To Settle Infinity Dispute, a New Law of Logic

Image of spiral staircase.

As incomprehensible as it may seem, infinity comes in many measures. A new axiom is needed to make sense of its multifaceted nature.

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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.

Hugh Woodin

Courtesy of Hugh Woodin

Hugh Woodin, 58, is the leading proponent of an axiom called V=ultimate L that could help decide the fuller nature of infinity.

“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.”

Georg Cantor, shown here circa 1870, proved that infinite sets come in different sizes. Many of his contemporaries despised his work, but it had a prevailing influence on mathematics.

Georg Cantor, shown here circa 1870, proved that infinite sets come in different sizes. Many of his contemporaries despised his work, but it had a prevailing influence on mathematics.

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.

Diagram of one-to-one correspondence.

Because each point on the interval (0, 1) corresponds to the point on (0, 2) that lies on the same red line, there are just as many real numbers between 0 and 1 as there are between 0 and 2. This “one-to-one correspondence” proves both infinite sets are the same size.

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.”

Stevo Todorcevic

George M. Bergman, Berkeley

Stevo Todorcevic has helped show that forcing axioms bestow many mathematical structures with user-friendly properties.

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.”

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  • This reminds me of Euclid’s parallel postulate, and the particle-wave duality of quantum mechanics. More than one solution may be correct, and useful, depending on context.

  • That is an interesting article about a subject that is just not really very important. It is rather like people debating the desirability of the designated hitter. It provokes fascinating discussion and even passion, but does not actually matter much to anyone at all, except those few who pitch but do not bat or bat but do not field. One could go on infinitely about it, I imagine, were one so disposed.

  • It’s funny how the ZFC set of infinities is itself an infinity. If it will never be able to solve all the “holes” in mathematics, then wouldn’t it always needed to be updated in an infinite cycle like Godel was saying? Interesting that Simpson says there are no infinites in our world. I agree. Just things that have yet to be discovered. How can something not be infinite if it is never truly 100% discovered?

  • I clearly know nothing on the subject but it does seem like a question how many angels can dance on the head of a pin. i.e. a logical construct with no basis in reality.

    For example, the triangular diagram is interesting in that each point on line 1 corresponds to a similar point on line 2. However, since the lines converge to a point, could you not just as easily conclude that each point on both line 1 and line 2 have a corresponding point on the convergent point and therefore a point has infinite points associated with it? At what point as the two lines converge is that true and when does it become false?

  • The human mind is a survival strategy that has been shaped by natural selection within the milieu of the planet Earth. As a result, the mind does not encompass infallible ‘logic’ – it encompasses delusions that are useful approximations (sometimes) within the limited framework of survival on Earth. There is no a priori reason to believe our ‘logical’ systems are related at every level to any natural ‘reality’ or that ‘reality’ is knowable by the human mind.

    Someone once suggested that what we see is illusion and what we don’t see is reality. I would add that all our stories of ‘reality’ are hubristic nonsense.

  • The root of the problem is the conflict between infinite human imagination and finite physical world. So both could be right and useful, depending on what you are talking about.

  • FTA: “But without real-world infinite objects upon which to base abstractions . . . ”
    IMHO, that’s exactly where there is a hole in our knowledge of the real world too. At the macro level, we don’t know whether the universe is infinite or not. We don’t know if there are other universes out there. At the micro level, the deeper we go, the tinier sub-atomic particles we discover and there does not seem to be a foreseeable end to it. Maybe some day we can find out if the universe is finite or infinite. Only then we can say for sure if there are any real world infinite objects or not.

  • @Chris G
    What the diagram is meant to illustrate is that it is possible to construct a rule that connects every every element of one set with an unique element of the other set and vice-versa. In the case of the set with one point, the map fails to be a such a correspondence because that one point is connected to each and every element of the other sets. The notion of ‘one-to-one correspondence’ is conceptually prior to counting. One can be a person with no knowledge of numbers but can still keep track of, say, her/his personal belongings by creating a ‘one-to-one correspondence’ between the set of stuff she/he own and a set of m&ms.

    Talk of ‘infinte sets’ may seem far-removed from the real world (not that this is a problem: math is an artform with a larger audience than poetry) but they were developed when Cantor was thinking about the convergence of fourier series (which is as practical as math can get).

  • @ChrisG: The point of the triangular diagram was to point out that the two lines have points in one-to-one correspondence with each other: each point on the first line corresponds to exactly one point on the second line, and vice versa. The definition of two sets having the same size (more formally, “cardinality”) is that there’s a one-to-one correspondence between them. If you look at the single point where those triangles converge, it’s no longer a one-to-one correspondence, because that one point corresponds to more than one point on the lines. So those sets don’t necessarily have the same size.

  • Despite objections by the admittedly ignorant that this is an unimportant venture, it is in fact of fundamental importance as it concerns the very foundations of mathematics.
    ZFC is very powerful, but IMHO it encounters difficulties when the Axiom of Choice is applied to uncountable sets. It simply doesn’t make sense to discuss an uncountable choice set.

  • Godel showed any system as complicated as Peano’s axioms (a very simple system that can count) could be used to construct the logical system that defined itself. Self-referential systems have problems (constructing the set of all sets that don’t contain themselves, for example). In Godel’s case, there are a set of operations representing logical actions, and a correspondence between theorems and numbers (Godel numbers). Godel showed there are numbers that correspond to statements like “this number is not a theorem.” It doesn’t mean you cannot count, or do integrals. It does impose some limits for whether you can be consistent or complete for any system complicated enough to count (there’s a pun there). Also, while the idea of the Axiom of Choice (C in ZFC) seems simple enough, its consequences can be downright weird. As it stands, there are a zoology of systems that can be constructed with different viewpoints, which may or may not be consistent with each other, or may have limits and quirks. To conclude one or the other is ultimately wrong would be to provide a proof that the ‘axiom’ was not true… or maybe both axioms could be shown to be not true.

  • Let V = ultimate L remain busy building a castle of unimaginable infinities, and forcing axioms continue to fill some problematic potholes in everyday mathematics. Mathematics will surely gain much more in this way. After all, why should one be so obsessed with some elusive Ultimate Mathematics. There is no final, absolute Mathematics as such. The very concept of infinity forecloses that possibility.
    Yes, the debate itself reveals a lack of human intuition regarding the concept of infinity. One cannot agree with Simpson’s argument for keeping actual infinity out of mathematics altogether simply because ‘there are no finite objects in our world’. Infinity pervades all apparently finite objects, from sub-atomic particles to galaxies. So I am more interested in set theorists efforts to find an inner model that is as pristine and analyzable as L but incorporates large cardinals ‘to keep this symphony of infinities’.

  • Since ‘infinity’ is part of the ZFC axioms, can we really talk about it consistently and completely without violating Godel’s theorem ? If true then we can neither prove nor disprove the continuum hypothesis. Anything wrong in this line of argument ?

  • The first thing to notice is that this debate can not possibly be about “truth.” Since we know that both the continuum hypothesis and its negation are consistent with ZFC, we know that both kinds of structures are legitimate topics of exploration. Presumably the same goes for structures based on other advanced axioms as well (though as I understand it that has yet to be proven). The current debate therefore has to do with which kind of structure it is in some sense “better” for mathematicians to focus on. The real debate ought to be: why focus on just one?
    Practically speaking, the debate will be settled if structures of one particular type ever turn out to be useful in physics or (less likely) in some other science. At that point there will be a real payoff to focusing in one direction rather than another. Until then, I think it makes sense to explore a variety of structures.
    In other words, I conjecture that the more types of structures we know about, the greater the chances that one of them will be useful. The alternative hypothesis is that extremely arcane exploration of a single structure is more productive than moderately arcane explorations of a variety of structures—a hypothesis which strikes me as dubious, given the extreme difficulty any physicist would have in mastering excessively arcane structures of no known utility while simultaneously pursuing a career in physics.

  • The alternative between Martin’s maximum and V = ultimate L is misleading, since the new axiom should be an open condition: V >= ultimate L (meaning that the Universe contains the yet-to-be-constructed large class, including the supercompact cardinals).

    The point is that neither of the two considered axioms is known to be consistent, but the consistency of a supercompact cardinal is known to imply the consistency of Martin’s maximum. So, if the ultimate L is constructed, set theorists will still have the comfort of two choices: either have the cake (assuming V = ultimate L, and hence, the continuum hypothesis) or eat it (consistently assuming Martin’s axiom, and hence, V > ultimate L and no continuum hypothesis). What is more, each of the assumptions can then be used to prove the consistency of various practical constructions, according to the situation. Mathematics will become extremely rich. If, on the other hand, Martin’s axiom is assumed without a supercompact cardinal, mathematicians may remain in the dark about the consistency of both for a long time.

  • First, good job with the article – this is a difficult piece to write. I think Quantum Mechanic takes the right view at the end of his comment (at least a popular view among set theorists): more than one “solution” may be correct and useful depending on the contexts. I have a hard time calling anything after Cohen/Godel a “solution” to something like the Continuum Hypothesis. It’s also worth pointing out that most set theorists would say it is rather limiting to consider only the two options for foundations mentioned here.

  • @David Burgess: good comment. I’ve said myself in the past that the ultimate vindication of any foundational theory for mathematics would be its ability to provide a mathematical model for something which can be experimentally verified. Presumably that would be something in physics. We’re a long way from that and maybe it is something which will never come to be. Maybe something less would suffice. I think it is fair to say that, today, the Axiom of Choice is widely accepted as being part of the axioms of mathematics. That wasn’t always the case, but it has played such an extensive role in developing so many different parts of mathematics that it is harder to go without it than to get comfortable with accepting it as part of the axioms. Maybe something like that will happen with extensions of ZFC in the future (one might argue large cardinal axioms already fall into this category). We can only know by working out the theory of different axioms systems and “experimenting” with them. Much the way geometers started to experiment with different models of geometry over a century ago. To come full circle, is the parallel postulate “true”? One might take that to mean “is it true in the physical model we live in?” The question about “truth” in set theory is really a question of whether there is some mathematical analog of the physical world. That’s really a philosophical question, not a mathematical one (and I’m a mathematician, not a philosopher).

  • This is an expansion on some remarks already made by others, making explicit some points that are only implicit in those remarks.

    There is really no problem. Any postulate that is independent of ZFC can be added to the ZFC postulates or not, depending on what you want to do with the resulting system.

    Is the amended system true in some general sense? Not necessarily, but it could be true of something that we want to study.

    Another way to look at this is to understand that for each independent new postulate, we can get two systems: One in which it is assumed and one in which its negation is assumed. Each system may be true of something, even if we don’t know what at moment.

    And, of course, each system may lead to new insights and understanding.

    It would actually be a problem if mathematical completeness were possible, because this would limit the applicability of mathematics. What if we adopt a postulate that supports the continuum hypothesis and then find that we need some mathematics for cases where this doesn’t work?

    We need to be able to make new systems to suit what we want them to apply to, just as we need to be able to make geometries in which the parallel postulate does not apply. Euclidean geometry applies to things in flat space, so we want a postulate that distinguishes flat space from curved “spaces.” This is what the parallel postulate does.

    It is convenient that the ZFC postulates are all intuitively true of numbers, but, as long as a new postulate is independent of the ZFC postulates and true of what we want to apply the resulting system to, there is really no reason to see such new postulates as problematic (unless they are logically problematic in their own right, independently of their role in the system being constructed, of course).

    For all I know (I’m no mathematician), the two approaches described in the article may both be problematic for their own reasons. But, assuming that each does not break consistency when assumed as a postulate, we are free to use either one (or both, if they are consistent with each other) according to our needs.

  • Physically, the continuum seems to be the biggest existing cardinal (somewhat analogous to the speed of light). Intuitively (when we free ourselves from the formal-axiomatic bias) it can be argued, amazingly, that continuum=aleph_2. In other words, the special continuum hypothesis should be false, specifically: aleph_0<aleph_1<aleph_2=c.

    In most mathematical applications (such as measure theory) the continuum and aleph_1 are the only used and clearly visible uncountable infinities (the latter consisting of all countable well-orderings). However, their roles are so disparate that they deserve separate cardinals to play these roles. At the same time, any well-ordering of the reals seems absurd, again making it absurd to equate c and aleph_1. Is the continuum very large? Probably not, since it appears to us in a simple linear order. So, if there were anything between aleph_1 and c, why is it not apparent? Probably because there *is* nothing in between.

    Remarkably, the same conclusion is provided by certain large cardinal axioms, as well as Martin's maximum. Thus, the special CH settled, we should rather be concerned with the size of 2^c, the powers of aleph_omega, and so on. Can these numbers be rationalized, too?

  • @Justin Moore Would you be willing to expand on the idea that truth in set theory is connected to truth in the physical world? I can think of two extremely simple objections, although they don’t pertain exactly to set theory. I think you are right that they are more appropriately philosophical questions. First, is the mathematical conception of a sphere threatened by the fact that there are not true spheres in the physical world? And second – you mention the parallel postulate. To be pedantic, it could be more complex – we exist on the surface of sphere, so perhaps it doesn’t exactly hold in the physical world. The geometry of the universe is also still debated, so maybe neither Euclidean nor spherical geometry are true in the sense of corresponding to the physical world. Yet it is still possible to derive theorems that work for a system with the parallel postulate or something similar.

    My guess is that these points have extremely simple rebuttals, so my apologies. Does one’s view on the connection of mathematical logic to the physical universe simply depend on one’s philosophical beliefs about logical systems? I don’t study math but I attended the Woodin lecture in your department a few years ago. I probably only understood the first 5 minutes, but I was happy to be there.

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