definition of continuum hypothesis

Continuum Hypothesis

Gödel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory . However, using a technique called forcing , Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory . Together, Gödel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the axiom of choice ).

Woodin (2001ab, 2002) formulated a new plausible "axiom" whose adoption (in addition to the Zermelo-Fraenkel axioms and axiom of choice ) would imply that the continuum hypothesis is false. Since set theoreticians have felt for some time that the Continuum Hypothesis should be false, if Woodin's axiom proves to be particularly elegant, useful, or intuitive, it may catch on. It is interesting to compare this to a situation with Euclid's parallel postulate more than 300 years ago, when Wallis proposed an additional axiom that would imply the parallel postulate (Greenberg 1994, pp. 152-153).

Portions of this entry contributed by Matthew Szudzik

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8.5: The Continuum Hypothesis and The Generalized Continuum Hypothesis

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The word “continuum” in the title of this section is used to indicate sets of points that have a certain continuity property. For example, in a real interval it is possible to move from one point to another, in a smooth fashion, without ever leaving the interval. In a range of rational numbers this is not possible, because there are irrational values in between every pair of rationals. There are many sets that behave as a continuum – the intervals \((a, b)\) or \([a, b]\), the entire real line \(\mathbb{R}\), the \(x\)-\(y\) plane \(\mathbb{R} × \mathbb{R}\), a volume in \(3\)-dimensional space (or for that matter the entire space \(\mathbb{R}^3\)). It turns out that all of these sets have the same size.

The cardinality of the continuum, denoted c , is the cardinality of all of the sets above.

In the previous section, we mentioned the continuum hypothesis and how angry Cantor became when someone (König) tried to prove it was false. In this section, we’ll delve a little deeper into what the continuum hypothesis says and even take a look at CH’s big brother, GCH. Before doing so, it seems like a good idea to look into the equivalences we’ve asserted about all those sets above which (if you trust us) have the cardinality c .

We’ve already seen that an interval is equivalent to the entire real line but the notion that the entire infinite Cartesian plane has no more points in it than an interval one inch long defies our intuition. Our conception of dimensionality leads us to think that things of higher dimension must be larger than those of lower dimension. This preconception is false as we can see by demonstrating that a \(1×1\) square can be put in one-to-one correspondence with the unit interval. Let \(S = \{(x, y) 0 < x < 1 ∧ 0 < y < 1\}\) and let \(I\) be the open unit interval \((0, 1)\). We can use the Cantor-Bernstein-Schroeder theorem to show that \(S\) and \(I\) are equinumerous – we just need to find injections from \(I\) to \(S\) and vice versa. Given an element \(r\) in \(I\) we can map it injectively to the point \((r, r)\) in \(S\). To go in the other direction, consider a point \((a, b)\) in \(S\) and write out the decimal expansions of \(a\) and \(b\):

\(a = 0.a_1a_2a_3a_4a_5 . . .\)

\(b = 0.b_1b_2b_3b_4b_5 . . .\)

as usual, if there are two decimal expansions for \(a\) and/or \(b\) we will make a consistent choice – say the infinite one.

From these decimal expansions, we can create the decimal expansion of a number in \(I\) by interleaving the digits of \(a\) and \(b\). Let

\(s = 0.a_1b_1a_2b_2a_3b_3 . . \).

be the image of \((a, b)\). If two different points get mapped to the same value s then both points have \(x\) and \(y\) coordinates that agree in every position of their decimal expansion (so they must really be equal). It is a little bit harder to create a bijective function from \(S\) to \(I\) (and thus to show the equivalence directly, without appealing to C-B-S). The problem is that, once again, we need to deal with the non-uniqueness of decimal representations of real numbers. If we make the choice that, whenever there is a choice to be made, we will use the non-terminating decimal expansions for our real numbers there will be elements of \(I\) not in the image of the map determined by interleaving digits (for example \(0.15401050902060503\) is the interleaving of the digits after the decimal point in \(π = 3.141592653\). . . and \(\dfrac{1}{2} = 0.5\), this is clearly an element of \(I\) but it can’t be in the image of our map since \(\dfrac{1}{2}\) should be represented by \(0.49\) according to our convention. If we try other conventions for dealing with the non-uniqueness it is possible to find other examples that show simple interleaving will not be surjective. A slightly more subtle approach is required.

Presume that all decimal expansions are non-terminating (as we can, WLOG) and use the following approach: Write out the decimal expansion of the coordinates of a point \((a, b)\) in \(S\). Form the digits into blocks with as many \(0\)’s as possible followed by a non-zero digit. Finally, interleave these blocks.

For example if

\(a = 0.124520047019902 . . .\)

\(b = 0.004015648000031 . . .\)

we would separate the digits into blocks as follows:

\(a = 0.1 \;\;\;\;2 \;\;\;\;4 \;\;\;\;5 \;\;\;\;2\;\;\;\; 004\;\;\;\; 7\;\;\;\; 01 \;\;\;\;9 \;\;\;\;9 \;\;\;\;02 . . .\)

\(b = 0.004\;\;\;\; 01\;\;\;\; 5 \;\;\;\;6\;\;\;\; 4\;\;\;\; 8 \;\;\;\;00003 \;\;\;\;1 . . .\)

and the number formed by interleaving them would be

\(s = 0.10042014556240048 . . .\)

We’ve shown that the unit square, \(S\), and the unit interval, \(I\), have the same cardinality. These arguments can be extended to show that all of R×R also has this cardinality ( c ).

So now let’s turn to the continuum hypothesis.

We mentioned earlier in this chapter that the cardinality of \(\mathbb{N}\) is denoted \(ℵ_0\). The fact that that capital letter aleph is wearing a subscript ought to make you wonder what other aleph-sub-something-or-others there are out there. What is \(ℵ_1\)? What about \(ℵ_2\)? Cantor presumed that there was a sequence of cardinal numbers (which is itself, of course, infinite) that give all of the possible infinities. The smallest infinite set that anyone seems to be able to imagine is \(\mathbb{N}\), so Cantor called that cardinality \(ℵ_0\). What ever the “next” infinite cardinal is, is called \(ℵ_1\). It’s conceivable that there actually isn’t a “next” infinite cardinal after \(ℵ_0\) — it might be the case that the collection of infinite cardinal numbers isn’t well-ordered! In any case, if there is a “next” infinite cardinal, what is it? Cantor’s theorem shows that there is a way to build some infinite cardinal bigger than \(ℵ_0\) — just apply the power set construction. The continuum hypothesis just says that this bigger cardinality that we get by applying the power set construction is that “next” cardinality we’ve been talking about.

To re-iterate, we’ve shown that the power set of \(\mathbb{N}\) is equivalent to the interval \((0, 1)\) which is one of the sets whose cardinality is \(\text{c}\) . So the continuum hypothesis, the thing that got Georg Cantor so very heated up, comes down to asserting that

\(ℵ_1 =\) c .

There really should be a big question mark over that. A really big question mark. It turns out that the continuum hypothesis lives in a really weird world. . . To this day, no one has the least notion of whether it is true or false. But wait! That’s not all! The real weirdness is that it would appear to be impossible to decide. Well, that’s not so bad – after all, we talked about undecidable sentences way back in the beginning of Chapter 2 . Okay, so here’s the ultimate weirdness. It has been proved that one can’t prove the continuum hypothesis. It has also been proved that one can’t disprove the continuum hypothesis.

Having reached this stage in a book about proving things I hope that the last two sentences in the previous paragraph caused some thought along the lines of “well, ok, with respect to what axioms?” to run through your head. So, if you did think something along those lines pat yourself on the back. And if you didn’t then recognize that you need to start thinking that way — things are proved or disproved only in a relative way, it depends what axioms you allow yourself to work with. The usual axioms for mathematics are called ZFC; the Zermelo-Frankel set theory axioms together with the axiom of choice. The “ultimate weirdness” we’ve been describing about the continuum hypothesis is a result due to a gentleman named Paul Cohen that says “CH is independent of ZFC.” More pedantically – it is impossible to either prove or disprove the continuum hypothesis within the framework of the ZFC axiom system.

It would be really nice to end this chapter by mentioning Paul Cohen, but there is one last thing we’d like to accomplish — explain what GCH means. So here goes.

The generalized continuum hypothesis says that the power set construction is basically the only way to get from one infinite cardinality to the next. In other words, GCH says that not only does \(\mathcal{P}(\mathbb{N})\) have the cardinality known as \(ℵ_1\), but every other aleph number can be realized by applying the power set construction a bunch of times. Some people would express this symbolically by writing

\[∀n ∈ \mathbb{N}, \;\;\;\;\; ℵ_{n+1} = 2^{ℵ_n} .\]

I’d really rather not bring this chapter to a close with that monstrosity so instead I think I’ll just say

Paul Cohen.

Hah! I did it! I ended the chapter by sayi. . . Hunh? Oh.

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Continuum hypothesis

The hypothesis, due to G. Cantor (1878), stating that every infinite subset of the continuum $\mathbf{R}$ is either equivalent to the set of natural numbers or to $\mathbf{R}$ itself. An equivalent formulation (in the presence of the axiom of choice ) is: $$ 2^{\aleph_0} = \aleph_1 $$ (see Aleph ). The generalization of this equality to arbitrary cardinal numbers is called the generalized continuum hypothesis (GCH): For every ordinal number $\alpha$, \begin{equation} \label{eq:1} 2^{\aleph_\alpha} = \aleph_{\alpha+1} \ . \end{equation}

In the absence of the axiom of choice, the generalized continuum hypothesis is stated in the form \begin{equation} \label{eq:2} \forall \mathfrak{k} \,\,\neg \exists \mathfrak{m}\ (\,\mathfrak{k} < \mathfrak{m} < 2^{\mathfrak{k}}\,) \end{equation} where $\mathfrak{k}$,$\mathfrak{m}$ stand for infinite cardinal numbers. The axiom of choice and (1) follow from (2), while (1) and the axiom of choice together imply (2).

D. Hilbert posed, in his celebrated list of problems, as Problem 1 that of proving Cantor's continuum hypothesis (the problem of the continuum). This problem did not yield a solution within the framework of traditional set-theoretical methods of solution. Among mathematicians the conviction grew that the problem of the continuum was in principle unsolvable. It was only after a way had been found of reducing mathematical concepts to set-theoretical ones, axioms had been stated in set-theoretical language which could be placed at the foundations of mathematical proofs actually encountered in real life and logical derivation methods had been formalized, that it became possible to give a precise statement, and then to solve the question, of the formal unsolvability of the continuum hypothesis. Formal unsolvability is understood in the sense that there does not exist a formal derivation in the Zermelo–Fraenkel system ZF either for the continuum hypothesis or for its negation.

In 1939 K. Gödel established the unprovability of the negation of the generalized continuum hypothesis (and hence the unprovability of the negation of the continuum hypothesis) in the system ZF with the axiom of choice (the system ZFC) under the hypothesis that ZF is consistent (see Gödel constructive set ). In 1963 P. Cohen showed that the continuum hypothesis (and therefore also the generalized continuum hypothesis) cannot be deduced from the axioms of ZFC assuming the consistency of ZF (see Forcing method ).

Are these results concerning the problem of the continuum final? The answer to this question depends on one's relation to the premise concerning the consistency of ZF and, what is more significant, to the experimental fact that every meaningful mathematical proof (of traditional classical mathematics) can, after it has been found, be adequately stated in the system ZFC. This fact cannot be proved nor can it even be precisely stated, since each revision raises a similar question concerning the adequacy of the revision for the revised theorem.

In model-theoretic language, Gödel and Cohen constructed models for ZFC in which $$ 2^{\mathfrak{k}} = \begin{cases} \mathfrak{m} & \text{if}\ \mathfrak{k} < \mathfrak{m}\,; \\ \mathfrak{k}^{+} & \text{if}\ \mathfrak{k} \ge \mathfrak{m} \ . \end{cases} $$

where $\mathfrak{m}$ is an arbitrary uncountable regular cardinal number given in advance, and $\mathfrak{k}^{+}$ is the first cardinal number greater than $\mathfrak{k}$. What is the possible behaviour of the function $2^{\mathfrak{k}}$ in various models of ZFC?

It is known that for regular cardinal numbers $\mathfrak{k}$, this function can take them to arbitrary cardinal numbers subject only to the conditions $$ \mathfrak{k} < \mathfrak{k}' \Rightarrow 2^{\mathfrak{k}} < 2^{\mathfrak{k}'} \,,\ \ \ \mathfrak{k} < \text{cf}(\mathfrak{k}) $$ where $\text{cf}(\mathfrak{a})$ is the smallest cardinal number cofinal with $\mathfrak{a}$ (see Cardinal number ). For singular (that is, non-regular) $\mathfrak{k}$, the value of the function $2^{\mathfrak{k}}$ may depend on its behaviour at smaller cardinal numbers. E.g., if \eqref{eq:1} holds for all $\alpha < \omega_1$, then it also holds for $\alpha = \omega_1$.

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• Continuum Hypothesis
• 1.1 Generalized Continuum Hypothesis
• 2 Hilbert $23$
• 3 Historical Note

There is no set whose cardinality is strictly between that of the integers and the real numbers .

Symbolically, the continuum hypothesis asserts:

Generalized Continuum Hypothesis

The Generalized Continuum Hypothesis is the proposition:

Let $x$ and $y$ be infinite sets .

In other words, there are no infinite cardinals between $x$ and $\powerset x$.

Hilbert $23$

This problem is no. $1$ in the Hilbert $23$ .

Historical Note

The Continuum Hypothesis was originally conjectured by Georg Cantor .

In $1940$, Kurt Gödel showed that it is impossible to disprove the Continuum Hypothesis (CH for short) in Zermelo-Fraenkel set theory (ZF) with or without the Axiom of Choice ( ZFC ).

In $1963$, Paul Cohen showed that it is impossible to prove CH in ZF or ZFC .

These results together show that CH is independent of both ZF and ZFC .

Note, however, that these results do not settle CH one way or the other, nor do they establish that CH is undecidable.

They merely indicate that CH cannot be proved within the scope of ZF or ZFC , and that any further progress will depend on further insights on the nature of sets and their cardinality .

It has been suggested that a key factor contributing towards the difficulty in resolving this question may be the fact that Gödel's Incompleteness Theorems prove that there is no possible formal axiomatization of set theory that can represent the entire spread of possible properties that can uniquely specify any possible set .

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Title: independence of the continuum hypothesis: an intuitive introduction.

Abstract: The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the main workhorse of set theory; and it has broad implications on mathematical foundations and on the role of syntax versus semantics. Despite its broad impact, it is not broadly taught. A main reason is the lack of accessible expositions for nonspecialists, because the mathematical structures and techniques employed in the proof are unfamiliar outside of set theory. This manuscript aims to take a step in addressing this gap by providing an exposition at a level accessible to advanced undergraduate mathematicians and theoretical computer scientists, while covering all the technically challenging parts of the proof.

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Notes to Continuity and Infinitesimals

1. The word “continuous” derives from the Latin continēre meaning “to hang together” or “to cohere”; this same root gives us the nouns “continent”—an expanse of land unbroken by sea—and “continence”—self-restraint in the sense of “holding oneself together”. Synonyms for “continuous” include: connected, entire, unbroken, uninterrupted.

2. The word “discrete” derives from a Latin discernere meaning “to separate”. This same root yields the verb “discern”—to recognize as distinct or separate—and the cognate “discreet”—to show discernment, hence “well-behaved”. It is a curious fact that, while “continuity” and “discreteness” are antonyms, “continence” and “discreetness” are synonyms. Synonyms for “discrete” include separate, distinct, detached, disjunct.

3. Of course, this presupposes that there are no “gaps” between the elements or points, which is implicit in the assumption that the points have been obtained by complete division of a continuum.

4. It should also be mentioned that the German philosopher Johann Friedrich Herbart (1776–1841) introduced the term synechology for the part of his philosophical system concerned with the continuity of the real.

5. According to the Oxford English Dictionary the term infinitesimal was originally

an ordinal, viz. the “infinitieth” in order… but, like other ordinals, also used to name fractions, thus infinitesimal part or infinitesimal came to mean unity divided by infinity \((1/\infty)\), and thus an infinitely small part or quantity.

6. For the doctrines of the presocratic philosophers see Kirk, Raven, & Schofield 1983 and Barnes 1982.

7. That this was the Eleatic position may be inferred from Plato’s Parmenides .

8. For the history of the doctrine of atomism see especially Pyle 1997.

9. In Book VI of the Categories , Quantity (ποσόν is introduced by Aristotle as the category associated with how much . In addition to exhibiting continuity and discreteness, quantities are, according to Aristotle, distinguished by the feature of being equal or unequal.

10. Here it must be noted that for Aristotle, as for ancient Greek thinkers generally, the term “number”— arithmos —means just “plurality”.

11. Aristotle points out that (spoken) words are analyzable into syllables or phonemes, linguistic “atoms” themselves irreducible to simpler linguistic elements.

12. For an account of Epicurus’s doctrines, see Furley 1967.

13. He seems to have refrained, however, from subjecting the continuum to his celebrated “razor”.

14. See, e.g., the papers of Murdoch and Stump in Kretzmann 1982.

15. Hermann Weyl makes a similar suggestion in connection with Galileo’s “bending” procedure:

If a curve consists of infinitely many straight “line elements”, then a tangent can simply be conceived as indicating the direction of the individual line segment; it joins two “consecutive ” points on the curve. (Weyl 1926 [1949: 44])

16. This conception was to prove fruitful in the later development of the calculus and to achieve fully rigorous formulation in the smooth infinitesimal analysis of the later twentieth century. See Section 8 .

17. On Barrow, see Child 1916 and Boyer 1939 [1959].

18. On Newton’s contributions to the calculus see Baron 1969 [1987] and Boyer 1939 [1959].

19. On Leibniz see especially Russell 1900 [1937].

20. On Nieuwentijdt and other critics of Leibniz see Mancosu 1996.

21. But the other properties have resurfaced in the theories of infinitesimals which have emerged over the past several decades. Appropriately defined, the relation \(\approx\), property 1 holds of the differentials in nonstandard analysis, while properties 1, 2 and 3 hold of the differentials in smooth infinitesimal analysis. See section 6 and section 8 .

22. On Euler, see especially Truesdell 1972 [1984].

23. Or, to put it another way, (real) numbers are just the ratios of infinitesimals: this is a reigning principle of smooth infinitesimal analysis, see Section 8 below.

24. Likely the astronomer Edmund Halley (1656–1742).

25. Kant would probably maintain the truth of the Thesis in that event.

26. This had been previously given by Bolzano.

27. Fisher argues that here and there in his work Cauchy did “argue directly with infinitely small quantities treated as actual infinitesimals” (1978:315)

28. According to Hobson,

the term “arithmetization” is used to denote the movement which has resulted in placing analysis on a basis free from the idea of measurable quantity, the fractional, negative, and irrational numbers being so defined that they depend ultimately upon the conception of integral number. (1907: 21)

29. The concept of function had by this time been greatly broadened: in 1837 Dirichlet suggested that a variable y should be regarded as a function of the independent vatiable x if a rule exists according to which, whenever a numerical value of x is given, a unique value of y is determined. (This idea was later to evolve into the set-theoretic definition of function as a set of ordered pairs.) Dirichlet’s definition of function as a correspondence from which all traces of continuity had been purged, made necessary Weirstrass’s independent definition of continuous function.

30. The notion of uniform continuity for functions was later introduced (in 1870) by Heine: a real valued function \(f\) is uniformly continuous if for any \(\varepsilon \gt 0\) there is \(\delta \gt 0\) such that \(|f(x) - f(y)| \lt \varepsilon\) for all \(x\) and \(y\) in the domain of \(f\) such that \(|x - y| \lt \delta\). In 1872 Heine proved the important theorem that any continuous real-valued function defined on a closed bounded interval of real numbers is uniformly continuous.

31. On Cantor, see Dauben 1979 and Hallett 1984.

32. This, Cantor’s continuum hypothesis , is actually stated in terms of the transfinite ordinal numbers introduced in previous sections of the Grundlagen.

33. In the terminology of general topology, a set is perfect if it is closed and has no isolated points.

34. This set later became known as the Cantor ternary set or the Cantor discontinuum.

35. Cantor later turned to the problem of characterizing the linear continuum as an ordered set. His solution was published in 1895 in the Mathematische Annalen (Dauben 1979: Chapter 8.) For a modern presentation, see §3 of Ch. 6 of Kuratowski-Mostowski (1968).

36. For du Bois-Reymond’s theory of infinitesimals see Fisher 1981; for Veronese’s, see Fisher 1994. The introduction to Ehrlich 1994a provides an overview of these “non-Cantorian” theories of infinitesimals and the continuum.

37. In a letter to Husserl drafted in 1905, Brentano asserts that “I regard it as absurd to interpret a continuum as a set of points”. (Brentano 1905 [1966: 95])

38. For an account of Peirce’s view of the continuum, see Ketner and Putnam 1992.

39. For Poincare’s philosophy of mathematics see Folina 1992.

40. The failure of these important results of classical analysis caused most mathematicians of the day to shun intuitionistic, and even constructive mathematics. It was not until the 1960s that adequate constructive versions were worked out. See Section 7 .

41. So-called, Robinson says, because his theory

involves and was, in part, inspired by the so-called Non-standard models of Arithmetic whose existence was first pointed out by T. Skolem. (1966: vii [1996: xiii])

42. It follows that \(\hat{\Re}\) is a nonarchimedean ordered field. One might question whether this is compatible with the facts that \(\hat{\Re}\) and \(\Re\) share the same first-order properties, but the latter is archimedean. These data are consistent because the archimedean property is not first-order. However, while \(\hat{\Re}\) is nonarchimedean, it is *- archimedean in the sense that, for any \(a \in \hat{\Re}\) there is \(n \in \hat{\bbN}\) for which \(a \lt n\).

43. Robinson (1966 [1996: Ch. 3]). A number of “nonstandard” proofs of classical theorems may also be found there.

44. Here “nonempty” has the stronger constructive meaning that an element of the set in question can be constructed.

45. This may be seen to be plausible if one considers that the according to Brouwer the construction of a choice sequence is incompletable; at any given moment we can know nothing about it outside the identities of a finite number of its entries. Brouwer’s principle amounts to the assertion that every function from \(\bbN^{\bbN}\) to \(\bbN\) is continuous.

46. For an explicit statement of the principle of Bar Induction, see Ch. 3 of Dummett (1977), or Ch. 5 of Bridges and Richman (1987).

47. See Kock (1981), Lavendhomme (1996), Lawvere (1980, 1998 [ Other Internet Resources ]), McLarty (1992), Moerdijk and Reyes (1991). For an elementary account of smooth infinitesimal analysis see Bell (1998).

48. For any \(f \in(\Delta^{\Delta})_0\), the microaffineness axiom ensures that there is a unique \(b \in \bR\) for which \(f(\varepsilon) = b\varepsilon\) for all \(\varepsilon\), and conversely each \(b \in \bR\) yields the map \(\varepsilon \mapsto b\varepsilon\) in \((\Delta^{\Delta})_0\).

49. A monoid is a multiplicative system (not necessarily commutative) with an identity element.

50. The domain of \(f\) is in fact \((\bR - \{0\}) \cup \{0\}\), which, because of the failure of the law of excluded middle in SIA, is provably unequal to \(\bR\).

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