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Why is Cantor’s Absolute Inherently Inaccessible?

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Abstract

In this article, as implied by the title, I intend to argue for the unattainability of Cantor’s Absolute at least in terms of the proof-theoretical means of set-theory and of the theory of large cardinals. For this reason a significant part of the article is a critical review of the progress of set-theory and of mathematical foundations toward resolving problems which to the one or the other degree are associated with the concept of infinity especially the one beyond that of the natural intuition of natural numbers. Naturally the review includes the foundation and development of the theory of large cardinals, especially after Cohen’s revolutionary introduction of the forcing method, insofar as it paved the way toward a transcendence of the delimitative character of Gödel’s constructible universe L and further toward ever stronger infinity assumptions. Given that Cantor in his theory of transfinite numbers defined the mathematical absolute in ontological rather than concrete mathematical terms, I proceed in the last section to a philosophical discussion with certain prompts from phenomenology regarding the transposability of the formal conception of the infinite to the level of subjective constitution and argue in these terms on the infeasibility of acceding to an ‘absolute’ infinite cardinality. Of course the argumentation is strongly based on a view of the set-theoretical universe V of von Neumann’s cumulative hierarchy as a formal representative of the essential traits one would seek from a model of Cantor’s Absolute. It is also based on the conclusions reached from the whole discussion within the context of formal-theoretical structures as such.

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Notes

  1. The term large cardinals will be used to characterize those cardinals larger than ‘small’ large cardinals, e.g., those in the Mahlo hierarchies of inaccessible cardinals. These cardinals correspond to existence axioms based on extrinsic grounds, that is, on axioms not necessarily consistent with \(V=L\), which is what happens with the axiom of existence of a measurable cardinal. Large cardinals are often termed in bibliography as ‘large’ large cardinals.

  2. A proper class M is an inner model iff M is a transitive \(\in\)-model of ZF with \(\text{ ON }\subseteq M\) where \(\text{ ON }\) is the class of all ordinal numbers. A transitive set is intuitively a set that preserves the notion of \(\in\)-inclusion and is defined to be a set in which each of its elements is a subset of it.

  3. An extensive, though quite technical, look into the research results in inner models theory in the last decades of 20th century and a brief reference to prospective results in the beginning of 21st is found in Mitchell’s Inner Models for Large Cardinals, Mitchell (2012).

  4. U is an ultrafilter over a set S iff it is a maximal filter over S, i.e. for any \(X\subseteq S\) either \(X\in U\) or else \(S-X\in U\). Given that the paper includes quite a few technical mathematical terms I chose for reasons of cohesion and economy of space to be selective about definitions. The interested reader may in any case consult Kanamori’s The Higher Infinite about relevant definitions, Kanamori (2009).

  5. This is a reference to the statement of existence of a measurable cardinal, MC, which was proved back in the sixties by L\(\acute{e}\)vy and Solovay, in Lévy and Solovay (1967), to have no influence on the decidability of CH. More precisely, it was proven that if ZFM is set theory with the axiom asserting the existence of a measurable cardinal, then if ZFM is consistent then it is consistent with both the continuum hypothesis CH and its negation.

  6. Woodin was able to show that if an inner model could be found compatible with supercompact cardinals, then, contrary to what holds for smaller large cardinals, this inner model would be compatible with all other large cardinals which are consistent with ZFC. This significant result could lead to a new kind of axiom, the axiom \(V=\text{ Ultimate }-L\). In turn, this axiom would decide independent statements such as CH. However given that such an inner model has not yet been found, there has been a turn of Woodin’s toward the idea of applying strategic extender models which in case they can accommodate a supercompact cardinal they can accommodate all large cardinals consistent with ZFC. These extender models are of the form \(\text{ HOD }^{L(A,R)}\) where A is a universally Baire set. Proving the existence of such models it would be, thanks to the axiom \(V=L_{S}^{\Omega }\) (i.e., \(V=\text{ Ultimate }-L\)), possible to achieve the global consistency requirements of the model as above and the truth of CH. As a matter of fact there are no formal proofs as yet relative to the existence of such models and the ensuing conjecture. A more detailed technical exposition of these matters can be found in Rittberg (2015) (pp. 140–146).

  7. This is done in arguing for the Inner Model Hypothesis, an alternative infinity hypothesis essentially ensuring a concept of maximization based on qualitative rather than quantitative criteria. This is the following statement: If a statement \(\varphi\) without parameters holds in an inner universe of some outer universe of V (i.e., in some universe compatible with V), then it already holds in some inner universe of V (Arrigoni and Friedman 2012, p. 1364). See also Friedman (2006).

  8. Let it be noted that while IMH has served to settle some independent questions of ZFC it still leaves the CH question unresolved (Arrigoni and Friedman 2012, p. 1365).

  9. See: Feferman (1990), fn 23, p. 262.

  10. The construction of the ultrapower of V (or generally of any inner model of ZFC) by ultrafilter U hinges principally on the following satisfaction formula and is based on Lós theorem:

    $$\begin{aligned} <^{S}V/ U, E_{U}>\models \varphi [(f_{1})_{U},\ldots , (f_{n})_{U}]\;\;\text{ iff }\;\;\{i\in S;\; V\models \varphi [f_{1}(i),\ldots ,f_{n}(i)]\}\in U \end{aligned}$$

    where S is a subset of the universal set V, U an ultrafilter over S and \(<^{S}V/ U, E_{U}>\) the ultrapower of V by U. For more details on these matters look into: Kanamori (2012b), pp. 365–367 and Kanamori (2009), pp. 47–49.

  11. Footnote 14 refers essentially to the intuitive, iterative concept ‘set of x’s’, even though the operation ‘set of x’s’ itself seemed to Gödel somewhat circular in that it can only be paraphrased by other expressions involving again the concept of set, e.g., ‘multitude of x’s’, ‘combination of any number of x’s’, etc. (Feferman 1990, p. 263).

  12. The Shoenfield’s Absoluteness Lemma states that all \(\sum _{2}^{1}(\alpha )\) and \(\prod _{2}^{1}(\alpha )\) relations are absolute for inner models M of ZF theory (plus the weaker dependent choice maxim) which contain the real number \(\alpha\) as an element. In other words, Shoenfield’s Absoluteness Lemma shows that in the analytical hierarchy, \(\sum _{2}^{1}\) and \(\prod _{2}^{1}\) sentences are absolute between the model V of ZF and the constructible universe L, when interpreted as statements about the natural numbers in each model or about sets of natural numbers as parameters from V.

  13. As commonly understood this term refers to a concept of the set of all subsets of a given set envisaging all possibilities in the iteration of this mathematical act irrespectively of how a set is defined, e.g. by a predicative definition or not.

  14. \(J_{x}\) is an idealized infinite game associated with each subset X of the set of real numbers, or what essentially amounts to the same thing, of the set \(2^{N}\) of all infinite sequences of 0’s and 1’s. This game has two idealized players, I and II, where at each stage of the play, I plays first by choosing a 0 or a 1; player II responds in the same way. The play terminates with a sequence s in \(2^{N}\), for which player I wins if \(s\in X\) and player II wins if not. The game \(J_{x}\) is said to be determined if one or the other of the players has a winning strategy, i.e. a rule for how to play at each stage in order to win no matter what choices are made by the opposite player. AD says that \(J_{x}\) is determined for each set X of reals.

  15. For structures \({\mathcal {M}} = \{M, . . .\}\) and \({\mathcal {N}} = \{N, . . .\}\) and a language L, an injective function \(j : {\mathcal {M}}\longrightarrow {\mathcal {N}}\) is an elementary embedding of \({\mathcal {M}}\) into \({\mathcal {N}}\) denoted \(j: {\mathcal {M}}\prec {\mathcal {N}}\), iff it satisfies the elementarity schema: for any formula \(\varphi (v_{1}, . . . , v_{n})\) of L and \(x_{1},\ldots ,x_{n}\in {\mathcal {M}}\)

    $$\begin{aligned} {\mathcal {M}}\models \varphi (x_{1},\ldots ,x_{n})\;\;\text{ iff }\;\;{\mathcal {N}} \models \varphi (j(x_{1}),\ldots ,j(x_{n})) \end{aligned}$$

  16. Kunen’s proof is based on the Erdös-Hajnal theorem which is dependent in a substantial way on the application of the Axiom of Choice (Kunen 1971). The problem is open without this assumption. On the other hand, the question of whether a weaker version of Reinhardt cardinals is consistent with the axioms ZF, and thus not dependent on AC, proves to be a controversial issue too. This owes to the results of Woodin’s, namely, the theory ZF + ‘There is a weak Reinhardt cardinal’ proves the formal consistency of the theory ZFC + ‘There is a proper class of strongly (\(\omega\) + 1)-huge cardinals’ with the latter cardinals being the strongest large cardinals not known to be refuted by the Axiom of Choice (Woodin 2011, p. 99).

  17. \({\mathcal {N}}\) is a common symbol for the Baire space \(\omega ^{\omega }\).

  18. For a filter F over cardinal \(\lambda\), F is normal iff for any class \(\{X_{\alpha };\;\alpha < \lambda \}\; \in \;^{\lambda }F\) its diagonal intersection

    $$\begin{aligned} \Delta _{\alpha< \lambda }X_{\alpha }=\{\xi< \lambda ;\;\xi \in \cap _{\alpha < \xi } X_{\alpha }\}\in F \end{aligned}$$

    By setting \(X_{\alpha }=\lambda\) for \(\alpha\) sufficiently large it follows that normality subsumes \(\lambda\)-completeness. See: Kanamori (2009), p. 26 & pp. 52–53.

  19. By Zorn’s lemma it is proved that any partially ordered set in which each chain (i.e., a linearly ordered subset) has an upper bound, has a maximal element. Historically, versions of what is known as Zorn’s lemma were given prior to Zorn by Haudorff in connection with the maximality aspect of the Well-Ordering Theorem (Kanamori 2012a, p. 23.)

  20. The existence of the set \(0^{\sharp }\) has a significant role to play in the development of large cardinals theory and the quest for an ever closer approximation to the set-theoretic universe as due to Jensen’s covering theorem it expresses a sense of closeness between V and L by its non-existence!

  21. Cantor used the Burali-Forti paradox positively to give a mathematical expression to his Absolute. The ‘totality of everything thinkable’ was noted to be an absolutely infinite or inconsistent multiplicity (Kanamori 2012a, p. 18). An excellent exposition of the theoretical debate that followed the Burali-Forti paradox which is essentially reducible to the ambivalences of the well-ordering concept along infinite classes is presented in Moore’s and Garciadiego’s Burali-Forti’s Paradox: A Reappraisal of its Origins, Moore and Garciadiego (1981).

  22. Call \(\kappa\) a Woodin cardinal if for each class \(A\subseteq V_{\kappa }\) there exists a cardinal \(r< \kappa\) which is strong with respect to A in \(V_{\kappa }\). See also definition in Kanamori (2009), p. 360.

  23. A cardinal \(\kappa\) is \(\gamma\)-supercompact iff there is an elementary embedding \(j: V\longrightarrow M\) for some inner model M, with critical point \(\kappa\), \(\text{ crit }(j)=\kappa\), and \(\gamma < j(\kappa )\) such that \(^{\gamma }M\subseteq M\), i.e. M is closed under the taking of arbitrary \(\gamma\)-sequences. \(\kappa\) is supercompact iff \(\kappa\) is \(\gamma\)-supercompact for every \(\gamma \ge \kappa\) (Kanamori 2012b, p. 379).

  24. Such instances are, to cite a few, Foreman’s, Magidor’s and Shelah’s result that if there is a supercompact cardinal \(\kappa\), then there is a forcing extension in which \(\kappa =\omega _{2}\) and where Martin’s Maximum holds, Kanamori (2012b), p. 398, the same set-theorists’ result that the Levy collapse of a supercompact cardinal to \(\omega _{2}\) generates a \(\aleph _{1}\)-complete \(\aleph _{2}\)-saturated ideal over \(\omega _{1}\), ibid., p. 398, and Magidor’s result that if \(\kappa\) is supercompact there is a forcing extension in which \(\kappa\) is \(\aleph _{\omega }\) as a strong limit cardinal, yet it violates the Singular Cardinal Hypothesis, i.e. \(2^{\aleph _{\omega }}> \aleph _{\omega +1}\), ibid., p. 387.

  25. Of course \(\mathbf{ZF} + AC\) is also proved consistent with the negation of \(V=L\) so there is no apparent contradiction concerning the consistency of AC with \(\mathbf{ZF} + V = L\), at least on epistemological grounds.

  26. Immanence or immanent object is roughly meant something correlative (or co-substantial) to the stream of consciousness and therefore not ‘external’ or transcendent to it.

  27. Here I must refer to some phenomenological concepts I apply, for which the reader may further consult Husserl’s Ideas I, Husserl (1976), Experience and Judgment, Husserl (1939), as well as other Husserlian works. Summarily, a noematic object corresponding to the phenomenological notion of noema is an object as meant, said to be constituted by certain modes as a well-defined object immanent to the temporal flux of a subject’s consciousness (Husserl 1976, Engl. transl., pp. 240–245). By eidetic laws or eidetic attributes in the world of phenomena one can roughly communicate to a non-phenomenologist what relates to the existence of objects or states-of-affairs as regularities by essential necessity and not by mere facticity. One may also consult E. Husserl’s Ideas I; Husserl (1976), Engl. transl., pp. 12–15.

  28. Even though Maddy claims to be driven by naturalistic motivations in the general sense that “the entities to be admitted are just those posited by and studied in the natural sciences, and that the methods of justification and explanation are somehow continuous with those of the natural sciences”, she argues against \(V=L\) and in favor of the acceptance of various large cardinal axioms from a point to view that does not always conform to a conventional naturalistic attitude. I refer in particular to her claim that mathematics in general and set theory in particular are susceptible of interpretation in their own right independently of relationships to natural sciences (Feferman et al. 2000, p. 409).

  29. By the first principle of generation a given number \(\gamma\) is followed by its successor \(\gamma + 1\). By he second principle to any given ‘definite succession’ of numbers, e.g. an integer sequence, is assigned a least upper bound. See also Sect. 1.

  30. Cantor remarked, in a letter to Hilbert in 1897, that a collection of well-defined, distinguished elements of our intuition or thought conceived as a whole is “only possible if a ‘being together’ is possible” without clarifying the conditions or laws under which the distinct elements can be united into a whole (Hauser 2013, p. 11). Given his metaphysically leaning attitude, however, one can safely assume that these conditions were taken to be of an ontological nature, therefore presumably not subjectively based.

  31. See e.g., Husserl (1973), pp. 205–206.

  32. See: Husserl (1939), pp. 246–247.

  33. This means that one does not perform a reduction of the intuitive continuum to a constitutive consciousness in terms of the biological-neuronic structure of the brain possibly down to the quantum level.

  34. In this sense I cite, for instance, the ‘curtailing’ of uncountable inconsistent conditions by the Countable Chain Condition restriction to preserve the power of cardinals along forcing extensions, (Kunen 1982, p. 205) or the diagonalizing through all strategies, in assuming AC, to prove that there is an undetermined set \(A\subseteq \omega ^{\omega }\) (Kanamori 2012b, p. 381).

  35. Prompted by this controversy Oskar Becker set out to argue in Becker (1927) that the indefinite even in the metamathematical assumptions of Hilbert’s theory is not avoided and is not avoidable. See Becker (1927), pp. 485–494.

  36. For details on the being of absolute consciousness, as well as its temporal essence and its proper intentional forms the interested reader may look into Husserl’s On the Phenomenology of Inner-Time Consciousness, Husserl (1966).

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    Stathis Livadas

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Livadas, S. Why is Cantor’s Absolute Inherently Inaccessible?. Axiomathes 30, 549–576 (2020). https://doi.org/10.1007/s10516-020-09474-y

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