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continuum hypothesis

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continuum hypothesis , statement of set theory that the set of real number s (the continuum) is in a sense as small as it can be. In 1873 the German mathematician Georg Cantor proved that the continuum is uncountable—that is, the real numbers are a larger infinity than the counting numbers—a key result in starting set theory as a mathematical subject. Furthermore, Cantor developed a way of classifying the size of infinite sets according to the number of its elements, or its cardinality . ( See set theory: Cardinality and transfinite numbers .) In these terms, the continuum hypothesis can be stated as follows: The cardinality of the continuum is the smallest uncountable cardinal number.

In Cantor’s notation, the continuum hypothesis can be stated by the simple equation 2 ℵ 0  = ℵ 1 , where ℵ 0 is the cardinal number of an infinite countable set (such as the set of natural numbers), and the cardinal numbers of larger “ well-orderable sets ” are ℵ 1 , ℵ 2 , …, ℵ α , …, indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2 ℵ 0 ; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.

A stronger statement is the generalized continuum hypothesis (GCH): 2 ℵ α  = ℵ α + 1 for each ordinal number α. The Polish mathematician Wacław Sierpiński proved that with GCH one can derive the axiom of choice .

Since ZF neither proves nor disproves the continuum hypothesis, there remains the question of whether to accept the continuum hypothesis based on an informal concept of what sets are. The general answer in the mathematical community has been negative: the continuum hypothesis is a limiting statement in a context where there is no known reason to impose a limit. In set theory, the power-set operation assigns to each set of cardinality ℵ α its set of all subsets, which has cardinality 2 ℵ α . There seems to be no reason to impose a limit on the variety of subsets that an infinite set might have.

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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To Settle Infinity Dispute, a New Law of Logic

November 26, 2013

In the course of exploring their universe, mathematicians have occasionally stumbled across holes: statements that can be neither proved nor refuted with the nine axioms, collectively called “ZFC,” that serve as the fundamental laws of mathematics. Most mathematicians simply ignore the holes, which lie in abstract realms with few practical or scientific ramifications. But for the stewards of math’s logical underpinnings, their presence raises concerns about the foundations of the entire enterprise.

“How can I stay in any field and continue to prove theorems if the fundamental notions I’m using are problematic?” asks Peter Koellner , a professor of philosophy at Harvard University who specializes in mathematical logic.

Chief among the holes is the continuum hypothesis, a 140-year-old statement about the possible sizes of infinity. As incomprehensible as it may seem, endlessness comes in many measures: For example, there are more points on the number line, collectively called the “continuum,” than there are counting numbers. Beyond the continuum lie larger infinities still — an interminable progression of evermore enormous, yet all endless, entities. The continuum hypothesis asserts that there is no infinity between the smallest kind — the set of counting numbers — and what it asserts is the second-smallest — the continuum. It “must be either true or false,” the mathematical logician Kurt Gödel wrote in 1947, “and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality.”

Infinity has ruffled feathers in mathematics almost since the field’s beginning.

The decades-long quest for a more complete axiomatic system, one that could settle the infinity question and plug many of the other holes in mathematics at the same time, has arrived at a crossroads. During a recent meeting at Harvard organized by Koellner, scholars largely agreed upon two main contenders for additions to ZFC: forcing axioms and the inner-model axiom “V=ultimate L.”

“If forcing axioms are right, then the continuum hypothesis is false,” Koellner said. “And if the inner-model axiom is right, then the continuum hypothesis is true. You go through a whole list of issues in other fields, and the forcing axioms will answer those questions one way, and ultimate L will answer them a different way.”

According to the researchers, choosing between the candidates boils down to a question about the purpose of logical axioms and the nature of mathematics itself. Are axioms supposed to be the grains of truth that yield the most pristine mathematical universe? In that case, V=ultimate L may be most promising. Or is the point to find the most fruitful seeds of mathematical discovery, a criterion that seems to favor forcing axioms ? “The two sides have a somewhat divergent view of what the goal is,” said Justin Moore , a mathematics professor at Cornell University.

Axiomatic systems like ZFC provide rules governing collections of objects called “sets,” which serve as the building blocks of the mathematical universe. Just as ZFC now arbitrates mathematical truth, adding an extra axiom to the rule book would help shape the future of the field — particularly its take on infinity. But unlike most of the ZFC axioms, the new ones “are not self-evident, or at least not self-evident at this stage of our knowledge, so we have a much more difficult task,” said Stevo Todorcevic , a mathematician at the University of Toronto and the French National Center for Scientific Research in Paris.

Proponents of V=ultimate L say that establishing an absence of infinities between the integers and the continuum promises to bring order to the chaos of infinite sets, of which there are, unfathomably, an infinite variety. But the axiom may have minimal consequences for traditional branches of mathematics.

“Set theory is in the business of understanding infinity,” said Hugh Woodin , who is a mathematician at the University of California, Berkeley; the architect of V=ultimate L; and one of the most prominent living set theorists. The familiar numbers relevant to most mathematics, Woodin argues, “are an insignificant piece of the universe of sets.”

Meanwhile, forcing axioms, which deem the continuum hypothesis false by adding a new size of infinity, would also extend the frontiers of mathematics in other directions. They are workhorses that regular mathematicians “can actually go out and use in the field, so to speak,” Moore said. “To me, this is ultimately what foundations [of mathematics] should be doing.”

New advances in the study of V=ultimate L and newfound uses of forcing axioms, especially one called “Martin’s maximum” after the mathematician Donald Martin, have energized the debate about which axiom to adopt. And there’s a third point of view that disagrees with the debate’s very premise. According to some theorists, there are myriad mathematical universes, some in which the continuum hypothesis is true and others in which it is false — but all equally worth exploring. Meanwhile, “there are some skeptics,” Koellner said, “people who for philosophical reasons think set theory and the higher infinite doesn’t even make any sense.”

Infinite Paradoxes

Infinity has ruffled feathers in mathematics almost since the field’s beginning. The controversy arises not from the notion of potential infinity —the number line’s promise of continuing forever — but from the concept of infinity as an actual, complete, manipulable object.

“What truly infinite objects exist in the real world?” asks Stephen Simpson , a mathematician and logician at Pennsylvania State University. Taking a view originally espoused by Aristotle, Simpson argues that actual infinity doesn’t really exist and so it should not so readily be assumed to exist in the mathematical universe. He leads an effort to wean mathematics off actual infinity, by showing that the vast majority of theorems can be proved using only the notion of potential infinity. “But potential infinity is almost forgotten now,” Simpson said. “In the ZFC set theory mindset, people tend not to even remember that distinction. They just think infinity means actual infinity and that’s all there is to it.”

Infinity was boxed and sold to the mathematical community in the late 19th century by the German mathematician Georg Cantor. Cantor invented a branch of mathematics dealing with sets — collections of elements that ranged from empty (the equivalent of the number zero) to infinite. His “set theory” was such a useful language for describing mathematical objects that within decades, it became the field’s lingua franca. A nine-item list of rules called Zermelo-Fraenkel set theory with the axiom of choice, or ZFC, was established and widely adopted by the 1920s. Translated into plain English, one of the axioms says two sets are equal if they contain the same elements. Another simply asserts that infinite sets exist.

Assuming actual infinity leads to unsettling consequences. Cantor proved, for instance, that the infinite set of even numbers {2,4,6,…} could be put in a “one-to-one correspondence” with all counting numbers {1,2,3,…}, indicating that there are just as many evens as there are odds-and-evens.

More shocking was his proving in 1873 that the continuum of real numbers (such as 0.00001, 2.568023489, pi and so on) is “uncountable”: Real numbers do not correspond in a one-to-one fashion with the counting numbers because for any numbered list of them, it is always possible to come up with a real number that isn’t on the list. The infinite sets of real numbers and counting numbers have different sizes, or in Cantor’s parlance, different “ cardinal numbers .” In fact, he found that there are not two but an infinite sequence of ever-larger cardinals, each new infinity consisting of the power set, or set of all subsets, of the infinite set before it.

Some mathematicians despised this mess of infinities. One of Cantor’s colleagues called them a “grave disease”; another called him a “corruptor of youth.” But by the logic of set theory, it was true.

Cantor wondered about the two smallest cardinals. “It’s in some sense the most fundamental question you can ask,” Woodin said. “Is there an infinity in between, or is the infinity of the real numbers the first infinity past the infinity of the counting numbers?”

All the obvious candidates for a mid-size infinity fail. Rational numbers (ratios of integers such as ½) are countable and thus have the same cardinality as the counting numbers. And there are just as many real numbers in any slice of the continuum (such as between 0 and 1) as there are in the whole set. Cantor guessed that there was no infinity in between countable sets and the continuum. But he couldn’t prove this “continuum hypothesis” using the axioms of set theory. Nor could anyone else.

Then, in 1931, Gödel, who had recently finished his doctorate at the University of Vienna, made an astounding discovery. With a pair of proofs, the 25-year-old Gödel showed that a specifiable yet sufficiently complex axiomatic system like ZFC could never be both consistent and complete. Proving that its axioms are consistent (that is, that they don’t lead to contradictions) requires an additional axiom not on the list. And to prove that ZFC-plus-that-axiom is consistent, yet another axiom is needed. “Gödel’s incompleteness theorems told us we are never going to be able to catch our own tail,” Moore said.

The incompleteness of ZFC means that the mathematical universe that its axioms generate will inevitably have holes. “There will be [statements] that cannot be decided by those principles,” Woodin said. It soon became clear that the continuum hypothesis, “the most fundamental question you can ask” about infinity, was such a hole. Gödel himself proved that the truth of the continuum hypothesis is consistent with ZFC, and Paul Cohen, an American mathematician, proved the opposite, that the negation of the hypothesis is also consistent with ZFC. Their combined results demonstrated that the continuum hypothesis is actually independent of the axioms. Something beyond ZFC is needed to prove or refute it.

With the hypothesis unresolved, many other properties of cardinal numbers and infinity remain uncertain too. To set theory skeptics like Solomon Feferman , a professor emeritus of mathematics and philosophy at Stanford University, this doesn’t matter. “They’re simply not relevant to everyday mathematics,” Feferman said.

But to those who spend their days wandering in the universe of sets known as “V,” where almost everything is infinite, the questions loom large. “We don’t have a clear vision of the universe of sets,” Woodin said. “Almost any question you write down about sets is unsolvable. It’s not a satisfactory situation.”

Universe of Sets

Gödel and Cohen, whose combined work led to the current crossroads in set theory, happen to be the founders of the two schools of thought about where to go from here.

Gödel conceived of a small and constructible model universe called “L,” populated by starting with the empty set and iterating it to build bigger and bigger sets. In the universe of sets that results, the continuum hypothesis is true: There is no infinite set between that of the integers and the continuum. “Unlike the chaos of the universe of sets, you can really analyze L,” Woodin said. This makes the axiom “V=L,” or the statement that the universe of sets V is equal to the “inner model” L, appealing. According to Woodin, there’s only one problem: “It severely limits the nature of infinity.”

L is too small to encompass “large cardinals,” infinite sets that ascend in a never-ending hierarchy, with levels named “inaccessible,” “measurable,” “Woodin,” “supercompact,” “huge” and so on, altogether composing a cacophonous symphony of infinities. Discovered periodically over the 20th century, these large cardinals cannot be proved to exist with ZFC and instead must be posited with additional “large cardinal axioms.” But over the decades, they have been shown to generate rich and interesting mathematics. “As you climb up the large cardinal hierarchy, you get more and more significant consequences,” Koellner said.

As many of the mathematicians pointed out, the debate itself reveals a lack of human intuition regarding the concept of infinity.

To keep this symphony of infinities, set theorists have striven for decades to find an inner model that is as pristine and analyzable as L but incorporates large cardinals. However, constructing a universe of sets that included each type of large cardinal required a unique tool kit. For each larger, more inclusive inner model, “you had to do something completely different,” Koellner said. “Since the large cardinal hierarchy just goes on and on forever, it looked like we had to go on and on forever too, building as many new inner models as there are transition points in the large cardinal hierarchy. And that kind of makes it look hopeless because, you know, life is short.”

Because there was no largest large cardinal, it seemed like there could be no ultimate L, an inner model that encompassed them all. “Then something very surprising happened,” Woodin said. In work that was published in 2010 , he discovered a breakaway point in the hierarchy.

“Woodin showed that if you can just reach the level of the supercompacts, then there’s an overflow and your inner model picks up all the bigger large cardinals as well,” Koellner explained. “That was a sort of landscape shift. It provided this new hope that this approach can work. All you have to do is hit one supercompact and then you’ve got it all.”

Although it has not yet been constructed, ultimate L is the name for the hypothetical inner model that includes supercompacts and therefore all large cardinals. The axiom V=ultimate L asserts that this inner model is the universe of sets.

Woodin, who is moving from Berkeley to Harvard in January, recently completed the first part of a four-stage proof of the ultimate L conjecture and is now vetting it with a small group of colleagues. He says he is “very optimistic about stage two” of the proof and hopes to finish it by next summer. “It all comes down to this conjecture, and if one can prove it, one proves the existence of ultimate L and verifies it is compatible with all notions of infinity, not only that we have thought of today but that we could ever think of,” he said. “If the ultimate L conjecture is true, then there’s an absolutely compelling case that V is ultimate L.”

Expanding the Universe

Even if ultimate L exists, can be constructed and is every bit as glorious as Woodin hopes, it isn’t everyone’s ideal universe. “There’s a contrary impulse running through much of set-theoretic history that tells us the universe should be as rich as possible, not as small as possible,” said Penelope Maddy , a philosopher of mathematics at the University of California, Irvine and the author of “Defending the Axioms,” published in 2011. “And that’s what motivates the forcing axioms.”

To expand ZFC, address the continuum hypothesis and better understand infinity, advocates of forcing axioms put stock in a method called forcing, originally conceived of by Cohen. If inner models build a universe of sets from the ground up, forcing expands it outward in all directions.

Todorcevic, one of the method’s leading specialists, compares forcing to the invention of complex numbers, which are real numbers with an extra dimension. But instead of starting with real numbers, “you are starting with the universe of sets, and then you extend it to form a new, bigger universe,” he said. In the extended universe created by forcing, there is a larger class of real numbers than in the original universe defined by ZFC. This means the real numbers of ZFC constitute a smaller infinite set than the full continuum. “In this way, you falsify the continuum hypothesis,” Todorcevic said.

A forcing axiom called “Martin’s maximum,” discovered in the 1980s, extends the universe as far as it can go. It is the most powerful rival for V=ultimate L, albeit much less beautiful. “From a philosophical point, it is much harder to justify this axiom,” Todorcevic said. “It could only be justified in terms of the influence it has on the rest of mathematics.”

This is where forcing axioms shine. While V=ultimate L is busy building a castle of unimaginable infinities, forcing axioms fill some problematic potholes in everyday mathematics. Work over the past few years by Todorcevic, Moore, Carlos Martinez-Ranero and others shows that they bestow many mathematical structures with nice properties that make them easier to use and understand.

To Moore, these sorts of results give forcing axioms the advantage over inner models. “Ultimately, the decision has to be grounded in: ‘What does it do for mathematics?’ ” he said. “Aside from its own intrinsic interest, what good mathematics does it produce?”

“My response would be, it’s certainly true that Martin’s maximum is great for understanding structures in classical mathematics,” Woodin said. “That’s not what set theory is about, to me. It’s not clear how Martin’s maximum is going to lead to a better understanding of infinity.”

At the recent Harvard meeting, researchers from both camps presented new work on inner models and forcing axioms and discussed their relative merits. The back-and-forth will likely continue, they said, until one or the other candidate falls by the wayside. Ultimate L could turn out not to exist, for example. Or perhaps Martin’s maximum isn’t as beneficial as its proponents hope.

As many of the mathematicians pointed out, the debate itself reveals a lack of human intuition regarding the concept of infinity. “Until you further investigate the consequences of the continuum hypothesis, you don’t have any real intuition as to whether it’s true or false,” Moore said.

Mathematics has a reputation for objectivity. But without real-world infinite objects upon which to base abstractions, mathematical truth becomes, to some extent, a matter of opinion — which is Simpson’s argument for keeping actual infinity out of mathematics altogether. The choice between V=ultimate L and Martin’s maximum is perhaps less of a true-false problem and more like asking which is lovelier, an English garden or a forest?

“It’s a personal thing,” Moore said.

However, the field of mathematics is known for its unity and cohesion. Just as ZFC came to dominate alternative foundational frameworks in the early 20th century, firmly embedding actual infinity in mathematical thinking and practice, it is likely that only one new axiom to decide the fuller nature of infinity will survive. According to Koellner, “one side is going to have to be wrong.”

This article was reprinted on ScientificAmerican.com .

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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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Notes to The Continuum Hypothesis

1. See Hallett (1984) for further historical information on the role of CH in the early foundations of set theory.

2. We have of necessity presupposed much in the way of set theory. The reader seeking additional detail—for example, the definitions of regular and singular cardinals and other fundamental notions—is directed to one of the many excellent texts in set theory, for example Jech (2003).

3. To say that GCH holds below δ is just to say that 2 ℵ α = ℵ α+1 for all ω ≤ α < δ and to say that GCH holds at δ is just to say that 2 ℵ δ = ℵ δ+1 ).

4. To see this argue as follows: Assume large cardinal axioms at the level involved in (A) and (B) and assume that there is a proper class of Woodin cardinals. Suppose for contradiction that there is a prewellordering in L (ℝ) of length ℵ 2 . Now, using (A) force to obtain a saturated ideal on ℵ 2 without collapsing ℵ 2 . In this forcing extension, the original prewellordering is still a prewellordering in L (ℝ) of length ℵ 2 , which contradicts (B). Thus, the original large cardinal axioms imply that Θ L (ℝ) ≤ ℵ 2 . The same argument applies in the more general case where the prewellordering is universally Baire.

5. For more on the topic of invariance under set forcing and the extent to which this has been established in the presence of large cardinal axioms, see §4.4 and §4.6 of the entry “ Large Cardinals and Determinacy ”.

6. The non-stationary ideal I NS is a proper class from the point of view of H (ω 2 ) and it manifests (through Solovay’s theorem on splitting stationary sets) a non-trivial application of AC. For further details concerning A G see §4.6 of the entry “ Large Cardinals and Determinacy ”.

7. Here are the details: Let A ∈ Γ ∞ and M be a countable transitive model of ZFC. We say that M is A - closed if for all set generic extensions M [ G ] of M , A ∩ M [ G ] ∈ M [ G ]. Let T be a set of sentences and φ be a sentence. We say that T ⊢ Ω φ if there is a set A ⊆ ℝ such that

• L ( A , ℝ) ⊧ AD + ,
• 𝒫 (ℝ) ∩ L ( A , ℝ) ⊆ Γ ∞ , and

M ⊧ “ T ⊧ Ω φ”,

where here AD + is a strengthening of AD.

8. Here are the details: First we need another conjecture: (The AD + Conjecture) Suppose that A and B are sets of reals such that L ( A , ℝ) and L ( B , ℝ) satisfy AD + . Suppose every set

X ∈ 𝒫 (ℝ) ∩ ( L ( A , ℝ) ∪ L ( B , ℝ))

is ω 1 -universally Baire. Then either

(Δ̰ 2 1 ) L ( A ,ℝ) ⊆ (Δ̰ 2 1 ) L ( B ,ℝ)

(Δ̰ 2 1 ) L ( B ,ℝ) ⊆ (Δ̰ 2 1 )} L ( A ,ℝ) .

(Strong Ω conjecture) Assume there is a proper class of Woodin cardinals. Then the Ω Conjecture holds and the AD + Conjecture is Ω-valid.

9. As mentioned at the end of Section 2.2 it could be the case (given our present knowledge) that large cardinal axioms imply that Θ L (ℝ) < ℵ 3 and, more generally, rule out the definable failure of 2 ℵ 0 = ℵ 2 . This would arguably further buttress the case for 2 ℵ 0 = ℵ 2 .

Copyright © 2013 by Peter Koellner < koellner @ fas . harvard . edu >

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Can the Continuum Hypothesis Be Solved?

In 1900, David Hilbert published a list of twenty-three open questions in mathematics , ten of which he presented at the International Congress of Mathematics in Paris that year. Hilbert had a good nose for asking mathematical questions as the ones on his list went on to lead very interesting mathematical lives. Many have been solved, but some have not been, and seem to be quite difficult. In both cases, some very deep mathematics has been developed along the way. The so-called Riemann hypothesis, for example, has withstood the attack of generations of mathematicians ever since 1900 (or earlier). But the effort to solve it has led to some beautiful mathematics. Hilbert’s fifth problem turned out to assert something that couldn’t be true, though with fine tuning the “right” question—that is, the question Hilbert should have asked—was both formulated and solved. There is certainly an art to asking a good question in mathematics.

The problem known as the continuum hypothesis has had perhaps the strangest fate of all. The very first problem on the list, it is simple to state: how many points on a line are there? Strangely enough, this simple question turns out to be deeply intertwined with most of the interesting open problems in set theory , a field of mathematics with a very general focus, so general that all other mathematics can be seen as part of it, a kind of foundation on which the house of mathematics rests. Most objects in mathematics are infinite, and set theory is indeed just a theory of the infinite.

How ironic then that the continuum hypothesis is unsolvable—indeed, “provably unsolvable,” as we say. This means that none of the known mathematical methods—those that mathematicians actually use and find legitimate—will suffice to settle the continuum hypothesis one way or another. It seems odd that being unsolvable is the kind of thing one can prove about a mathematical question. In fact, there are many questions of this type, particularly about sets of real numbers—or sets of points on a line, if you like—that we know cannot be settled using standard mathematical methods.

Now, mathematics is not frozen in time or method—to the contrary, it is a very dynamic enterprise, each generation expanding and building on what went before. This process of expansion has not always been easy; sometimes it takes a while before new methods are accepted. This was true of set theory in the late nineteenth century. Its inventor, Georg Cantor , met with serious opposition on the part of those who were hesitant to admit infinite objects into mathematics.

What concerns us here is not so much the prehistory of the continuum hypothesis, but the present state of it, and the remarkable fact that mathematicians are in the midst of developing new methods by which the continuum hypothesis could be solved after all.

I will explain some of these developments, along with some of the more recent history of the continuum hypothesis, from the point of view of Kurt Gödel’s role in them. Gödel , a Member of the Institute’s School of Mathematics on several occasions in the 1930s, and then continuously from 1940 until 1976, 1 was a relative newcomer to the problem. But it turns out that Gödel’s hand is visible in virtually every aspect of the problem, from the post-Cantorian period onward. Curiously enough, this is even more true now than it was at the time of Gödel’s death nearly thirty-five years ago.

What is the Continuum Hypothesis?

Mathematics is nowadays saturated with infinity . There are infinitely many positive whole numbers 0, 1, 2, 3 . . . . There are infinitely many lines, squares, circles in the plane, balls, cubes, polyhedra in the space, and so on. But there are also different degrees of infinity. Let us say that a set—a collection of mathematical objects such as numbers or lines—is countable if it has the same number of elements as the sequence of positive whole numbers 1, 2, 3 . . . . The set of positive whole numbers is thus countable, and so is the set of all rational numbers. In the early 1870s , Cantor made a momentous discovery: the set of real numbers (such as 5, 17, 5/12, √–2, π, e, . . . ) sometimes called the “continuum,” is uncountable . By uncountable, we mean that if we try to count the points on a line one by one, we will never succeed, even if we use all of the whole numbers. Now it is natural to ask the following question: are there any infinities between the two infinities of whole numbers and of real numbers?

This is the continuum hypothesis, which proposes that if you are given a line with an infinite set of points marked out on it, then just two things can happen: either the set is countable, or it has as many elements as the whole line. There is no third infinity between the two.

At first, Cantor thought he had a proof of the continuum hypothesis; then he thought he could prove it was false; and then he gave up. This was a blow to Cantor, who saw this as a defect in his work—if one cannot answer such a simple question as the continuum hypothesis, how can one possibly go forward?

Some History

The continuum hypothesis went on to become a very important problem, so much so that in 1900 Hilbert listed it as the first on his list of open problems, as previously mentioned. Hilbert eventually gave a proof of it in 1925—the proof was wrong, though it contained some important ideas.

Around the turn of the century, mathematicians were able to prove that the continuum hypothesis holds for a special class of sets called the Borel sets. 2 This is a concrete class of sets, containing, for the most part, the usual sets that mathematicians work with. Even with this early success in the special case of Borel sets though, and in spite of Hilbert’s attempted solution, mathematicians began to speculate that the continuum hypothesis was in general not solvable at all. Hilbert , for whom nothing less than “the glory of human existence” seemed to depend upon the ability to resolve all such questions, was an exception. “ Wir müssen wissen . Wir werden wissen ,” 3 he said in 1930 in Königsberg . In a great irony of history, at the very same meeting, but on the day before, the young Gödel announced his first incompleteness theorem. This theorem, together with Gödel’s second incompleteness theorem, is generally thought to have dealt a death blow to Hilbert’s idea that every mathematical question that permits an exact formulation can be solved. Hilbert was not in the room at the time.

Gödel , however, became a strong advocate of the solvability of the continuum hypothesis, taking the view that his incompleteness theorems , though they show that some provably undecidable statements do exist, have nothing to do with whether the continuum hypothesis is solvable or not. Like Hilbert , Gödel maintained that the continuum hypothesis will be solved.

What is Provable Unsolvability Anyway?

We arrive at an apparent conundrum. On the one hand, the continuum hypothesis is provably unsolvable, and on the other hand, both Gödel and Hilbert thought it was solvable. How to resolve this difficulty? What does it mean for something to be provably unsolvable anyway?

Some mathematical problems may be extremely difficult and therefore without a solution up to now, but one day someone may come up with a brilliant solution. Fermat’s last theorem, for example, went unsolved for three and a half centuries. But then Andrew Wiles was able to solve it in 1994. The continuum hypothesis is a problem of a very different kind; we actually can prove that it is impossible to solve it using current methods , which is not a completely unknown phenomenon in mathematics. For example, the age-old trisection problem asks: can we trisect a given angle by using just a ruler and compass? The Greeks of the classical period were very puzzled by how to make such a trisection, and no wonder, for in the nineteenth century it was proved that it is impossible—not just very difficult but impossible. You need a little more than a ruler and compass to trisect an arbitrary angle—for example, a compass and a ruler with two marks on it.

It is the same with the continuum hypothesis: we know that it is impossible to solve using the tools we have in set theory at the moment. And up until recently nobody knew what the analogue of a ruler with two marks on it would be in this case. Since the current tools of set theory are so incredibly powerful that they cover all of existing mathematics, it is almost a philosophical question: what would it be like to go beyond set-theoretic methods and suggest something new? Still, this is exactly what is needed to solve the continuum hypothesis.

Consistency

Gödel began to think about the continuum problem in the summer of 1930, though it wasn’t until 1937 that he proved the continuum hypothesis is at least consistent . This means that with current mathematical methods, we cannot prove that the continuum hypothesis is false .

Describing Gödel ’s solution would draw us into unneeded technicalities, but we can say a little bit about it. Gödel built a model of mathematics in which the continuum hypothesis is true. What is a model? This is something mathematicians build with the purpose of showing that something is possible, even if we admit that the model is just what it is, a kind of artificial construction. Children build model airplanes; architects draw up architectural plans; mathematicians build models of the mathematical universe. There is an important difference though, between mathematicians’ models and architectural plans or model airplanes: building a model that has the exact property the mathematician has in mind, is, in all but trivial cases, extremely difficult. It is like a very great feat of engineering.

The idea behind Gödel’s model, which we now call the universe of constructible sets , was that it should be made as small as is conceivably possible by throwing everything out that was not absolutely essential. It was a tour de force to show that what was left was enough to satisfy the requirements of mathematics, and, in addition, the continuum hypothesis. This did not show that the continuum hypothesis is really true, only that it is consistent, because Gödel’s universe of constructible sets is not the real universe, only a kind of artifact. Still, it suffices to demonstrate the consistency of the continuum hypothesis.

Unsolvability

After Gödel’s achievement, mathematicians sought a model in which the continuum hypothesis fails, just as Gödel found a model in which the continuum hypothesis holds. This would mean that the continuum hypothesis is unsolvable using current methods. If, on the one hand, one can build a picture of the mathematical universe in which it is true, and, on the other hand, if one can also build another universe in which it is false, it would essentially tell you that no information about the continuum hypothesis is lurking in the standard machinery of mathematics.

So how to build a model for the failure of the continuum hypothesis? Since Gödel’s universe was the only nontrivial universe that had been introduced, and, moreover, it was the smallest possible, mathematicians quickly realized that they had to find a way to extend Gödel’s model, by carefully adding real numbers to it. This is hair-raisingly difficult. It is like adding a new card to a huge house of cards, or, more exactly, like adding a new point to a line that already is—in a sense—a continuum. Where do you find the space to slip in a few new real numbers?

Looking back at Paul Cohen ’s solution, a logician has to slap her forehead, not once, but a few times. His idea was that the real numbers one adds should have “no properties,” as strange as this may sound; they should be “generic,” as he called them. In particular, a Cohen real , as they came to be called, should avoid “saying anything” nontrivial about the model. How to make this idea mathematically precise? That was Paul Cohen’s great invention: the forcing method, which is a way to add new reals to a model of the mathematical universe.

Even with this idea, serious obstacles now stood in the way of a full proof. For example, one has to prove an extremely delicate metamathematical theorem—as these are called—that even though forcing extends the universe to a bigger one, one can still talk about it in the first universe; in technical terms, one has to prove that forcing is definable . Moreover, to violate the continuum hypothesis, we have to add a lot of new points to the continuum, and what we believe is “a lot” may in the final stretch turn out to be not so many after all. This last problem—the technical term is preserving cardinals —was a very serious matter. Cohen later wrote of his sense of unease at that point, “given the rumors that had circulated that Gödel was unable to handle the CH.” 4 Perhaps Cohen sensed, while on the brink of his great discovery, the almost physical presence of the one mathematician who had walked the very long way up to that very door, but was unable to open it.

Two weeks later, while vacationing with his family in the Midwest, Cohen suddenly remembered a lemma from topology (due to N. A. Shanin ), and this was just what was needed to show that everything falls into place. The proof was now finished. It would have been an astounding achievement for any set theorist, but the fact that it was solved by someone from a completely different field—Paul Cohen was an analyst after all, not a set theorist—seemed beyond belief.

Writing the Paper

The story of what happened in the immediate aftermath of Cohen’s announcement of his proof is very interesting, also from the point of view of human interest, so we will permit ourselves a slight digression in order to touch upon it here.

The announcement seems to have been made at a time when the extent of what had been shown was not clear, and the proof, though it was finished in all the essentials, was not in all details completely finished. In a first letter to Gödel , dated April 24, 1963, Cohen communicated his results. But about a week later, he wrote a second, more urgent letter, in which he expressed his fear that there might be a hidden flaw in the proof, and, at the same time, his exasperation with logicians, who could not believe that he was able to prove that very delicate theorem on the definability of forcing.

Cohen confessed in the letter that the situation was wearing, also considering “the unexpected interest my work has aroused among the general (non-logical) mathematical world.”

Gödel replied with a very friendly letter, inviting Cohen to visit him, either at his home on Linden Lane or in his office at the Institute, writing, “You have just achieved the most important progress in set theory since its axiomatization . So you have every reason to be in high spirits.”

Soon after receiving the letter, Cohen visited Gödel at home, whereupon Gödel checked the proof, and pronounced it correct.

What followed over the next six months is a voluminous correspondence between the two, centered around the writing of the paper for the Proceedings of the National Academy of Sciences . The paper had to be carefully written; but Cohen was clearly impatient to go on to other work. It therefore fell to Gödel to fine tune the argument, as well as simplify it, all the while keeping Cohen in good spirits. The Gödel that emerges in these letters—sovereign, generous, and full of avuncular goodwill, will be unfamiliar to readers of the biographies—especially if one keeps in mind that by 1963 Gödel had devoted a good part of twenty-five years to solving the continuum problem himself, without success. “Your proof is the very best possible,” Gödel wrote at one point. “Reading it is like reading a really good play.”

Gödel and Cohen bequeathed to set theorists the only two model construction methods they have. Gödel’s method shows how to “shrink” the set-theoretic universe to obtain a concrete and comprehensible structure. Cohen’s method allows us to expand the set-theoretic universe in accordance with the intuition that the set of real numbers is very large. Building on this solid foundation, future generations of set theorists have been able to make spectacular advances.

There was one last episode concerning Gödel and the continuum hypothesis. In 1972, Gödel circulated a paper called “Some considerations leading to the probable conclusion that the true power of the continuum is ℵ 2 ,” which derived the failure of the continuum hypothesis from some new assumptions, the so-called scale axioms of Hausdorff . The proof was incorrect, and Gödel withdrew it, blaming his illness. In 2000, Jörg Brendle , Paul Larson, and Stevo Todorcevic 5 isolated three principles implicit in Gödel’s paper, which, taken together, put a bound on the size of the continuum. And subsequently Gödel’s ℵ 2 became a candidate of choice for many set theorists, as various important new principles from conceptually quite different areas were shown to imply that the size of the continuum is ℵ 2 .

Currently, there are two main programs in set theory. The inner model program seeks to construct models that resemble Gödel’s universe of constructible sets, but such that certain strong principles, called large cardinal axioms, would hold in them. These are very powerful new principles, which go beyond current mathematical methods (axioms). As Gödel predicted with great prescience in the 1940s , such cardinals have now become indispensable in contemporary set theory. One way to certify their existence is to build a model of the universe for them—not just any model, but one that resembles Gödel’s constructible universe, which has by now become what is called “canonical.” In fact, this may be the single most important question in set theory at the moment—whether the universe is “like” Gödel’s universe, or whether it is very far from it. If this question is answered, in particular if the inner model program succeeds, the continuum hypothesis will be solved.

The other program has to do with fixing larger and larger parts of the mathematical universe, beyond the world of the previously mentioned Borel sets. Here also, if the program succeeds, the continuum hypothesis will be solved.

We end with the work of another seminal figure, Saharon Shelah . Shelah has solved a generalized form of the continuum hypothesis, in the following sense: perhaps Hilbert was asking the wrong question! The right question, according to Shelah , is perhaps not how many points are on a line, but rather how many “small” subsets of a given set you need to cover every small subset by only a few of them. In a series of spectacular results using this idea in his so-called pcf-theory , Shelah was able to reverse a trend of fifty years of independence results in cardinal arithmetic, by obtaining provable bounds on the exponential function. The most dramatic of these is 2 ℵω ≤ 2 ℵ0 + ℵ ω4 . Strictly speaking, this does not bear on the continuum hypothesis directly, since Shelah changed the question and also because the result is about bigger sets. But it is a remarkable result in the general direction of the continuum hypothesis.

In his paper, 6 Shelah quotes Andrew Gleason, who made a major contribution to the solution of Hilbert’s fifth problem:

Of course, many mathematicians are not aware that the problem as stated by Hilbert is not the problem that has been ultimately called the Fifth Problem. It was shown very, very early that what he was asking people to consider was actually false. He asked to show that the action of a locally-euclidean group on a manifold was always analytic, and that’s false . . . you had to change things considerably before you could make the statement he was concerned with true. That’s sort of interesting, I think. It’s also part of the way a mathematical theory develops. People have ideas about what ought to be so and they propose this as a good question to work on, and then it turns out that part of it isn’t so .

So maybe the continuum problem has been solved after all, and we just haven’t realized it yet.

1 Appointed to the permanent Faculty in 1953; 2 This was extended to the so-called analytic sets by Mikhail Suslin in 1917. Borel sets are named for Emile Borel, uncle of the late mathematician (and IAS Faculty member) Armand Borel.; 3 “We must know. We will know.”; 4 P. J. Cohen, “The Discovery of Forcing”; 5 In their "Rectangular Axioms, Perfect Set Properties and Decomposition"; 6 "The Generalized Continuum Hypothesis Revisited"

Some Mathematical Details

Intuitively, the set-theoretic universe is the result of iterating basic constructions such as products ∏ i ∈ I A i , unions U i ∈ I A i , and power sets P(A) . In addition, the universe is assumed to satisfy so-called reflection : any property that it has is already possessed by some smaller universe, the domain of which is a set. The process starts from some given urelements , objects that are not sets, i.e., do not consist of elements, but it has been proven that the urelements are unnecessary and the process can be started from the empty set. Iterating this process into the transfinite , we obtain the cumulative hierarchy V of sets. Transfinite iterations are governed by ordinals , canonical representatives of well-ordered total orders, denoted by lower-case Greek letters α , β , etc. The hierarchy V is defined recursively by V α = U β < α P ( V β ). The fact that V = U α V α is the entire universe of sets is the intuitive content of the axioms of Zermelo-Frankel set theory with the Axiom of Choice, or ZFC , the basic system we have been working with all along.

Now Gödel’s model of the ZFC axioms, the constructible hierarchy L = U α L α , where L α = U β <α P L ( V β ), is built up not by means of the unrestricted power set operation P ( A ), but by the restricted operation P L ( A ), which takes from P ( A ) only those sets that are definable in ( A , ∈). Gödel showed that we can consistently assume V = L , but Cohen showed that it is consistent to assume that there are real numbers that are not in L .

The Borel sets of reals are obtained from open sets by means of iterating complements and countable unions. If we enlarge the set of Borel sets by including images of continuous functions, we obtain the analytic sets; a set is coanalytic if its complement is analytic.

Finally, the projective sets are obtained from analytic sets by iterating complements and continuous images. The field of descriptive set theory asks, among other questions, whether the classical theory of analytic and coanalytic sets can be extended to the projective sets; in particular, whether the projective sets are Lebesgue measurable, and have the perfect set property and the property of Baire . This was settled in the 1980s with the work of Shelah and Woodin , building on earlier work of Solovay , who showed that the projective sets have these three properties as a consequence of the existence of certain so-called large cardinals. This also follows from projective determinacy , a principle that was shown by Martin and Steel to follow from the existence of such large cardinals. A cardinal α is called a large cardinal if V α behaves in certain ways like V itself. For example, in that case, V α is a model of ZFC , but more is assumed. A famous large cardinal is a measurable cardinal, introduced by Stanislaw Ulam , an example of which is the smallest cardinal that admits a nontrivial countably additive two-valued measure.

What a State Mathematics Would Be In Today . . .

Before coming to the Institute where he was appointed as one of its first Professors in 1933, John von Neumann was a student of David Hilbert’s in Göttingen . Von Neu­mann worked on Hilbert’s program to find a complete and consistent set of axioms for all of mathematics. In addition to his many other contributions to mathematics and physics, von Neumann defined Hilbert space (unbounded operators on an infinite dimensional space), which he used to formulate a mathematical structure of quantum mechanics. Below, the late Herman Goldstine , a former Member in the Schools of Mathematics, Natural Sciences, and Historical Studies, recalls von Neumann’s working dreams about Kurt Gödel’s incompleteness theorem(s). ( Ex­cerp­ted from an oral history transcript available at www.prince­ton.edu/%7Emudd/finding_aids/math­oral/pmc15.htm ; more information about von Neumann and Gödel is available at www.ias.edu/people/noted-figures .)

His work habits were very methodical. He would get up in the morning, and go to the Nassau Club to have breakfast. And then from the Nassau Club he’d come to the Institute around nine, nine-thirty, work until lunch, have lunch, and then work until, say, five, and then go on home. Many evenings he would entertain. Usually a few of us, maybe my wife and me. We would just sit around, and he might not even sit in the same room. He had a little study that opened off of the living room, and he would just sit in there sometimes. He would listen, and if something interested him, he would interrupt. Otherwise he would work away.

At night he would go to bed at a reasonable hour, and he would waken, I think, almost every night, judging from the things he told me and the few times that he and I shared hotel rooms. He would waken in the night, two, three in the morning, and would have thought through what he had been working on. He would then write. He would write down the things he had worked on. . . .

He, under Hilbert’s tutelage, was trying to prove the opposite of the Gödel theorem. He worked and worked and worked at this, and one night he dreamed the proof. He got up and wrote it down, and he got very close to the end. He went and worked all day on that part, and the next night he dreamed again. He dreamed how to close the gap, and he got up and wrote, and he got within epsilon of the end, but he couldn’t make the final step. So he went to bed. The next day he worked and worked and worked at it, and he said to me, “You know, it was very lucky, Herman, that I didn’t dream the third night, or think what a state mathematics would be in today.” [Laughter.]

Juliette Kennedy is Associate Professor in the Department of Mathematics and Statistics at the University of Helsinki and a Member (2011–12) in the School of Historical Studies. In the history and foundations of mathematics, she has worked extensively on a project that attempts to put Kurt  Gödel  in full perspective, historically and foundationally. Her project at the Institute this year is centered on  Gödel’s  notion of semantic content. The mathematical aspect of the project involves the question of how many of the larger “large cardinals” can be captured with a newly discovered class of L-like inner models of set theory.

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What does the continuum hypothesis of fluid mechanics mean?

I'm a bit confused by the continuum hypothesis stating that fluid are continuous objects rather than made out of discrete objects.

Say for $\rho (x,t)$ (density) is there more than one fluid particle at $x$ or less than one.

• fluid-dynamics
• continuum-mechanics

• $\begingroup$ It'd depend on the fluid you are modeling and the density at $x$. If you have a hydrogen plasma ($m_H\approx1.67\times10^{-24}\,\rm g$) and the density is $\rho\leq10^{-25}\,\rm g/cm^3$ then you have less than one particle per cubic centimeter, but if you have $\rho\geq10^{-24}\,\rm g/cm^3$, then you have more than one. $\endgroup$ –  Kyle Kanos Commented Mar 28, 2015 at 18:45

The continuum hypothesis means the following: at each point of the region of the fluid it is possible to construct one volume small enough compared to the region of the fluid and still big enough compared to the molecular mean free path.

Why is that important? Because of two things. First, since the volume you can build at each point is very small compared to the size of the region of the fluid, you can think of the volume as located at a point instead of considering it as a collection of points. Imagine Earth for instance. If you build a small volume of $1 m^3$ somewhere on the surface of the Earth it is so small compared to Earth's size it can be considered to be associated with a particular point.

The secont thing is that since the volume is big enough compared to the molecular mean free path, this means it contains a large enough number of molecules. Why is that something you would want? Because containing a reasonable number of molecules allows you to take means on the volume and those means will make sense.

So, for instance, you can go there, compute the mass of each molecule, sum them up and divide by the volume. Given this hypothesis, this mean makes sense. And given the first hypothesis, you can think about this mean as associated with the point.

Because of that it makes sense of talking about fields defined on the region of the fluid. The mass density for example or the velocity field. They are in truth, means of quantities associated with the molecules, but that on the macroscopic point of view, can be considered just as fields associating quantities to points of the region.

On the density case, it really means: if $x$ is a point on the region of the fluid and $t$ an instant of time, $\rho(x,t)$ is the mean value of the mass of molecules contained inside one such small volume associated at $x$ at time $t$. From the macroscopic point of view, it is just a density that allows you to get mass through integration.

• $\begingroup$ Fantastic. Do these individual volumes that have been constructed around each point neccesarily represent individual fluid particles? $\endgroup$ –  usainlightning Commented Mar 28, 2015 at 19:31
• $\begingroup$ We usually call this volume containing a number of molecules a fluid particle. That is a terminology used, where one might think "well a fluid particle is at a point", but what one really means is "a very small volume, which we think as located at a point with many molecules inside". So yes, they are fluid particles, but a fluid particle contains many true molecules of the fluid. $\endgroup$ –  Gold Commented Mar 28, 2015 at 19:38
• $\begingroup$ I'm not sure what is meant by 'taking means on the volume'. Do you mean taking the mean of the molecular masses inside the volume around a point and requiring there to be a large number of molecules so this mean is representative of the fluid? $\endgroup$ –  usainlightning Commented Mar 28, 2015 at 20:40
• $\begingroup$ Yes that is the idea. If such small volume according to the hypothesis exists at each point, you simply take the mean of the mass of the molecules in the volume. Then intuitively for this mean to make sense you need a big enough number of molecules on the volume. $\endgroup$ –  Gold Commented Mar 28, 2015 at 20:50
• $\begingroup$ Would you not take the mean and then multiply it by the number of molecules inside to get the mass of that volume as opposed to just taking the mean of the masses? $\endgroup$ –  usainlightning Commented Mar 28, 2015 at 20:56

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Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis ( There does not exist a set with a cardinality less than the reals and no set strictly greater than the natural numbers. ) is neither true or false.

This is utterly baffling to me, If it's possible to construct a set between $\mathbb{N}$ and $\mathbb{R}$ then this statement is demonstrably false, but if not then the statement is true.

This seems to be a straitforward deduction, but many with a more advanced understanding of the topic matter believe CH to be neither.

How can this be?

• incompleteness

• 10 $\begingroup$ A related previous question: "Impossible to prove" vs "neither true nor false" $\endgroup$ –  user856 Commented Sep 1, 2012 at 0:12
• 2 $\begingroup$ This type of thing makes me like intuitionistic logic more. $\endgroup$ –  Tunococ Commented Sep 1, 2012 at 0:14
• 4 $\begingroup$ "If it's possible to construct a set between $\mathbb{N}$ and $\mathbb{R}$ then this statement is demonstrably false, but if not then the statement is true." The predominant attitude among modern mathematicians is that math doesn't have to be constructive: that we can prove things exist without constructing them explicitly, and that such a proof is satisfactory. It then becomes possible to have things that you can prove exist, and can also prove can't be constructed explicitly; this is the situation, e.g., with ultrafilters. Independence of CH from ZFC says we can't rule this out for CH. $\endgroup$ –  user13618 Commented Sep 2, 2012 at 14:54

7 Answers 7

Set theory is much more complicated than "common" mathematics in this aspect, it deals with things which you can often prove that are unprovable.

Namely, when we start with mathematics (and sometimes for the rest of our lives) we see theorems, and we prove things about continuous functions or linear transformations, etc.

These things are often simple and have a very finite nature (in some sense), so we can prove and disprove almost all the statements we encounter. Furthermore it is a good idea, often, to start with statements that students can handle. Unprovable statements are philosophically hard to swallow, and as such they should usually be presented (in full) only after a good background has been given.

Now to the continuum hypothesis. The axioms of set theory merely tell us how sets should behave. They should have certain properties, and follow basic rules which are expected to hold for sets. E.g., two sets which have the same elements are equal.

Using the language of set theory we can phrase the following claim:

If $A$ is an uncountable subset of the real numbers, then $A$ is equipotent with $\mathbb R$.

The problem begins with the fact that there are many subsets of the real numbers. In fact we leave the so-called "very finite" nature of basic mathematics and we enter a realm of infinities, strangeness and many other weird things.

The intuition is partly true. For the sets of real numbers which we can define by a reasonably simple way we can also prove that the continuum hypothesis is true: every "simply" describable uncountable set is of the size of the continuum .

However most subsets of the real numbers are so complicated that we can't describe them in a simple way. Not even if we extend the meaning of simple by a bit, and if we extend it even more, then not only we will lose the above result about the continuum hypothesis being true for simple sets; we will still not be able to cover even anything close to "a large portion" of the subsets.

Lastly, it is not that many people "believe it is not a simple deduction". It was proved - mathematically - that we cannot prove the continuum hypothesis unless ZFC is inconsistent, in which case we will rather stop working with it.

Don't let this deter you from using ZFC, though. Unprovable questions are all over mathematics, even if you don't see them as such in a direct way:

There is exactly one number $x$ such that $x^3=1$.

This is an independent claim. In the real numbers, or the rationals even, it is true. However in the complex numbers this is not true anymore. Is this baffling? Not really, because the real and complex numbers have very canonical models. We know pretty much everything there is to know about these models (as fields, anyway), and it doesn't surprise us that the claim is true in one place, but false in another.

Set theory (read: ZFC), however, has no such property. It is a very strong theory which allows us to create a vast portion of mathematics inside of it, and as such it is bound to leave many questions open which may have true or false answers in different models of set theory. Some of these questions affect directly the "non set theory mathematics", while others do not.

Some reading material:

• A question regarding the Continuum Hypothesis (Revised)
• Neither provable nor disprovable theorem
• Impossible to prove vs neither true nor false

• 3 $\begingroup$ It's always fun when everyone writes a short answer and I find myself writing a short story! :-) $\endgroup$ –  Asaf Karagila ♦ Commented Sep 1, 2012 at 0:59
• 7 $\begingroup$ This answer is illuminates a murky subject matter with exceptional clarity and wisdom. Thank you! $\endgroup$ –  zetavolt Commented Sep 1, 2012 at 1:16
• 12 $\begingroup$ "...the real and complex numbers have a very canonical model. We know pretty much everything there is to know about these models..." except for whether they have any subsets with cardinality strictly between that of the integers and the reals. And where to find the zeros of the zeta function. $\endgroup$ –  Gerry Myerson Commented Sep 1, 2012 at 1:21
• 2 $\begingroup$ @Mark: I think that writing a +17 answer about independence of CH from ZFC should hint that I am probably aware to that, among other things such as the :-) , however for the general point for those who might be unaware of this fact: thanks. $\endgroup$ –  Asaf Karagila ♦ Commented Sep 1, 2012 at 7:57
• 2 $\begingroup$ @Andrew: "Continuum" is a term for the real numbers. It has other, more general meanings, but in this context it simply means the real numbers. $\endgroup$ –  Asaf Karagila ♦ Commented Apr 21, 2014 at 19:50

A number of mathematicians have definite opinions about the truth of CH, the majority I believe opting for false, Kurt Gödel among them. What there is agreement on, because it is a theorem, is that CH is neither provable nor refutable in ZFC. But that is quite a different assertion than "neither true nor false."

The theory ZFC captures many common intuitions about sets. It has been the dominant "set theory" for many years. There is no good reason that it will remain that forever.

• 1 $\begingroup$ Doesn't one have to say, "neither provable nor refutable in ZFC, provided ZFC is consistent"? $\endgroup$ –  Gerry Myerson Commented Sep 1, 2012 at 1:16
• 2 $\begingroup$ @GerryMyerson: Yes. But the axioms (at least of ZF) are true in the universe of sets, and the relative consistency of ZFC is a theorem. So I thought that in the context of a question about truth, the obligatory "if ZFC is consistent" could be dispensed with. $\endgroup$ –  André Nicolas Commented Sep 1, 2012 at 1:20
• $\begingroup$ Would you know (trusted estimates of) the proportion of mathematicians with definite opinions about the truth of CH , and amongst them, the proportion of those opting for false ? $\endgroup$ –  Did Commented Sep 1, 2012 at 14:23
• 2 $\begingroup$ Should probably not have said mathematicians. For set theorists, one can make lists. The pro-CH list would be short. $\endgroup$ –  André Nicolas Commented Sep 1, 2012 at 17:58
• $\begingroup$ WP has a nice summary of this: en.wikipedia.org/wiki/… $\endgroup$ –  user13618 Commented Sep 2, 2012 at 14:46

One can construct a model of set theory in which CH is true, and one can construct a model in which CH is false.

• $\begingroup$ I think with and without axiom of choice? how is it? can you give a bit more info? thanks. $\endgroup$ –  Seyhmus Güngören Commented Sep 1, 2012 at 0:11
• $\begingroup$ Can you elaborate on this answer? $\endgroup$ –  zetavolt Commented Sep 1, 2012 at 0:14
• $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ –  Robert Israel Commented Sep 1, 2012 at 0:16
• $\begingroup$ @SeyhmusGüngören: The axiom of choice has nothing to do with this... $\endgroup$ –  Asaf Karagila ♦ Commented Sep 1, 2012 at 0:21
• 2 $\begingroup$ Asaf, I think that's what @Seyhmus was asking: there are models both with and without Choice in which CH is true, and models both with and without Choice in which CH is false. $\endgroup$ –  Gerry Myerson Commented Sep 1, 2012 at 1:14

It is not possible to explicitly "construct" such a set and prove (using the ZFC axioms of set theory) that its cardinality is strictly between those of $\mathbb N$ and $\mathbb R$. That doesn't mean that no such set exists.

As a Platonist, I would not say that CH is "neither true nor false", rather that we do not know (and in a certain sense we cannot know) which it is. Truth and provability are very different things.

• 5 $\begingroup$ We can actually prove that it is true or false... $\endgroup$ –  Michael Greinecker Commented Sep 1, 2012 at 0:18
• $\begingroup$ @MichaelGreinecker, could you explain that? $\endgroup$ –  user56834 Commented Jan 23, 2018 at 19:31
• 3 $\begingroup$ @Programmer2134 For every statement $A$, $A\vee\neg A$ is a true statement in classical logic. But that does not mean there is a proof of $A$ or a proof of $\neg A$. $\endgroup$ –  Michael Greinecker Commented Jan 23, 2018 at 20:31
• $\begingroup$ That assumes you can have this as a statement $\endgroup$ –  Gabriel Tellez Commented Jun 2 at 1:10
• $\begingroup$ @GabrielTellez I'm not sure what your objection is. CH is a statement of set theory. $\endgroup$ –  Robert Israel Commented Jun 3 at 2:09

If you take the parallel axiom away from Euclidean geometry, you cannot prove (using the remaining axiom system) whether it is true or false. But even in a geometry without parallel axiom, you can have interesting results (see http://en.wikipedia.org/wiki/Absolute_geometry ).

• 1 $\begingroup$ Maybe add a sample of said interesting results . $\endgroup$ –  Did Commented Sep 1, 2012 at 11:39
• 1 $\begingroup$ So you added a link to a WP page on Absolute geometry. Fine. But could you be more specific about interesting results ? $\endgroup$ –  Did Commented Sep 2, 2012 at 12:53
• 1 $\begingroup$ WP mentions the first 28 propositions from Euclid's Elements, the exterior angle theorem and the Saccheri-Legendre theorem. Maybe this isn't terrible interesting, but I guess good enough to get my point across. It's just an analogy, dude... $\endgroup$ –  Landei Commented Sep 2, 2012 at 13:18

I beleive that your confusion rise from bad definition of CH. It is not "there does not exist a set with a cardinality less than the reals and no set strictly greater than the natural numbers" as you stated it, but rather "there does not exist a set with a cardinality less than the reals AND strictly greater than that of the natural numbers.".

(1)To make your wording more accurate, I assume you mean "there exists" when you state it as "It's possible to construct."

(2)Truth and Falsity in logic might have a different meaning than you think. Truth and Falsity here can only be talked within a scope, which is given by ZFC axiom system. Under the consistency assumption of ZFC, the system is incomplete, meaning that there are statements there not having a T/F answer.

(3) When people say CH "is neither true or false" in this context, they really just mean that such Truth or Falsity cannot be deduced from ZFC system. More precisely, they mean that under the consistency assumption of ZFC, if you add CH or its negation to ZFC, the system remains consistent, and therefore CH and its negation cannot be deduced from ZFC.

(4) In a larger system, it is possible to give a definitive answer to CH. As a trivial example, if we add CH to ZFC, then CH would be true.

• $\begingroup$ You missed "HC" in the last sentence of the 3rd point. $\endgroup$ –  Asaf Karagila ♦ Commented Nov 18, 2014 at 21:25
• $\begingroup$ Thank you. I did misspelled it. $\endgroup$ –  Deadwood Kumu Commented Nov 18, 2014 at 23:32

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• Published: 03 September 2024

Stage dependence of Elton’s biotic resistance hypothesis of biological invasions

• Kun Guo   ORCID: orcid.org/0000-0001-9597-2977 1 ,
• Petr Pyšek   ORCID: orcid.org/0000-0001-8500-442X 2 , 3 ,
• Milan Chytrý   ORCID: orcid.org/0000-0002-8122-3075 4 ,
• Jan Divíšek   ORCID: orcid.org/0000-0002-5127-5130 4 , 5 ,
• Martina Sychrová 4 , 5 ,
• Zdeňka Lososová 4 ,
• Mark van Kleunen   ORCID: orcid.org/0000-0002-2861-3701 6 , 7 ,
• Simon Pierce 8 &
• Wen-Yong Guo   ORCID: orcid.org/0000-0002-4737-2042 1 , 9 , 10  

Nature Plants ( 2024 ) Cite this article

Metrics details

• Biodiversity
• Invasive species

Elton’s biotic resistance hypothesis posits that species-rich communities are more resistant to invasion. However, it remains unknown how species, phylogenetic and functional richness, along with environmental and human-impact factors, collectively affect plant invasion as alien species progress along the introduction–naturalization–invasion continuum. Using data from 12,056 local plant communities of the Czech Republic, this study reveals varying effects of these factors on the presence and richness of alien species at different invasion stages, highlighting the complexity of the invasion process. Specifically, we demonstrate that although species richness and functional richness of resident communities had mostly negative effects on alien species presence and richness, the strength and sometimes also direction of these effects varied along the continuum. Our study not only underscores that evidence for or against Elton’s biotic resistance hypothesis may be stage-dependent but also suggests that other invasion hypotheses should be carefully revisited given their potential stage-dependent nature.

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Native diversity buffers against severity of non-native tree invasions

Invasion intensity influences scale-dependent effects of an exotic species on native plant diversity

The impact of land use on non-native species incidence and number in local assemblages worldwide

Data availability.

The data used in this study were obtained from these sources: the data on vegetation plots were from the Czech National Phytosociological Database 54 ( https://botzool.cz/vegsci/phytosociologicalDb/ ); species’ statuses along the invasion continuum were extracted from Pyšek et al. 55 ; the three leaf traits required for CSR calculation were collected from the Pladias Database of the Czech Flora and Vegetation 58 and other publications 61 , 62 , 63 , 64 , 65 , 66 ; species CSR scores were calculated using the StrateFy tool 60 ; climatic variables were extracted from Tolasz 73 ; soil pH was collected from the Land Use/Land Cover Area Frame Survey 74 ; and the human population density of the cadastral area where each plot located was obtained from the Digital Vector Database of Czech Republic ArcČR v.4.0 (ref. 75 ). The data that support the findings of this study are available via GitHub at https://github.com/kun-ecology/BioticResistance_InvasionContinuum and via Zenodo at https://doi.org/10.5281/zenodo.12818669 (ref. 79 ).

Code availability

R functions for the computation of phylogenetic and functional metrics have been deposited on GitHub ( https://github.com/kun-ecology/ecoloop ). R scripts for reproducing the analyses and figures are available via GitHub at https://github.com/kun-ecology/BioticResistance_InvasionContinuum and via Zenodo at https://doi.org/10.5281/zenodo.12818669 (ref. 79 ).

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Acknowledgements

K.G. and W.-Y.G. were supported by the Natural Science Foundation of China (grant no. 32171588, awarded to W.-Y.G.) and the Shanghai Pujiang Program (grant no. 21PJ1402700, awarded to W.-Y.G.). K.G. was also supported by the Shanghai Sailing Program (grant no. 22YF1411700) and the Natural Science Foundation of China (grant no. 32301386). P.P. was supported by the Czech Science Foundation (EXPRO grant no. 19-28807X) and the Czech Academy of Sciences (long-term research development project RVO 67985939). M.C. and Z.L. were supported by the Czech Science Foundation (EXPRO grant no. 19-28491X). J.D. was supported by the Technology Agency of the Czech Republic (grant no. SS02030018). M.S. was funded by the project GEOSANT with the funding organization Masaryk University (MUNI/A/1469/2023).

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Zhejiang Tiantong Forest Ecosystem National Observation and Research Station, Shanghai Key Lab for Urban Ecological Processes and Eco-Restoration & Research Center for Global Change and Complex Ecosystems, School of Ecological and Environmental Sciences, East China Normal University, Shanghai, People’s Republic of China

Kun Guo & Wen-Yong Guo

Department of Invasion Ecology, Institute of Botany, Czech Academy of Sciences, Průhonice, Czech Republic

Department of Ecology, Faculty of Science, Charles University, Prague, Czech Republic

Department of Botany and Zoology, Faculty of Science, Masaryk University, Brno, Czech Republic

Milan Chytrý, Jan Divíšek, Martina Sychrová & Zdeňka Lososová

Department of Geography, Faculty of Science, Masaryk University, Brno, Czech Republic

Jan Divíšek & Martina Sychrová

Ecology, Department of Biology, University of Konstanz, Konstanz, Germany

Mark van Kleunen

Zhejiang Provincial Key Laboratory of Plant Evolutionary Ecology and Conservation, Taizhou University, Taizhou, People’s Republic of China

Department of Agricultural and Environmental Sciences (DiSAA), University of Milan, Milan, Italy

Simon Pierce

Zhejiang Zhoushan Island Ecosystem Observation and Research Station, Zhoushan, People’s Republic of China

Wen-Yong Guo

State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai, People’s Republic of China

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K.G., P.P. and W.-Y.G. conceptualized the research. P.P., M.C., J.D., M.S. and W.-Y.G. provided the data. K.G. analysed the data and drafted the paper, with substantial contributions from all authors.

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Correspondence to Wen-Yong Guo .

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Guo, K., Pyšek, P., Chytrý, M. et al. Stage dependence of Elton’s biotic resistance hypothesis of biological invasions. Nat. Plants (2024). https://doi.org/10.1038/s41477-024-01790-0

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Received : 18 April 2024

Accepted : 15 August 2024

Published : 03 September 2024

DOI : https://doi.org/10.1038/s41477-024-01790-0

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