“Metaphysics and theology have, I confess, taken hold of my soul to such an extent that I have relatively little time left for my first flame. If my wishes had been fulfilled fifteen, or even eight years ago, I would have been given a larger mathematical sphere of influence…But now I thank God, the All-Wise and All-Good, that he has forever denied me the fulfillment of these wishes, for in this way he has forced me to serve Him and the Holy Roman Catholic Church better by penetrating deeper into theology than I could have done with apparently weak mathematical powers through an exclusive occupation with mathematics.”
— Georg Cantor, letter to Charles Hermite, 1894 [1]
When Georg Cantor (1845-1918) initiated the systematic investigation of infinities in set theory, mathematicians were forced to confront ontological issues that they had previously never considered. Infinitary questions at the time mainly touched upon the Calculus; indeed, it was his doctoral work on trigonometric series in the 1870s that first led Cantor to set-theoretic questions. At the time, mathematicians were fearful of possible paradoxes of “actual infinities,” and preferred to work with “potential infinities.” They therefore regarded their use of infinitesimals in the Calculus as an example of potential infinites, despite the confusion and murkiness such a position might have caused. This orthodoxy was deeply rooted, dating back to Aristotle. But as Cantor exposed various issues concerning potential infinities, such as the incommensurability of different infinite sets, the potential-actual distinction became untenable.
Eventually, a later generation proposed axioms are today largely agreed upon as a reasonable foundation of mathematics, called ZFC. The ZFC axioms explicitly include an Axiom of Infinity.
Indeed, by Cantor’s diagonalization argument, they seem to commit us to the existence of infinitely many sizes of infinity.
Subsequent developments in set theory illustrated that several important mathematical questions, such as the Continuum Hypothesis or Projective Determinacy, are not decided by the ZFC axioms. This motivated the search for additional axioms, which naturally led to a family of candidate axioms that postulate “higher” infinities. These axioms of higher infinity are called large cardinal axioms.
The program for large cardinals can be traced back to 1947, in a paper by Godel on the continuum hypothesis [2]. There, Godel laid out a program for seeking new axioms (a project I have written more on here). The program is decidedly Platonist in nature; Godel wants to find the true axioms, not just axioms that work well enough for some purposes. In the decades since, set theory has answered Godel’s call with a plethora of new axiomatic worlds. These new worlds, each given by a new set of axioms, form a rich and intricate web of theories that show remarkable connections and interdependencies. These inter-relationships are far more dense than one would expect a priori. Just as Godel had predicted, these axioms are so abundant in their verifiable consequences, and shed so much light upon the field of set theory, that we may have to accept them.
Figure: Various large cardinal axioms organized in a dependency tree. Axioms get stronger from bottom to top. We see a rich web of inter-related candidate axioms, despite different cardinalities defined in distinct ways. We will discuss inaccessible cardinals, which are at the very bottom.
By the criteria we have previously laid out (Second philosophy and/or Quinean holism), large cardinals seem to be very strong contenders for new axioms. In particular, they seem to cohere into the broader mathematical edifice, as well as have logical consequences that are already reasonable to us. But are we committing to too much? How far can we stretch the limits of infinity? Is there a non-arbitrary size of infinity to “cut off” the set-theoretic universe at?
Laying out the stakes
We are engaged in a project of discovering additional axioms beyond ZFC [3]. To a Platonist these axioms are True; to a formalist or logicist these are simply useful scientific assumptions or theories. It is worth stressing the point that I am not making ontological commitments here; even the most conservative formalist would agree that mathematical investigations are useful for science, and to make progress in mathematics we sometimes need new axioms. So, we then ask what methods there are to judge new axioms, and as I have written, there are reasonable paradigms such as Second philosophy and Quinean holism.
Now, we know this project of discovering additional axioms can never end [4], by Godelian incompleteness. So there is an additional question lurking in the background: what are the overall limits and possible end points of this project? In particular, is there an “absolutely undecidable” sentence [5]?
The smallest large cardinal axiom
The most concrete example of a large cardinal is of the weakly inaccessible cardinals, which are the smallest among the “large” cardinals. This is still quite abstract, but it can be very roughly described as follows.
Let omega be the size of the counting numbers {0,1,2,}. This is a certain size of infinity, and in fact omega is the smallest infinity, period.
For a set X let P(X) be the power set, meaning the set of all subsets of X. For example if X = {1,2} then P(X) = {{}, {1}, {2}, {1, 2}}. Notice X has size 2 and P(X) has size 4. Power sets are always larger than the original sets; when X is an infinite set, P(X) is a strictly larger infinite set.
We can successively apply P to the infinite set {0,1,2,...} to get a chain of increasingly large infinite sets. We can denote this “ladder” of increasingly large sets as:
omega, P(omega), P(P(omega)), …
Let kappa be the “limit” of this process (see limit cardinal for a formal definition). Then kappa is a WI (Weakly Inaccessible) cardinal.
Kappa is the smallest of the large cardinals; of course, it is still bizarrely and unimaginatively big. The physical universe can be said to have size P(omega) or P(P(omega)), depending on how we count it (as a Euclidean space, or as a Hilbert space). Already P(P(P(omega))), which is only “level 3” of this infinitely long ladder, is unimaginably huge, and kappa is infinitely larger still. This point was not lost on the mathematicians at the time of WI’s introduction. Quoting Kanamori [6],
Hausdorff was to write in his classic text that if weakly inaccessible cardinals did exist, “the least among them has such an exorbitant magnitude [exorbitanten Grösse] that it will hardly ever come into consideration for the usual purposes of set theory.” It is now well-known that the existence of weakly inaccessible cardinals cannot be established in ZFC.
Note that kappa is analogous to omega, the smallest infinity, in the sense that both involve completing a potential infinity. Just as we take the limit:
0, 1, 2, … —> omega
We can take:
omega, P(omega), P(P(omega)), … —> kappa
In both cases we move from a “potential infinity” which is only ever finite (or a “small infinity”) at every step, to an “actual infinity” which looks on from above at all of the smaller sets below it.
Completed infinity as a tool for consistency proofs
The reason we care about large cardinal axioms is that they resolve mathematical questions that are independent of ZFC. For example, Hugh Woodin has proposed a program of large cardinals that would resolve the Continuum Hypothesis [7]. But more strikingly, large cardinals also resolve consistency questions.
Theorem (Informal): The axioms set “ZFC + There exists a weakly inaccessible cardinal” proves the sentence “ZFC is consistent.”
To prove the statement, one argues that the axiom set “ZFC + there exists a WI cardinal” give a model (a possible world) that satisfy each of the axioms of ZFC. It is a theorem that if a set of axioms T has a model, then T is consistent (this is the soundness theorem). This is intuitive, because possible worlds cannot instantiate contradictions.
Because of the existence of the WI cardinal, we simply argue that the model has “enough” sets to satisfy the ZFC axioms. The only difficult step is to verify the Axiom of Replacement, and this is done (roughly speaking) by moving to the “actual infinity” of the WI cardinal, which ensures that all of the objects that we need to be sets are actually sets. This argument crucially considers the WI cardinal as a completed entity in itself, not a “potential” infinity that is approached but never reached.
Obviously, the argument generalizes beyond just WI cardinals. Large cardinals are evidently useful in proving consistency of theories that are “just below” them. Put another way, universes with large cardinals can prove the consistency of set-theoretic universes that are strictly smaller in size.
Universes with large cardinals can prove the consistency of set-theoretic universes that are strictly smaller in size.
The Absolute Infinite
While we have described WI cardinals, there is no reason we cannot continue up the hierarchy of infinite sets by taking the powerset P(kappa), then P(P(kappa)), then limits, and so on. All of these extensions are studied in modern set theory, giving rise to increasingly higher infinities and therefore increasingly stronger mathematical theories.
Source: Wylie Beckert for Quanta Magazine.
This tower of infinities goes on forever. Naturally, it is fascinating to push the concept of the infinite as far as we can, and see what deductive consequences follow. We should take care, however, to not become lotus eaters [8] and pursue such abstractions for their own sake. At each successive large cardinal, it is worth asking what the motivation for studying the object is. By my amateur estimation, it does seem that even the “larger” large cardinals are well-motivated by “actual” mathematical questions; for example Woodin cardinals are motivated by the resolution of the Continuum Hypothesis [7], and the even-larger huge cardinals were originally motivated by applications to knot theory.
If we are willing to accept that even the higher reaches of the large cardinal hierarchy are well-motivated, the natural question is what the end point of such a process might be. Peter Koellner has written about such absolute undecidability questions; but as to an endpoint, he simply states: “There is at present no solid argument to the effect that a given statement is absolutely undecidable. We do not even have a clear scenario for how such an argument might go” [5].
The only place left to go, then, would be to Cantor’s “Absolute,” which I alluded to in the beginning of this note. Properly speaking, such a concept is more theology than mathematics. Readers less tolerant of such speculation are invited to close the tab now.
The Absolute
Cantor spoke of the Absolute as an infinity lying beyond all infinities. Such a concept cannot hope to be rigorously defined; indeed, the WI cardinal we defined earlier seems to fit this description, but of course there are larger cardinals still. Therefore, we will not attempt to explicate the concept further, and instead quote Cantor’s own description, from a letter to Grace Chisholm Young [9].
I have never assumed a “Genus Supremum” of the actual infinite. Quite on the contrary I have proved that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”, which is by many called “God”.
The religious analogy is striking; I am reminded of the Hindu concept of Nirguna Brahman (“nirguna” meaning “indescribable”, and “Brahman” meaning “ultimate reality”). Quoting Sri Ramakrishna [11],
"What Brahman is cannot be described. All things in the world - the Vedas, the Puranas, the Tantras, the six systems of philosophy - have been defiled, like food that has been touched by the tongue for they have been read or uttered by the tongue. Only one thing has not been defiled in this way, and that is Brahman. No one has ever been able to say what Brahman is."
Cantor, for his part, corresponded regularly with the Catholic theologians of the day concerning his conceptions of infinity and the Absolute [10]. At the time, newly installed Pope Leo XIII (installed 1878) had issued the Encylical Aeterna Patris (“of the Eternal Father”), which sought to revive Scholastic philosophy in relation to the Church. This encouraged a generation of neo-Thomists, who argued that both atheism and materialism were the result of philosophical error [10].
Notably, Cantor corresponded with philosopher and theologian Cardinal Franzelin, who contributed to the former’s evolving views on the natural of potential and absolute infinites. Cantor argued that proofs referring to potential infinites necessarily assume the existence of the completed collection of numbers above a certain number. Taken as a totality, he referred to this completed whole as the Transfinitum. For example, the statement: “for all natural numbers N there exists an n > N” presupposed the existence of all n > N.
Cantor originally argued for the existence of the Transfinitum in natura naturata (that is, concrete existence); however, in his correspondence with Franzelin, who raised the concern that this view might be equated with Pantheism, Cantor refined his view of the infinite to account for theological objections. In particular, he introduced a distinction between the Absolute-Infinite and Actual-Infinite [13], the former having a religious significance ascribable to God. Cardinal Franzelin encouraged this distinction and in fact went further than Cantor, writing to the latter [14]:
I observe with satisfaction how you distinguish very well the Absolute-Infinite and that which you call the Actual Infinite in the created…thus the two concepts of the Absolute-Infinite and the Actual-Infinite in the created, or Transfinitum, are essentially different, so that when both are compared, only the one must be characterized as genuine Infinite [eigentlich Unendliches], the other as non-genuine [uneigentlich] and equivocal Infinite…Nevertheless, in one respect you most certainly go astray against the unquestionable truth; this error, however, does not follow from your concept of the Transfinitum, but from the deficient conception of the Absolute….According to your conclusion of the necessity of a creation of the Transfinitum, you ought to go much further yet. Your Transfinitum Actuale is an increasable; now if God's infinite Benevolence and Magnificence really demands with necessity the creation of the Transfinitum, so, for entirely the same reason of the infinite ness of His Benevolence and Magnificence, the necessity of increase until it would be no longer increasable follows, which contradicts your own concept of the Transfinitum. In other words: he who infers the necessity of a creation from the infiniteness of the Benevolence and Magnificence of God, must maintain, that everything creatable is indeed created from eternity[.]” [13]
Recall that the usefulness of large cardinal axioms is that they prove the consistency of set-theoretic universes strictly weaker than them. In this regard, we may have to agree with Cardinal Franzelin, who points out that creation (our universe) must stem from the Absolute (the higher infinity).
Conclusion
From these brief investigations into new axioms, I have become convinced that the deductive method is fundamentally limited. This is the lesson of Godelian incompleteness, but also a broader point about the project of logic and truth seeking. Historically, even rock-solid axioms that we completely endorse today were only chosen for a pragmatic reason. It’s possible that e.g. the Axiom of Choice is historically contingent. Therefore, as much as we are interested in the concrete question of whether large cardinals truly exist, we also want to test the limits of our intuition and understand how we may develop new methods of even seeking axioms in the first place.
As far as the truth of large cardinals, I feel that their use in establishing consistency proof is strong evidence in their favor; but the choice of which large cardinals to accept seems arbitrary. While we can look to “ordinary" mathematics as a guide for what levels of infinity to explore, these so-called ordinary fields also look to set theory for objects of study; therefore, these “natural” motivations for higher cardinals may ultimately end up being circular. Of course, there is much more texture and richness to the world of large cardinals than the naive linear ordering picture that I may have presented. For example, Woodin’s program has implications not only for large cardinals but also model theory, and in particular the theory of inner models and the forcing technique [7].
Saharon Shelah himself has asked, regarding the linear ordering of large cardinals, “Is there some theorem explaining this, or is our vision just more uniform than we realize?” [12]. Certainly a theorem explaining the usefulness and ordering structure of large cardinals would be satisfying, but absent such an explanation we are left to ask what is so special about sizes of sets, and why they have such remarkable implications for the consistency of mathematical foundations as a whole. Such questions haunted Cantor, and they continue to bedevil us more than a century later.
End Notes
[1] Meschkowski, Herbert. "Aus den briefbüchern georg cantors." Archive for History of Exact Sciences 2.6 (1965): 503-519.
[2] Gödel, Kurt. “What Is Cantor’s Continuum Problem?” The American Mathematical Monthly, vol. 54, no. 9, 1947, pp. 515–25. JSTOR, https://doi.org/10.2307/2304666.
[3] In fact, we’re more generally interested in plausible axiomatic systems that can foreground all of modern mathematics, including the process of reasoning itself. ZFC happens to be particularly popular and well-studied, but it is entirely possible that the “true” axioms are completely different (e.g. homotopy type theory).
[4] As I have written elsewhere, Chaitin’s constant is “pure information” with respect to the ZFC axioms. Even knowing the first 1,000 bits will not help us deduce the 1001th bit of the constant. Therefore, we may speculate that “irreducible mathematical information” will never run out, no matter how many axioms we add.
[5] Koellner, Peter. "On the question of absolute undecidability." Philosophia Mathematica 14.2 (2006): 153-188.
[6] Kanamori, Akihiro. The higher infinite: large cardinals in set theory from their beginnings. Springer Science & Business Media, 2008.
[7] Woodin, W. Hugh. "The continuum hypothesis, the generic-multiverse of sets, and the Ω conjecture." Set theory, arithmetic, and foundations of mathematics: theorems, philosophies 36 (2011): 13-42.
Note that Woodin’s approach requires much more than just the existence of Woodin cardinals; the overall program is called the inner model program, or V = Ultimate L.
[8] A phrase coined by algebraic geometer Ravi Vakil to disparage students who rush into abstraction without concrete motivations. Quoting Vakil: “Before discussing details, I want to say clearly at the outset: the wonderful machine of modern algebraic geometry was created to understand basic and naive questions about geometry (broadly construed). The purpose of this book is to give you a thorough foundation in these powerful ideas. Do not be seduced by the lotus-eaters into infatuation with untethered abstraction. Hold tight to your geometric motivation as you learn the formal structures which have proved to be so effective in studying fundamental questions. When introduced to a new idea, always ask why you should care. Do not expect an answer right away, but demand an answer eventually.” Source: The Rising Sea.
[9] Source: https://pointatinfinityblog.wordpress.com/2017/06/12/cantor-and-the-absolute-universal-structures-iv/
[10] Dauben, Joseph W. "Georg Cantor and Pope Leo XIII: Mathematics, theology, and the infinite." Journal of the History of Ideas 38.1 (1977): 85-108.
[11] The Gospel of Sri Ramakrishna, Visit to Vidyasagar, August 5, 1882. Source: https://hinduism.stackexchange.com/questions/38339/understanding-nirguna-brahman.
[12] Saharon Shelah. The Future of Set Theory. arXiv, 2002. https://arxiv.org/abs/math/0211397
[13] “[H]e further distinguished between an "Infinitum aeternum increatum sive Absolutum," reserved for God and his attributes, and an "Infinitum creatum sive Transfinitum," evidenced throughout created nature and exemplified in the actually infinite number of objects in the universe.” Source.
[14] On the Theory of the Transfinite: Correspondence of Georg Cantor and J.B. Cardinal Franzelin. Springer-Verlag 1994. Link here.