to represent a linear continuum, this hypothesis is called "the Continuum Hypothesis" or CH.

Let c be the cardinality of (i.e., number of points in) a continuum, aleph0 be the cardinality of any countably infinite set, and aleph1 be the next level of infinity above aleph0. Here are six ways to state CH:

Sets-of-Reals Versions of CH

Cardinal-Number Versions of CH

The Continuum Hypothesis has been, and continues to be, one of the most hotly pursued problems in mathematics. It was the first problem in Hilbert's list of 23 important unsolved problems, ten of which he presented to the Second International Congress of Mathematicians at Paris in 1900. Pursuit of the Continuum Hypothesis has motivated a lot of useful and interesting mathematics in real analysis, topology, set theory, and logic.


1.2 Current Status of CH

Despite nearly 120 years of investigation, CH is still debated and continues to motivate a lot of mathematics, especially in set theory and logic. Like the Axiom of Choice (AC), Gödel showed that CH is consistent with standard set theory and Cohen showed that ~CH is consistent with standard set theory (and thus CH is independent of standard set theory). But, unlike AC, CH has not been adopted as an axiom of set theory. Instead, mathematicians either live with this incompleteness in set theory or try to find more intuitive axioms that will help to decide CH. In the "Introduction" of the Israel Mathematical Conference Proceedings, Vol. 6, 1993 Haim Judah said:

We still think that the study of the size of the continuum should be our guiding light for further research in set theory.


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1a. Alternate Overview


For a concise discussion of the continuum hypothesis, see my draft continuum hypothesis `FAQ,' which I'm writing for the sci.math FAQ.


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2. Style, Assumptions, and Terminology



2.1 Style

I link only the first occurrence of a word or phrase.

I use a minimal amount of graphics in order to:

I'm trying to figure out the best way to display the aleph character and some other mathematical symbols. Possibilities include: Some good information about using extended character sets with web browsers is at Alan Flavell's Notes on Internationalization.

 

2.2 Assumptions


2.2.1 Mathematical Assumptions

For most of this article, I assume: In the metamathematics section I discuss CH in the absence of some of these assumptions.


2.2.2 Audience Assumptions


2.3 Terminology


2.3.1 The Word "continuum"

Many people use the phrase "the continuum" to mean the real number line but be aware that there are many types of continua, including:


2.3.2 Ordered Sets

Ordered sets are usually surrounded by angle brackets rather than square brackets. Unfortunately, it's hard to produce good angle brackets in HTML. The best I could come up with is something like this:
<set, order relation>
but these angle brackets look too much like less than and greater than signs (which is what they are!), so, I'm using square brackets for ordered sets, i.e.:
[set, order relation]


2.3.3 More Terms and Notation

  TermMeaning
  ACThe Axiom of Choice
  CHThe Continuum Hypothesis
  GCHThe Generalized Continuum Hypothesis
  ZFZermelo Fraenkel set theory without Choice
  ZFCZermelo Fraenkel set theory with Choice
  A#BA is not equal to B, e.g.: V#L
  ~A not A, e.g.: ~CH
  T |- SStatement S is a theorem of theory T or "T entails S." E.g.: ZF+GCH |- AC
  Nset of natural numbers = {0, 1, 2, ...} (order doesn't matter)
  Rset of real numbers
  P(X)power set of X = set of all subsets of X
  card(X)   cardinality of X
  w first infinite ordinal = set of natural numbers in their natural order = [{0, 1, 2, ...}, <] = [N, <]; pronounced "omega"
  w1 first uncountable ordinal = set of all countable ordinals in their natural order = [{0, 1, 2, ..., w, w+1,..., w+w,..., wxw,...}, <]
  aleph0 first infinite cardinal = card(N) = card(w) = card(any countably infinite set)
  aleph1first uncountable cardinal = card(w1)
  ccardinality of a continuum = card(R) = card(P(N))
  countable   finite or countably infinite
  isomorphic order isomorphic
  measure   Lebesgue Measure



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3. Mathematics of the Continuum and CH


The existence of irrational numbers, the uncountability of the reals, and the self-repeating, fractal-nature of the continuum show us that our intuition can't always be trusted. To avoid problems with intuition, we formalize our intuitive notions and try to rigorously prove or disprove statements about these notions. Notions related to the continuum and CH include measure, Baire category, density, separability, connectedness, continuity, completeness, compactness, and of course cardinality. We start with cardinality, the main concept related to CH.

3.1 Sizes of Sets: Cardinal Numbers

Before Cantor, collections were either finite or infinite and there was no notion of different levels of infinity. Only Wallis's infinity symbol, oo, was needed to represent the notion of infinity. While investigating questions about singularities of Fourier series, Cantor made the revolutionary discovery that not all infinite sets are the same size.

Cantor showed that the natural numbers (N), the integers (Z), and the rational numbers (Q) are all the same size by constructing one-to-one correspondences between them (for details see Seton Hall University's Interactive Real Analysis). Cantor described these as "countably infinite" or having "cardinality aleph0." Cantor also showed that the real numbers can not be put into one-to-one correspondence with a countably infinite set and thus are not countably infinite. After this surprising discovery, Cantor proposed the Continuum Hypothesis and developed transfinite set theory, the "paradise of the infinite" from which Hilbert hoped we would never be driven!


3.1.1   aleph0 < c = 2aleph0

We now step through a sketch of the proof of the following, which will help us to define the continuum hypothesis (CH) and the generalized continuum hypothesis (GCH).

aleph0 < card(R) = c = card((0,1)) = card(P(N)) = 2aleph0


3.1.1.1   aleph0 < card(R) = c

Cantor showed, using his famous diagonal argument, that the real numbers (R) cannot be put into one-to-one correspondence with the natural numbers. Since the reals are a superset of N, their cardinality must be larger than aleph0. He called the cardinality of the real numbers c for "continuum."


3.1.1.2  card(R) = card((0,1))

When proving things about c it's sometimes easier to work with a set other than R that has cardinality c. An example of such a set is the set of reals between 0 and 1, which is called "the open interval zero one" and denoted:

(0,1)
There are many ways to show that all the reals can be put into one-to-one correspondence with (0,1). One way to do this mapping is to use the great and powerful arctan function. You can use arctan with different multipliers and shifters to construct a one-to-one correspondence between R and any open interval of reals.

Exercise
f(x)=arctan(x) maps R continuously one-to-one and onto (-pi/2, pi/2). What function of arctan maps R continuously one-to-one and onto (0,1)?

Since this function of arctan maps R continuously one-to-one and onto (0,1) and its inverse -- a function of tan -- maps (0,1) continuously one-to-one and onto R, we see that R and (0,1), in addition to having the same cardinality, are homeomorphic.


3.1.1.3   card((0,1)) = card(P(N)) = 2aleph0

The set of reals between 0 and 1 can be represented by the set of all countably infinite sequences of 0's and 1's . Think of these as representing binary "decimals" between .000000... and .111111... . In this representation .1=1/2, .01=1/4, .11=3/4, etc.

The power set of the natural numbers, P(N), can also be represented by the set of all countably infinite sequences of 0's and 1's. Each sequence represents a subset of N by interpreting a 0 in position n to mean that the number n is not in the subset and a 1 in position n to mean the number n is in the subset. This way of specifying a set is called the "characteristic function" of the set.

One way to represent all countably infinite sequences of 0's and 1's is to use Cartesian product notation:

{0, 1} x {0, 1} x {0, 1} x ... = {0, 1}aleph0
Since in set theory {0, 1} = 2, we can also write this as:

2aleph0
So we now have:

aleph0 < c = card(R) = card((0,1)) = card(P(N)) = 2aleph0


3.1.2 CH and GCH

Since CH is the proposal that c is the next level of infinity above aleph0, namely aleph1, and we have just shown that c = 2aleph0, another way to state CH is:

2aleph0 = aleph1
This is sometimes called the "arithmetic version of CH." In 1908 Felix Hausdorff proposed the following generalization of CH:

For any cardinal alepha,   2alepha = alepha+1
This is the Generalized Continuum Hypothesis or GCH. Another way to state GCH is:

   {card(N), card(P(N)), card(P(P(N))), card(P(P(P(N)))), . . .} = {aleph0, aleph1, aleph2, aleph3, . . .}

Obviously CH follows from GCH and we have:

ZF+GCH |- CH

Note that ZF+GCH |- AC so it would be redundant (and misleading) to put ``ZFC+GCH'' on the left side of the turnstile.


3.1.3 Sample Cardinalities

After Cantor showed that there are different levels of infinity, Cantor and others quickly discovered the cardinality of many sets, including the following.


Cardinality    Samples
aleph0
  • N = natural numbers
  • w = [N, <] = natural numbers in their natural order
  • Z = integers
  • Q = rational numbers
  • algebraic numbers
  • set of all finite sets of natural numbers
  • set of all cofinite sets of natural numbers (sets whose complements are finite)
  • <ww = <wN = set of all finite sequences of natural numbers
aleph1
  • w1 = set of all countable ordinals
c = 2aleph0
  • R = real numbers
  • C = complex numbers
  • irrational numbers
  • transcendental numbers
  • R2 = 2-dimensional Euclidean space
  • R3 = 3-dimensional Euclidean space
  • Rn = n-dimensional Euclidean space
  • any non-empty open set of reals
  • any perfect set of reals
  • any Cantor set
  • any uncountable closed set of reals
  • any uncountable Borel set of reals
  • any uncountable analytic set of reals
  • any set of reals with positive measure
  • any set of reals of second category (not meager)
  • P(N) = set of all subsets of the natural numbers
  • ww = wN = set of all countably infinite sequences of natural numbers
  • w2 = set of all countably infinite sequences of 0's and 1's
  • set of all coinfinite sets of natural numbers (sets whose complements are infinite)
  • set of all infinite coinfinite sets of natural numbers
  • set of all open sets of reals
  • set of all closed sets of reals
  • set of all Borel sets of reals
  • set of all analytic sets of reals
  • C(R) = set of all continuous functions from R to R
  • set of all analytic functions from R to R
2c = 22aleph0
  • P(R) = set of all subsets of the real numbers
  • RR = set of all functions from R to R
  • P(P(N)) = set of all subsets of the power set of N
In this hierarchy of cardinalities, CH is the claim that the aleph1 and c levels should be merged into one level. And GCH would mean that 2c = aleph2, 22c = aleph3, etc.


3.2 Ordering Sets: Ordinal Numbers

To fully understand CH in the context of ZFC you need to understand the second infinite cardinal, aleph1, which is the cardinality of w1, the set of all countable ordinals. Ordinal numbers tell you both the size of a set and the way its members are ordered. Ordering sets is important because it is the way we extend the notion of counting to infinite sets.

For a finite set, ordering is a breeze. No matter how you order a set with n members, its order type is isomorphic to:

0  1  2  3  . . .  n-1
This is sometimes written as [{0, 1, 2, ..., n-1},<] or [n,<] to emphasize that it is an ordered set.

For infinite sets, the situation is quite different. It turns out that for a countably infinite set, there are uncountably many non-isomorphic ways to well order it. Here are some examples of ways to well order a countably infinite set. The letters listed in the right column are used as place holders -- no member of the set is actually repeated in its ordering.

Countable Ordinal Order Type (what it looks like)
w a b c...
w+1 a b c... a
w+2 a b c... a b
wx2 = w+w a b c... a b c...
w2 = wxw = w+w...+w...     a b c... a b c... . . . abc... . . .
The collection of all countable ordinals is w1 and the cardinality of this collection is the next level of infinity after aleph0, namely aleph1. (A proof that the cardinality of the set of all countable ordinals is the next cardinal after aleph0 is in Leary.) Now, try to wrap your mind around the question of whether a continuum has the same number of points as this!

For more information about cardinal and ordinal numbers, see Infinite Ink's Cardinal Numbers.

 

[Under Construction] 3.3 Analysis of the Continuum

One path to finding out if CH is true is to look for sets of real numbers that have cardinality greater than aleph0 and less than c. If such sets exist, then CH is false. While looking for these sets mathematicians have decomposed the reals into different types of sets and come up with characterizations of the reals, continua, and continuity.

3.3.1 Decomposing the Reals

There are many ways to decompose the reals. We can split the reals into two sets, such as the rational and irrational numbers or the algebraic and transcendental numbers, or we can look at types of subsets of the reals, such as Borel sets and non-Borel sets. The hope is that these decompositions will help us to characterize the entire set of reals. For example, the rationals provide a good approximation to the reals because any real number can be approximated "as closely as you please" by a rational number. This should serve as a warning though. Despite the rationals' great ability to represent the reals they are of a different cardinality than the reals. So, just because a collection of sets is a good approximation of the reals doesn't mean that it will give us much useful information about cardinality questions. Bearing this in mind, let's try to understand CH by looking at sets that, in some sense, represent the reals.


CH is true for closed, Borel, and analytic sets, i.e.:

This means that if there is a set that falsifies CH, i.e., a set with cardinality between aleph0 and c, it will not be one of these types of sets. Since there are only c of these types of sets and there are 2c subsets of R, there are plenty of sets left which might falsify CH!

3.3.2 Characterizing the Reals

R is a complete ordered field.

Any total ordering [X,<] satisfying the following three conditions is isomorphic to [R,<].

  1. X has no first or last element
  2. X is connected in the order topology
  3. X is separable in the order topology
(more to come)


3.3.3 Characterizing Continuity

Hermann Weyl said that "Precisely what eludes us is the nature of the continuity, the flowing form point to point; in other words, the secret of how the continually enduring present can continually slip away into the receding past."

The Hahn-Mazurkiewicz Theorem
A topological space X is a continuous curve iff X is a compact Hausdorff space which is second countable, connected, and locally connected.


3.4 What ZFC Does and Does Not Tell Us About c

ZFC does not tell us much about c, it only tells us that c:


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[Under Construction] 4. Metamathematics and CH


The results of Gödel and Cohen about the consistency and independence of CH are metamathematical theorems. Questions that are not within the framework of standard mathematics, but are rather about the framework of mathematics, are part of metamathematics. Today the standard framework for mathematics is first-order logic and ZFC.

Most interesting results in logic are about the interplay between formal theory and model theory. An example of this type of result is Gödel's First Incompleteness Theorem, which tells us that if we have:

then there is a statement U that is true in the model but cannot be proved in the formal theory. This type of statement is consistent with T but "undecidable" in T. Gödel's Completeness theorem tells us that if T is consistent then U is undecidable in T if and only if models exist for both T+U and T+~U. Since ZFC contains arithmetic, it has been known since 1938, when Gödel proved his incompleteness theorems, that it is incomplete. CH was suspected to be one of its undecidable statements but it wasn't until 1963 that Cohen proved the independence of CH. He did this by constructing a model of ZFC in which CH is false (i.e., a model of ZFC+~CH). The technique he used to construct the model is called "forcing." In 1966 Cohen received a Fields Medal for his work.

A good discussion of these topics is in What is Mathematical Logic by J.N. Crossley and others. The last chapter gives a nice description of forcing and the construction of a model of ZFC+~CH.


4.1 Models

(standard model, non-standard analysis and the hyperreal line, Hensel's p-adics, forcing models)

One way to resolve CH would be to come up with a model of the reals, which we all accept as being the "correct" interpretation of the reals, in which CH is true or false. A candidate is Robinson's hyperreal line.


4.2 Adding Axioms to ZFC

Most mathematicians work within ZFC but some, usually set theorists, venture into the strange world of ZFC with additional axioms. Many of these axioms were proposed because they shed light on the continuum and CH.


4.2.1 Why Not Just add GCH or ~GCH?

The most obvious axiom to add is one that explicitly asserts GCH or its negation, ~GCH. Since neither GCH nor its negation is "intuitively obvious," this is not done. Whenever someone proves something using either GCH or ~GCH it is explicitly stated. Contrast this with AC, which is used freely in mathematics and its use is often not mentioned.


4.2.2 Large Cardinal Axioms

Large cardinals are cardinals that cannot be constructed using only the axioms of ZFC. Examples of large cardinals are inaccessible cardinals, hyperinaccessible cardinals, Mahlo cardinals, measurable cardinals, and supercompact cardinals. There are lots more. Surprisingly, some of these shed light on questions about the real numbers and the continuum.

MC = there exists a measurable cardinal
SC = there exists a supercompact cardinal

ZFC+MC |- V#L
ZFC+SC |- Projective Determinacy


4.2.3 Martin's Axiom: A Weak Version of CH


ccc or countable chain condition or countable antichain condition
A topological space is ccc if every family of disjoint open sets is countable.

Example: R with the usual topology (open sets = open intervals) is ccc.

MA or Martin's Axiom
If X is a ccc compact Hausdorff space, then X is not the union of less than 2aleph0 nowhere dense sets.

Suslin line
A connected, linearly ordered, ccc space which is not separable

ZFC+CH |- MA
ZF+GCH |- (inaccessible = weakly inaccessible)
ZFC+MA+ ~CH |- There is no Suslin line.


4.3 ZF and Adding Axioms (other than Choice) to ZF

Without the Axiom of Choice, there is no guarantee that infinite cardinal numbers are comparable. Cardinal numbers can form a partially ordered set and aleph1 and c might be on two incomparable branches of the partial order.

(Note: c is comparable to aleph0 in ZF but in ZF there can be infinite sets that do not contain a countably infinite subset. These are called "Dedekind finite" sets.)


4.2.1 ZF+GCH


4.3.1 V=L: Shrinking the Set Theoretic Universe

ZF+V=L |- AC
ZF+V=L |- GCH
ZF+V=L |- Diamond
ZF+Diamond |- CH


4.3.2 ZF+AD: A Set Theoretic Universe that's Incompatible with ZFC

The Axiom of Determinacy...

In ZF+AD the Axiom of Choice is false and the reals cannot be well ordered and, thus, are not equal to any aleph. But, within this system every set of reals is either countable or has the cardinality of all the reals so the first three of the six versions of CH listed in section 1.1 hold.


4.4 Alternate Mathematical Frameworks

Another path to resolving CH is to view all of mathematics through a framework other than the standard first-order logic and Zermelo Fraenkel set theory. Possibilities include using Second-Order Logic, Linear Logic, Intuitionist Logic, or the meta framework, Category Theory, which can be used to encompass all frameworks.


Intuitionist Logic

If we restate the question in this form: "Is it impossible to construct infinite sets of real numbers between 0 and 1, whose power is less than that of the continuum, but greater than aleph-null?", then the answer must be in the affirmative; for the intuitionist can only construct denumerable sets of mathematical objects, and if on the basis of intuition of the linear continuum, he admits elementary series of free selections as elements of construction, then each non-denumerable set constructed by means of it contains a subset of the power of the continuum.

- Brouwer, 1912, p. 134 of   Collected Works


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[Under Construction] 5. Philosophy of CH


Is there a truth about CH and, if there is, can we know this truth? What does CH have to do with reality, both mathematical reality and physical reality? These type of philosophical questions have been debated since at least the time of Ancient Greece when the Pythagoreans discovered irrational numbers and the mystery of the continuum started to unfold.

The two main philosophical viewpoints are realist and antirealist. The quote below by René Thom characterizes pretty well these two views. Kronecker represents the antirealist view and Thom represents the realist view.

Much emphasis has been placed during the past fifty years on the reconstruction of the geometric continuum from the natural integers, using the theory of Dedekind cuts or the completion of the field of rational numbers. Under the influence of axiomatic and bookish traditions, man perceived in discontinuity the first mathematical Being: "God created the integers and the rest is the work of man." This maxim spoken by the algebraist Kronecker reveals more about his past as a banker who grew rich through monetary speculation than about his philosophical insight. There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity. If one has any consciousness at all, it is consciousness of time and space; geometric continuity is in some way inseparably bound to conscious thought.


5.1 Mathematical Realism: CH is True or False

As ??? said, most mathematicians are realists during the week and formalists on the weekend. The following are some mathematicians (with their realist hats on!) who've stated their opinion about the truth of CH: Here's a quote from Cohen. (He uses C, instead of c, for continuum.)

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now aleph1 is the set of countable ordinals and this is merely a special and the simplest way of generating a highter cardinal. The set C is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach C. Thus C is greater than alephn, alephw, alepha, where a= alephw, etc. This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently.

- Paul Cohen on p. 151 of   Set Theory and the Continuum Hypothesis


5.1.1 CH and the Physical World

For the advancing army of physics, battling for many a decade with heat and sound, fields and particles, gravitation and spacetime geometry, the cavalry of mathematics, galloping out ahead, provided what it thought to be the rationale for the real number system. Encounter with the quantum has taught us, however, that we acquire our knowledge in bits; that the continuum is forever beyond our reach. Yet for daily work the concept of the continuum has been and will continue to be as indispensable for physics as it is for mathematics. In either field of endeavor, in any given enterprise, we can adopt the continuum and give up absolute logical rigor, or adopt rigor and give up the continuum, but we can't pursue both approaches at the same time in the same application.

- John Archibald Wheeler on p. xii of Weyl's   The Continuum


5.2 Not Mathematical Realism

The two mainstream antirealist views are formalists and constructivists.

(For now see my descriptions of formalism and constructivism in the Axiom of Choice section of the sci.math FAQ)


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[Under Construction] 6. Conclusion


Maybe the conclusion from all this is that trying to reduce the continuum to a set of individual points is the wrong way to think about it. This reductionist strategy has also had problems in physics. Today there are many physicists moving away from reductionism and towards holism. Maybe a similar path should be taken in mathematics, away from thinking of point sets as the fundamental objects and towards thinking of structures and relations as the fundamental objects of mathematics.

There is a fundamental error in separating the parts from the whole, the mistake of atomizing what should not be atomized. Unity and complementarity constitute reality.

- Werner Heisenberg


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Appendix 1: Timeline


c.540 BCEPythagoras
c.515 BCEParmenides of Elea
c.490-c.435 BCEZeno of Elea
c.408-355 BCEEudoxos
384-322 BCEAristotle
c.287-212 BCEArchimedes
1564-1642Galilei, Galileo
1571-1630Kepler, Johannes
1642-1727Newton, Isaac
1646-1716Leibniz, Gottfried Wilhelm
1777-1855Gauss, Karl Friedrich
1781-1848Bolzano, Bernard
1789-1857Cauchy, Augustine Louis
1815-1897Weierstrass, Karl
1831-1916Dedekind, Julius
1845-1918Cantor, Georg
1862-1943Hilbert, David
December, 1873Cantor proved R uncountable
1877Cantor proposed CH
1878Cantor's proposal of CH published
1908Hausdorff proposed GCH
1934-Cohen, Paul born
April, 1963Cohen circulated notes about independence of CH
May 3, 1963Cohen lectured on independence of CH



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Appendix 2: Confusion About CH in Popular Literature


The most common confusions about CH in popular literature (and in Internet discussions!) are a writer:

  1. Assuming CH or GCH is true without stating this assumption. Since CH is not a standard assumption in mathematics and, in fact, most set theorists think it is false, it is important for a writer to state her assumptions about CH.

  2. Misstating CH. A common misstatement of CH is c = 2aleph0. As we saw in section 3.1.1, this statement is a fairly simple theorem to prove within ZFC.


An example of confusion is in George Gamow's One Two Three...Infinity. On page 34 he says:

The sequence of numbers (including the infinite ones!) now runs:

    1 2 3 4 5 ... aleph1 aleph2 aleph3 ...

and we say "there are aleph1 points on a line" or "there are aleph2 different curves" ...


Another example of confusion is on page 46 of Michael Guillen's Bridges to Infinity:

an aleph0 set ... has precisely 2aleph0 conceivable subsets ... It is the first stepping stone beyond infinity, the first transfinite number, which Cantor named aleph1 ... A set with aleph1 elements in turn, has precisely 2aleph1 conceivable subsets. This is the second stepping stone, the second transfinite number aleph2, and so forth.

Guillen continues his confusion on page 50:

To this day we still don't know exactly how many irrationals there are, although it has been established that the total number cannot be more than aleph1. Cantor himself guessed that the total number of irrationals is exactly aleph1, mainly since aleph1 is the next largest infinity after aleph0 defined by set theory. His guess came to be known as the Continuum Hypothesis. But there is still the uneliminated possibility that the number of irrationals actually lies between aleph0 and aleph1.


Your mission, should you choose to accept it, is to find all the mistakes in these quotes. After you've done that, take a look at the mistakes I found and let me know if you find any more (or disagree with the ones I found).

If you know of other misstatements of CH, please let me know.


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Glossaries

Eric's Treasure Trove of Mathematics - mirrored in U.S. and Italy

Eric's Treasure Trove of Physics - mirrored in U.S. and Italy

Infinite Ink's Glossary (very small right now!)

Interactive Real Analysis Glossary

Topology Atlas

UC Davis Math Department Glossary

(Please tell me about other mathematics, physics, and philosophy dictionaries and glossaries.)


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References


References are on a separate page now...


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Acknowledgments


Thanks to Jeremy Henty who found a mistake in my sample cardinalities list.

More thanks coming...


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Search the Net for "continuum hypothesis"


The following canned searches will give you a quick start at searching the Net for the phrase "continuum hypothesis" - have fun!

The next section gives links for searching Usenet news archives.


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Public Discussion


Good news groups for discussing these topics are:

You can also search the Deja News archives using the following canned searches:
continuum & hypothesis
((~g sci.math*) continuum) | ((~g sci.math*) continua)
((~g sci.logic) continuum) | ((~g sci.logic) continua)
((~g sci.physics*) continuum) | ((~g sci.physics*) continua)
((~g *philosophy*) continuum) | ((~g *philosophy*) continua)
(~s continuum) | (~s continua)



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