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Filozofski vestnik | Volume XLI | Number 2 | 2020 | 361–395 | doi: 10.3986/fv.41.2.14
Norma M. Hussey
A New Hope for the Symbolic, for the Subject
Symbolization and ontology
Tradition is the old world of castes, nobilities, religious obligation, … local my -
thology, the submission of women, the father’s absolute power over his children, 
and the official division between a small group of rulers and a condemned mass 
of toilers, in which the differences inherent to human life were regulated and sym -
bolized in a hierarchical form. The most important binaries, like old and young, 
men and women, … commoners and nobles, town and countryside – were all ad -
dressed (in language, in mythologies, in ideologies, and in the established reli-
gious models) by recourse to ordered structures setting everyone’s place in a set 
of overlapping hierarchical systems.
1
From the philosophical writings of Alain Badiou (illuminated by the mathemat -
ical ontology) it is recognized here that the hierarchical symbolization of ‘tradi-
tion’ found its guarantee, at the abstract level, in the supposition that beyond 
the multiple, the one is. In other words, the ontological basis for the hierarchical 
form of the traditional symbolic assumes the being of the one, and relies on it 
for the “normative power”
2
 of its deployment. As such, it is a symbolic based on 
finitude, which means that (in the mathematical ontology) it “resides within the 
elementary immanence of … finite ordinals.”
3
 This is the case whether the one 
be finite or infinite. For instance, “Christian monotheism, despite its designa-
tion of God as infinite, does not immediately and radically rupture with Greek 
finitism” – it can be based on a thinking in which being as such “remains essen-
tially finite;” the assumption of an infinite one does not yet break with the finite, 
1 Alain Badiou, “True and False Contradictions of the Crisis”, trans. David Broder, Verso 
blog, May 29, 2015. Emphasis added.
2 Alain Badiou, Theoretical Writings, trans. Ray Brassier and Alberto Toscano, Bloomsbury 
Academic, London and New York 2014, p. 44.
3 Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum, London and New York 
2007, p. 160.

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norma m. hussey
because it is still “hierarchically representable as beyond [or unknowable], yet 
deducible from,” the finite world.
4
 According to Badiou, the onto-theological 
device – trying to save truth – pays the price of the absolute transcendence of 
the one, thereby subordinating finite multiplicities to the formal authority of the 
One-infinity that has often been called God.
5
Modernity is effectively a break with tradition. In barely three centuries, this break 
with the world of tradition has swept aside forms of organization that had lasted 
for millennia. When this break takes a bourgeois, capitalist form, it opens up a 
gigantic crisis of humanity’s symbolic organization. It does not in fact propose any 
active new symbolization, but only the brutal, independent play of the economy: 
the neutral, a-symbolic reign of what Marx called “the icy water of egotistical 
calculation.”
6
It could be said that the breakdown of the traditional symbolic corresponds to 
modern ontological confusion owing to the death of God,
7
 i.e. the infinite One. 
Modernity – in a world structured by globalized capitalism – has revealed that 
being is essentially multiple; it has exposed the one as pure semblance, and 
therefore ruined the supposition on which the hierarchical symbolic depends. 
In his Manifesto for Philosophy, Badiou writes that 
The only thing we can and must welcome within Capital [is that]: it exposes the 
pure multiple as the foundation of presentation; it denounces every effect of One 
as a simple, precarious configuration … To think over and above Capital … we must 
still have as a departure point what it has revealed: Being is essentially multiple.
8
Symbolization and subjective orientation
Rejection of the traditional hierarchy and its oppressions exposes the being of 
the one as un-credible. But freedom from hierarchy, from subordination to the 
4 Ibid, p. 142.
5 Badiou, as reported by Jana Berankova in “The Immanence of Truths: the Absolute be -
tween the Singular and the universal”, Presentation at the International Conference 
“Thinking the Infinite”, April 11, Prague 2018.
6 Badiou, “True and False Contradictions of the Crisis”. Emphasis added.
7 Badiou, Theoretical Writings, pp. 27–30.
8 Alain Badiou, Manifesto for Philosophy, trans. Norman Madarasz, SUNY Press, Albany, 
New York 1996, pp. 56–57.

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a new hope for the symbolic, for the subject
normative power of the one, without symbolic replacement, “creates a subjective 
crisis … one of whose most remarkable aspects is the extreme and growing diffi-
culty that young people in particular face finding themselves a place in this new 
world.”
9
 Disorientation arises with the breakdown of symbolization, i.e. with 
the lack of a common understanding of the symbolic organization. Perhaps this 
is why the theme of the subject has been a constant for modernity. 
Faced with the crisis of humanity’s symbolic organization (and the associat -
ed crisis of general subjective disorientation) some would have us believe that 
there is no crisis, and that there is nothing better than this liberal ‘democratic’ 
model of freedoms weighed down by the neutrality of market calculations. This 
is Western capitalism’s a-symbolic vision, which creates monstrous inequalities 
and pathogenic upheavals; under the auspices of ‘freedom’ in the a-symbolic 
reign, exploitation replaces the oppressions of hierarchy, and often inspires the 
reactive desire to return to the traditional hierarchical symbolization.
10
With respect to the break with tradition then, the subjective orientations of mo -
dernity range from the embrace of this liberal a-symbolic model of freedoms 
(especially by the rich), to the reactive desire to return to the traditional sym-
bolization (especially by the ex-privileged), and in the wider context of general 
disorientation, from imaginary utopianism to hopeless nihilism. But there is an-
other subjectivity which, recognizing the non-being of the one and the unavoid -
able break with tradition, is convinced that a non-hierarchical symbolization 
is possible and that it must be invented, i.e. “an egalitarian symbolization that 
restructures differences – [recognized of equal subjective right] – based on a 
total sharing of resources.”
11
A new symbolic requires a coherent ontological referent
From reading Badiou, it is my understanding that a symbolic (defined as a uni-
versal structure encompassing the entire field of human action and existence)
12
 
can only be based (contrary, it would seem, to Lacan) on a credible thinking of 
being as such, i.e. a stable and consistent ontology. The normative power of an 
9 Badiou, “True and False Contradictions of the Crisis”.
10
 Ibid.
11
 Ibid.
12
 Encyclopedia.com, Symbolic, the (Lacan), available at: https://www.encyclopedia.
com>psychology>symbolic-lacan

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norma m. hussey
assumed one (finite or infinite) sustained the hierarchical symbolic for millen-
nia, but such an assumption is no longer credible – it has been exposed as pure 
semblance, and being is revealed as essentially multiple. If all is multiple, is a 
symbolic even possible? That is, a symbolic which recognizes the non-being of 
the one and therefore cannot rely on the deployment of its normative power, as 
such a non-hierarchical and therefore egalitarian symbolic? It would at least 
require a coherent, i.e. stable and consistent, ontology of multiplicity.
An ontology of multiplicity
The thesis of Alain Badiou which equates ontology with mathematics
13
 is under -
scored by the philosophical decision that being, or what we can think under the 
name of ‘being,’ is composed of pure multiplicities, and that “the one is not,” be 
it the one of an Idea (Plato), or a God, or Presence, or any-one; “the one exists 
solely as operation.”
14
 Since the axiomatic system of mathematical set theory 
delivers the multiple without implying the being of the one,
15
 Badiou adopts it 
as the thinking of being as such (ontology). Set theory is the area of modern 
mathematics devoted to the study of infinity; mathematics thinks (infinite) mul -
tiplicities in the form of the notion of a (infinite) set.
16
 It is based on a formal 
axiom system called ZFC (Zermelo-Fraenkel system with the axiom of choice). 
Quantitative infinity is not deducible from finitude, but is authorized solely by 
the axioms of mathematical set theory.
17
 It is by recognition of the mathematical 
existence of quantitatively distinct infinite multiples, that the ruin of the being 
of the infinite one is achieved.
18
 This thorough-going rationalization of actual 
infinities (as opposed to a transcendent one) teaches us that there is no reason 
to confine thinking within the ambit of finitude because we can have access to a 
rational, secular thinking of infinity.
19
 Mathematical ontology (i.e. set theory) is 
the proposal of an infinite ontology which is, finally, a radical rupture with the 
finitism of tradition, and in which “the finite is qualified as a region of being, a 
minor form of the latter’s presence.”
20
13
 Badiou, Being and Event, p. 4. 
14
 Ibid, p. 23.
15
 Ibid, p. 43.
16
 Ibid, p. 145.
17
 Ibid, p. 148.
18
 Ibid, p. 273.
19
 Badiou, Theoretical Writings, p. 19.
20
 Badiou, Being and Event, p. 159.

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a new hope for the symbolic, for the subject
Badiou, in fact, proposes the universe of set theory – conventionally denoted by 
V – as an absolute ontological referent, where V can be said “to formalize … the 
place … of everything that can be constructed by axioms.”
21
 He argues that “the 
absoluteness of the thinking of the pure multiple, in the atheistic takeover from 
monotheism … is feasible.”
22
But is it a coherent theory?
In Being and Event (1988),
23
 Badiou described the then prevailing situation of 
ontology (set theory), with all of its intricacies and its metaontological implica-
tions.  In particular, he highlighted the impasse concentrated in the continuum 
problem which, as Paul Cohen established in 1963, is undecidable from the con-
temporary ZFC axiomatic. Since then, the formal axiom system itself has been 
in crisis – which challenges the very conception of mathematical infinity,
24
 to -
gether with the credibility of set theory as a consistent ontology. It doesn’t auger 
well for the invention of an egalitarian symbolic which can only be based on an 
ontology of the multiple, if that ontology is inconsistent or uncredible. The pre -
vailing incoherence of the mathematical ontology underscores a contemporary 
deficit of symbolization which, in turn, yields confusion and conflict in terms of 
subjective orientation.
But something momentous is happening right now in set theory, in response to 
the recent surprise discovery now encapsulated in the universality theorem (see 
later); a new axiom, conjectured as yet, promises to resolve the crisis of its for -
malization,
25
 and achieve the pass of the impasse. The consequent realization of 
a coherent set theory – a stable and consistent ontology of multiplicity – would 
suggest that an egalitarian symbolization is possible, i.e. a modern symbolic 
which receives its guarantee solely, but transparently, in its mathematical con-
sistency. Indeed, “mathematics is the only discourse in which one has a com-
21
 Acheronta Movebo (Badiou), “Towards a New Thinking of the Absolute”, Crisis and Cri-
tique 1 (2/2014), p. 12. Badiou was one of the contributors to this journal as part of the 
anonymous collective Acheronta Movebo.
22
 Ibid., p. 21.
23
 The original edition of Being and Event, i.e. Badiou, L’Être et L’Événement, Éditions du 
Seuil, Paris, was published in January 1988.
24
 W. Hugh Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”, Presentation 
at the IMS Graduate Summer School in Logic, National University Singapore, June 2018.
25
 W . Hugh Woodin, “In Search of Ultimate-L, the 19
th
 Midrasha Mathematicae Lectures”, The 
Bulletin of Symbolic Logic 23 (1/2017), p. 2. (DOI:10.1017/bsl.2016.34).

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norma m. hussey
plete guarantee and a criterion of the truth of what one says, to the point that 
this truth is unique inasmuch as it is the only one ever to have been encountered 
which is fully transmissible.”
26
 
This remarkable development in set theory, led by W. Hugh Woodin,
27
 offers 
hope for the pursuit of a new symbolic in a modern age of actual infinity, and 
a consequent resolution of the general subjective disorientation which arises 
from our current crisis of symbolization.
The premise here, then, is that the process for the articulation of a coherent and 
transmissible ontology on which to ground an egalitarian symbolic has already 
begun; it originated with the Cantorian inventions of set theory and quantitative 
infinity, but it is still undergoing development as the contemporary set theoretic 
axiom system (ZFC) is not yet stable/complete.
Set theory – historical development
The saga surrounding the development of set theory over the past 150 years – its 
splits, its advances, its dramatic crises – reflects the extreme difficulty of in-
vention of an ontology of multiplicity which is not in the least intuitive as it 
depends on complex mathematical reasoning and innovation for the thinking 
and comparison of distinct quantitative infinities. Mathematical infinity is a 
counter-intuitive concept, even at its simplest level, for instance “the infinite set 
of even numbers {2,4,6,…} can be placed 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,”
28
 therefore we cannot rely on intuition, but require 
the patient and painstaking work of mathematical rigor, in order to understand 
it. The development of set theory also seems (almost uncannily) to have mir -
rored the general underlying situation in the social and political world over the 
same period (see below).
26
 Badiou, Being and Event, p. 9. Emphasis added.
27
 W . Hugh Woodin is the mathematical set theorist leading the remarkable developments in 
set theory referenced throughout this article, at Harvard University since 2014, and before 
that at the University of California, Berkeley.
28
 Natalie Wolchover, “To Settle Infinity Dispute, a New Law of Logic”, quantamagazine.org, 
November 26, 2013 (reprinted on ScientificAmerican.com, December 3, 2013).

367
a new hope for the symbolic, for the subject
The nine axioms of ZFC were laboriously developed around the turn of the 20
th
 
century, thereby putting Cantor’s still partly intuitive theory of sets on a secure 
axiomatic (or non-intuitive) footing.
29
 They accord with the iterative conception 
of sets as cumulative hierarchies, where sets are constructed by a combination 
of axiomatically prescribed procedures in successive steps from an assumed 
foundational empty set Ø, and the sets accumulate – hence ‘cumulative.’ The 
universe of sets V (as formalized by ZFC) is calibrated by the ordinals through 
the definition of the cumulative hierarchy of sets,
30
 in which the successor step 
is obtained by the power-set (i.e. the set of all subsets) of the preceding set. 
According to Badiou, the ordinals – as the ontological schema of natural multi -
plicity – “ say that nothingness is a form of natural being [the empty set Ø being 
an ordinal, and therefore a natural multiple], and that the infinite, far from be -
ing retained in the One of a God, is omnipresent in nature, and beyond that, in 
every situation-being.”
31
Constructibility and conservatism
Set theory was tentatively concluded with a tenth axiom, i.e. Kurt Gödel’s axi -
om of constructibility (1938), which asserts that the universe of sets V is the con -
structible universe L, i.e. V = L. Gödel’s definition of L arises from restricting the 
successor step in the definition of the cumulative hierarchy to the constructible 
power-set (also called the definable power-set), which is the set of all subsets of 
the preceding set such that each subset is logically definable (in the set struc -
ture) from parameters in the set.
32
29
 Peter Hallward, “On the development of Transfinite Set Theory”, in Peter Hallward, Ba -
diou: A Subject To Truth, University of Minnesota Press, Minnesota 2003, p. 337. He also 
notes that Cantor’s own position was anti-axiomatic.
30
 The cumulative hierarchy of sets V, is the collection of sets V
α
 indexed by the class of or -
dinal numbers. For each ordinal α, a set V
α
 is defined by induction on α as follows: 1. V
0
 = 
Ø, 2. (Successor step) V
α+1
 = P(V
α
), the power-set of V
α
, 3. (Limit step) If α is a limit ordinal, 
then V
α
 = U {V
β
|β<α}, the union of all the V-stages so far.
31
 Alain Badiou, Number and Numbers , Polity Press, Cambridge UK and Malden USA 2008, 
p. 84.
32
 W. Hugh Woodin, “The Transfinite Universe”, in Kurt Gödel and the Foundations of Math -
ematics. Horizons of Truth, ed. M. Baaz, C. H. Papadimitriou, H. W . Putnam, D. S. Scott, and 
C. L. Harper Jr., Cambridge University Press, Cambridge 2011, pp. 449–471.

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norma m. hussey
Gödel’s axiom “does provide a clear conception of the universe of sets V,”
33
 but 
as a minimal statement about the universe of sets (because if the set is infinite 
and the axiom of choice holds, then the definable power-set is never the set of all 
subsets of the set)
34
 it was never fully embraced as a true axiom for set theory.
35
 
In addition, the constructible universe cannot accommodate the large cardinals 
in the higher infinite which, although unprovable from ZFC, are now considered 
to be genuine.
36
 (Since the existence of such a set cannot be proven on the basis 
of ZFC, it has to form the object of a new axiom. What is then at stake is an axiom 
of infinity, stronger than that of ZFC which guarantees only the existence of a 
limit ordinal,
37
 i.e. a large cardinal axiom stating that there exists a cardinal (or 
perhaps many of them) with some specified large cardinal property. Such axi-
oms appear to occur in strict linear order by consistency strength, thus forming 
the large cardinal hierarchy. A measurable cardinal is a relatively ‘small’ large 
cardinal, i.e. it is low in the hierarchy.
38
) By Scott’s theorem (1961), there are no 
measurable cardinals in the constructible universe and therefore no stronger 
large cardinals either.
39
 As such, V = L “severely limits the nature of infinity.”
40
It could be said that Gödel’s constructible universe represents a first attempt at 
an abstract referent for an egalitarian symbolic, in that it does yield a model for 
ZFC, i.e. a stable and consistent framework (albeit restricted) for an ontology of 
multiplicity. It is a universe for all ordinals (i.e. natural multiplicities) indicating 
that “the infinite is omnipresent in nature,” but beyond that (i.e. with respect 
to non-natural multiplicities) it has nothing to say. As such it may be consid -
ered as a natural ontology, but it doesn’t account for every situation-being. Its 
mathematical experimentation coincided with the period of social and political 
conservatism of the 1940’s and ‘50’s, which is not to suggest that Gödel was re -
33
 Ibid.
34
 W. Hugh Woodin, “Strong Axioms of Infinity and the Search for V”, in Proceedings of the 
International Congress of Mathematicians, Hyderabad, India 2010. 
35
 Woodin writes: “We know that Gödel [himself] rejected the axiom V = L”, in “The Transfi-
nite Universe”.
36
 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”.
37
 Badiou, Being and Event, p. 513.
38
 Colin J. Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hy -
pothesis”, Archive for History of Exact Sciences 69 (2/2015), pp. 125–151.
39
 Ibid.
40
 Woodin, as quoted by Wolchover in “To Settle Infinity Dispute, a New Law of Logic”.

369
a new hope for the symbolic, for the subject
sponsible for that period of conservatism but, perhaps, that his work explains 
the abstract thinking behind what was going on at the time.
In The Immanence of Truths,
41
 Alain Badiou claims that we (still) live in a con-
structible universe without being aware that this is the case.
42
 He argues that the 
constructible universe largely equals … finitude,
43
 i.e. a modern finitude where 
the finite is defined (here) as a set that can be ‘covered’ by a constructible set 
with the same cardinality as the former.
44
 Badiou asserts that this modern form 
of finitude is one of ‘movement and innovation,’ as opposed to the classical 
finitude of ‘stability and repetition;’
45
 nevertheless it represents a return to the 
figure of oppression, to the normalization of the one, and a closure of thought 
to the realm of actual infinity. Such ‘finitude’ is precisely what Badiou rejects.
Independence and dissatisfaction
The axiom system (ZFC) itself has been in crisis since 1963 when Paul Cohen 
demonstrated that the most basic problem of set theory – that of Cantor’s con-
tinuum hypothesis – was not solvable on the basis of the ZFC axioms; it can 
neither be refuted nor proven from the axioms of set theory, it is an independent 
statement.
46
 This result established the impossibility, within ZFC, of determining 
the type of multiplicity of the continuum (which corresponds to the power-set 
of the very first infinity, i.e. all the subsets constituted from whole numbers); set 
theory could not determine “the measure of the set of parts of an infinite set,” 
or in Badiou’s metaontological terms, “the excess of the power of the state with 
respect to that of its situation.”
47
 Easton’s theorem (1970) extended this result 
41
 Alain Badiou, L ’Immanence des Vérités, Fayard, Paris 2018.
42
 Terence Blake, “My Path Through Badiou’s The Immanence of Truths”, available at: https://
terenceblake.wordpress.com/2018/10/06/my-path-through-badious-the-immanence-of-
truths. 
43
 Jana Berankova, Presentation at the Graduate Workshop with Alain Badiou on The Imma-
nence of Truths, November 13, Columbia University, New York 2019.
44
 Jana Berankova, “The attributes of the absolute and Alain Badiou’s response to Spinoza”, 
in Sometimes We Are Eternal, ed. J. N. Berankova and N. Hussey, Suture Press, Lyon 2019, 
pp. 174–175.
45
 Terence Blake, “Badiou’s The Immanence of Truths: Introduction (sketch)”, available at: 
https://terenceblake.wordpress.com/2017/01/19/ badious-the-immanence-of-truths-intro -
duction-sket/
46
 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”.
47
 Badiou, Being and Event, p. 275.

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norma m. hussey
and established the “quasi-total errancy of the excess of the state over the situ -
ation,” in that it is consistent with ZFC that the power-set could be any cardinal 
as immense as you like, provided that it is a successor.
48
 
Cohen’s method of forcing ushered in a new age of ‘freedom’ from the restricted 
formalization assumed by the constructible universe; it exposed the formal inde -
pendence of not just the continuum hypothesis, but also the axiom of constructi-
bility (Gödel’s constructible universe), from the set-theoretical framework
49
 – it 
inherently implied the non-obligation of conservatism. In the immediate term, 
however, the focus was on the unquantifiable excess of the infinite power-set it -
self, rather than on the implication of crisis for ZFC, owing to such independent 
statements. The recognition of the “exorbitant excess,” the “innumerable injus -
tice,” implied by the almost-total un-measure of the state with respect to the sit -
uation,
50
 seems to underlie the general dissatisfaction prevailing in the 1960’s & 
‘70’s and articulated against the state – whether in the domain of mathematics 
itself, or in the domains of art, or love, or politics.
Incoherence and disorientation
But today it’s a question of credibility with respect to the capacity of set theory to 
deliver a coherent framework for infinite multiplicities. The intractability of the 
independence phenomenon due to Cohen’s method of forcing, still unresolved 
more than fifty years later, challenges the foundational issues of truth in set 
theory (ZFC) itself, and raises the question of whether the continuum hypothesis 
has a truth value at all, or even whether one can “coherently talk of the realm 
of the infinite.”
51
 Hugh Woodin adds that the modern significance of Scott’s 
theorem – in light of significant advances in large cardinal theory since then –  
establishes that the axiom of constructibility is false, i.e. that V ≠ L, but he ac -
knowledges that this claim is not universally accepted.
52
 By Jensen’s dichotomy 
theorem,
53
 both hypotheses (i.e. V = L, and V ≠ L) are completely valid in con-
48
 Ibid, p. 280.
49
 Woodin, “The Transfinite Universe”. 
50
 Badiou, Being and Event, p. 282.
51
 W. Hugh Woodin, “The Realm of the Infinite”, in Infinity. New Research Frontiers, ed. M. 
Heller and W. H. Woodin Cambridge University Press, Cambridge 2011, pp. 89–118.
52
 Woodin, “The Transfinite Universe”.
53
 The Jensen dichotomy theorem shows that V must be either very close to L or very far from L. 
Refer to Woodin, “In Search of Ultimate-L, the 19
th
 Midrasha Mathematicae Lectures”, p. 2.

371
a new hope for the symbolic, for the subject
temporary mathematics, which contributes to the ambiguity of set theory. The 
prevailing incoherence of the frame of reference for thinking the infinite (or the 
crisis of formalization) coincides with contemporary widespread disorientation 
as manifested in dissatisfaction with society, mental ill-health, troubled youth, 
popular frustration, a feeling of obscure disorder, lack of orientation/stability.
Platonism, pluralism, and the skeptic
Set theory’s contemporary indeterminacy undermines its candidacy as an abso -
lute referent. As such, two major contradictory orientations arise; the universe 
view (or Platonism), and the multiverse view (or pluralism), together with a 
third: the reactionary orientation of the skeptic.
The universe view (or Platonism) holds the conviction that there is a unique 
absolute background concept of set, realizable in the corresponding absolute 
set-theoretic universe, V , in which every assertion has a definite truth value, i.e. 
that quantitative infinity is rationally thinkable in a coherent framework.  Ques -
tions such as the continuum hypothesis and others have definitive final answers, 
and it is considered that the pervasive independence phenomenon is due to an 
insufficiency of the theory in finding the truth, rather than about truth itself – 
because the independence of a statement from ZFC tells little about whether it 
holds or not in the universe.
54
 With regard to the continuum hypothesis, Gödel 
wrote that “it must be either true or false … and its undecidability from the axi-
oms as known today can only mean that these axioms do not contain a complete 
description of reality.”
55
 For Hugh Woodin, also Platonist, this undecidability is 
“perhaps tolerable on a temporary basis during a period of axiomatic discovery 
but it certainly cannot be the permanent state of affairs.”
56
 He also considers 
that, based on set theoretical work (his own and others), “there is the very real 
possibility that we can find a new optimal axiom (from structural and philo -
sophical considerations) that prevents us from undecidability of some impor -
tant properties of set. In which case, we will have returned, against all the odds 
or reasonable expectation, to the view of truth for set theory that was present 
54
 Joel David Hampkins, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”, 
(MR2857736 (2012h:03002)), based on a talk given by the author at the Philosophy of Math -
ematics Conference (New York University, April 2009).
55
 As quoted by Wolchover in “To Settle Infinity Dispute, a New Law of Logic”.
56
 Woodin, “Strong Axioms of Infinity and the Search for V”.

372
norma m. hussey
at the time when the investigation of set theory began.”
57
 Alain Badiou writes: 
“By saying that set theory constitutes an absolute reference, I am assuming that 
there exists a system of axioms, incompletely discovered as yet, which defines 
the universe V … and defines it alone. In other words, no important, significant, 
useful property of sets will remain undecidable once we have been able to fully 
identify the axioms.”
58
The opposing multiverse view (or pluralism) considers that there are many dis -
tinct concepts of set, each realized in their corresponding universes which, in 
turn, exhibit diverse truths. Its adherents argue that we may prefer some uni-
verses to others, and that there is no obligation to consider them all as some -
how equal. Here, the continuum problem is already settled, having a different 
answer in different universes, and it is considered that the Platonist’s ‘dream’ 
solution of a new axiom for set theory by which the continuum hypothesis is 
decided, is impossible. This view does not necessarily undermine the claim that 
set theory serves as an ontological foundation for mathematics, but rather is 
directed against the claim that there is an absolute frame of reference in which 
set-theoretic truths are immutable.
59
 Absolute, of course, means not relative, 
and for Badiou, the current of thought corresponding to this vision is relativism. 
He writes:
 60
 
Regarding the possibility that V could be an ontological referent, the most impor -
tant group of opponents is made up of those who have given up on any referent at 
all and claim that a truth is never anything but relative or local … This current of 
thought, suited like no other to representative democracy and cultural relativism, 
is the prevailing one today … [E]ven those who acknowledge a sort of practical 
value in it … have objected that the statement “there are only opinions” must be 
absolutely true, otherwise something other than opinions might well exist, and 
an absolute truth would therefore exist.
57
 W. Hugh Woodin, “The Axiom of Determinacy, Forcing Axioms, and the Nonstationary 
Ideal”, in De Gruyter Series in Logic and its Applications, ed. W.A. Hodges, R. Jensen, S. 
Lemp, M. Magidor, 2
nd
 ed., De Gruyter, Berlin 1999.
58
 Badiou, “Towards a New Thinking of the Absolute”, p. 16.
59
 Hampkins, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”.
60
 Badiou, “Towards a New Thinking of the Absolute”, p. 13.

373
a new hope for the symbolic, for the subject
The multiverse view – which ultimately corresponds to an a-symbolic vision 
since a symbolic requires an absolute referent – is dominant today so that diso -
rientation reigns as we are cast adrift in an a-symbolic world.
The skeptic refers to the meta-mathematical position which asserts that the only 
meaningful existence assertions are those that can be verified in the physical 
universe – which certainly excludes set theory (the mathematics of infinite 
sets).
61
 According to Woodin, it is a position which denies any genuine meaning 
to a conception of uncountable sets and considers that set theory and its exten-
sions do not reflect any mathematical reality – they are merely a reflection of 
the mathematician.
 62
 For the skeptic, the continuum hypothesis is neither true 
nor false, and consistency claims (e.g. for large cardinal axioms) can never be 
meaningfully made, because the entire concept of the universe of sets is a com-
plete fiction; as such, this view is simply a rejection of the infinite altogether.
63
 
For example, taking a view originally espoused by Aristotle, Stephen Simpson
64
 
argues that actual infinity doesn’t really exist and so it should not so readily be 
assumed to exist in the mathematical universe. As reported by Natalie Wolcho -
ver, 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,” he says. “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.”
65
The current of thought corresponding to the skeptical vision responds to the 
crisis of formalization (ontology’s contemporary indeterminacy) with a rejec -
tion of infinity altogether, and proposes a return to finitude, i.e. to the one of 
hierarchy, or a traditional symbolic. One might consider that such skepticism in 
mathematics is analogous with the political orientation which is very obviously 
61
 Andrés Eduardo Caicedo, reviewing Woodin’s “The Realm of the Infinite”, submitted to 
Mathematical Reviews/MathSciNet, MR 2767235, June 26, 2012.
62
 Woodin, “The Transfinite Universe”.
63
 Woodin, “Strong Axioms of Infinity and the Search for V”.
64
 Stephen Simpson is a mathematician and logician at Pennsylvania State University, USA.
65
 Wolchover, “To Settle Infinity Dispute, a New Law of Logic”.

374
norma m. hussey
gaining traction in the contemporary world, as manifested by popular support 
(reflecting obscure revolt)
66
 of Trump, Orban, Erdogan, Modi, etc.
67
Contemporary assessment of set theory (ontology)
Set theorists (today) are broadly drawn from three camps, which pursue the de -
velopment and refinement of set theory from different angles: the doctrine of 
constructibility (originating in Gödel’s constructible universe, and pursuing an 
inner model program for the incremental understanding of large cardinals),
68
 
the doctrine of genericity (originating in Cohen’s theory of generic extensions, 
and pursuing a forcing program for expanding the width of the universe of 
sets),
69
 and the doctrine of large cardinals, where the hierarchy of large cardi-
nals, while unmoored from ZFC, is increasingly emerging as an intrinsic, fun-
damental conception within set theory.
70
 As described in Being and Event, the 
orientations in thought corresponding to these three doctrines, hold mutually 
antagonistic positions, especially with respect to the question of the state.
71
  But 
in recent times perhaps, the ‘universists’ among them find more common cause 
in recognition of their shared conviction (albeit differing approaches) of the pos -
sibility of an absolute framework in which set-theoretic truths are immutable, 
and they continue its pursuit, often even against reasonable expectation.
Since the turn of the 21
st
 century, adherents of the universe view have insisted on 
a rational discussion and investigation based on a contemporary assessment of 
the facts and developments in set theory to date. For instance, they recognize that 
Cohen’s result (1963) and its mathematical descendants have severely challenged 
any hope for a concise conception of the universe of sets, since it is not provided 
by the axioms ZFC – as has been well-documented in the (more than) 50 years 
since.
72
 In fact Cohen’s method of forcing has been vastly developed and many 
66
 Badiou, “Reflections on the Recent Election”, Verso blog, November 15, 2016, which is a 
transcript of his Presentation at the University of California, November 9, Los Angeles 2016.
67
 These are the current (2020) democratically elected presidents of the USA, Hungary, Tur -
key, India, respectively – all with characteristics of proto-fascism.
68
 Woodin, “In Search of Ultimate-L, the 19
th
 Midrasha Mathematicae Lectures”. 
69
 Wolchover, “To Settle Infinity Dispute, a New Law of Logic”.
70
 Woodin, “The Transfinite Universe”.
71
 Badiou, Being and Event, chap. 27.
72
 Woodin, “Strong Axioms of Infinity and the Search for V”.

375
a new hope for the symbolic, for the subject
questions have been shown to be unsolvable; it’s not only about the continuum 
hypothesis which is just a statement about an infinitesimal fragment of V .
73
 
Also, large cardinal axioms are not provable from ZFC by Gödel’s 2
nd
 incomplete -
ness theorem.
74
 A large cardinal axiom is a statement that a ‘very large’ infinite 
set with certain properties exists; these cardinals carry names such as, for ex -
ample, inaccessible, weakly compact, Ramsey, measurable, strong, Woodin, 
superstrong, strongly compact, supercompact, extendible, huge, ɷ-huge, Rein-
hardt. “With the definition of huge cardinals, we approach the point of rupture 
presented by inconsistency,”
75
 but we cannot determine that point from ZFC.
All of this challenges the very conception of mathematical infinity; there is a 
temptation to yield, and to accept the skeptical assessment that the concep -
tion of the universe of sets is incoherent.
76
 The skeptic wants to reject actual 
infinity for the sake of certainty (ontological coherence); this represents an ex -
treme conservatism, to which adherents of the constructible universe are also 
vulnerable in the face of increasing recognition, among set theorists, of ‘large’ 
large cardinals without any formal consistency.
77
 Yet Woodin, for example, still 
insists that “there are statements (predictions) which we expect to be false (or 
true), but there does not seem to be a coherent argument that would explain 
‘why’ unless we adopt as meaningful the conception of a non-physical realm, 
a view where it makes sense to talk of the existence of at least some infinite 
objects.”
78
 For Badiou, in any case, it is meaningless to propound the theory of 
finitude since a rational, ramified thinking of infinity exists, i.e. large cardinal 
theory – which is the modern theory of infinity.
79
 Recall that the theory of fini -
tude (for him) extends beyond the classical form (embraced by the skeptic) to 
the modern form of finitude which is the constructible universe L.
73
 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”.
74
 Ibid.
75
 Badiou, Being and Event, p. 448. Badiou quoting set-theorist Thomas Jech.
76
 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”.
77
 The constructible universe cannot accommodate even a measurable cardinal – which is a 
relatively ‘small’ large cardinal in the hierarchy.
78
 Woodin, “The Realm of the Infinite”.
79
 Badiou, “Towards a New Thinking of the Absolute”, p. 14.

376
norma m. hussey
Over the last few decades it has become commonplace to doubt that an ultimate 
completion of set theory can be articulated, or that a natural completion will 
settle the continuum problem.
80
 The greater pressure then, is to conform to the 
dominant view that there can be no certainty with regard to infinity (relativ -
ism), that the arbitrariness is precisely what is so exciting about it (liberalism), 
and in any case the pursuit for a coherent conception of V goes against reason -
able expectation.
Forcing axioms
Meanwhile, forcing axioms (from the doctrine of genericity) continue to extend 
the frontiers of mathematics – the universe of sets is extended to form a new, 
wider universe; a forcing axiom called ‘Martin’s maximum’ (after the mathema -
tician Donald Martin) discovered in the 1980’s, extends the universe as far as it 
can go. Forcing axioms fill some problematic holes in everyday mathematics; 
work over the past few years (including new-found uses of Martin’s maximum) 
by Stevo Todorcevic
81
 and others shows that they bestow many mathematical 
structures with nice properties that make them easier to use and understand. 
They also solve the continuum hypothesis – deeming it false by adding a new 
size of infinity; in the extended universe created by forcing, there is a larger class 
of real numbers than in the original universe defined by ZFC, which means the 
real numbers of ZFC constitute a smaller infinite set than the full continuum. 
If the point is to find the most fruitful seeds of mathematical discovery (as op -
posed to the grains of truth that yield the most pristine mathematical universe), 
then forcing axioms emerge as a possible contender for addition to ZFC. Never -
theless, according to Todorcevic, “it is difficult to justify Martin’s maximum as 
an axiom from a philosophical point of view; it can only be justified in terms of 
the influence it has had on the rest of mathematics.” Woodin considers that it is 
a rich theory for understanding structures in classical mathematics but “it’s not 
clear how it would lead to a better understanding of infinity,” which (for him) is 
what set theory is about.
82
80
 Abstract for “The Joint Quest for Absolute Infinity and the Continuum – From Cantor to 
Woodin”, Series of (three) lectures by Hugh Woodin at the University of Turku, May 9-11, 
Finland 2019.
81
 Stevo Todorcevic is a mathematician at the University of Toronto, and the French National 
Center for Scientific Research in Paris.
82
 Paragraph paraphrased from Wolchover, “To Settle Infinity Dispute, a New Law of Logic”.

377
a new hope for the symbolic, for the subject
Large cardinal hierarchy
As already mentioned, large cardinal axioms are not provable from ZFC. Nor is 
it known whether the existence of such sets is even consistent with ZFC, with 
one exception; a basic theorem proved by Kunen (1971)
83
 shows that the exist -
ence of a Reinhardt cardinal is inconsistent with ZFC. The proof relies on the 
axiom of choice, and it is not known if the existence of a Reinhardt is inconsist -
ent with ZF (i.e. ZFC minus the axiom of choice).
84
 But large cardinal axioms can 
be well-ordered by their consistency strength, thus yielding the large cardinal 
hierarchy.
85
 Also, they can decide some of the problems which are unsolvable 
from ZFC alone, for example, the existence of a measurable cardinal implies 
that the universe of sets is larger than Gödel’s constructible universe.
86
 (This is 
because the existence of a measurable cardinal implies the existence of Silver’s 
0
#
 (zero sharp),
87
 where 0
#
 is the set of true formulae about indiscernibles and 
order-indiscernibles in the constructible universe L. The existence of 0
#
 howev -
er, like that of a measurable cardinal, is unprovable from ZFC).
88
But they cannot decide the continuum hypothesis. (This is because the contin -
uum hypothesis can be made true or false by forcings (Cohen) which are small 
in the sense that they only act on the lower (infinite) levels of the cumulative 
hierarchy, but these small kinds of forcings do not act on the level of a measura -
ble cardinal or beyond, therefore the existence of a measurable (or any stronger 
large cardinal) simply has no bearing on the truth value of the continuum hy -
pothesis).
89
83
 For the actual statement of Kunen’s theorem, see Akihiro Kanamori, “The Higher Infinite”, 
Perspectives in Mathematical Logic, Springer-Verlag, Berlin 1994.  This is Ref [5] in Woodin, 
“The Transfinite Universe”.
84
 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”.
85
 One theory has a higher consistency strength than another, if the consistency of the former 
implies the consistency of the latter. The order of consistency strength in the large cardinal 
hierarchy is not necessarily the same as the order of the size of the smallest witness to a 
large cardinal axiom. For example, the existence of a huge cardinal is much stronger, in 
terms of consistency strength, than the existence of a supercompact cardinal, but assum-
ing both exist, the first huge is smaller than the first supercompact.
86
 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”.
87
 0
#
 (zero sharp) was first introduced in Silver’s 1966 thesis, which was later published as: 
Jack H. Silver, “Some Applications of Model Theory in Set Theory”, Annals of Pure and Ap-
plied Logic 3 (1/1971), pp. 45–110.
88
 Woodin, “In Search of Ultimate-L”, p. 3.
89
 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”.

378
norma m. hussey
Yet the development of set theory after Cohen has led to the realization that 
formally unsolvable problems have degrees of unsolvability which can be ex -
actly calibrated by large cardinal axioms, which has proven to be useful. As a 
consequence of this calibration, it has been discovered that in many cases very 
different lines of investigation have led to problems whose degree of unsolva-
bility is the same. Thus the hierarchy of large cardinals emerges as an intrinsic, 
fundamental conception within set theory.
90
An interesting example in this context is that large cardinal axioms do not exist 
in the universe of number theory, yet their existence generates new truths of 
that universe; more precisely, assuming the large cardinals to be true, one can 
infer as true specific statements of number theory which arguably cannot other -
wise be proven. How then can the number theorist account for these truths? The 
skeptic of course will object here that all one is asserting about large cardinal 
axioms is formal consistency, and that such consistency claims cannot be mean-
ingfully made since the concept of the universe of sets is so ‘incomplete’ that it 
is a complete fiction.
91
 For so long as large cardinal axioms remain independent 
set-theoretical statements, this skeptical position cannot be refuted.
Exploring the universe from below (the local argument)
‘Universists’ however point to the increasingly stable consequences of the large 
cardinal hierarchy as evidence that we are on the right track towards final an-
swers to set-theoretical questions.
92
 Large cardinal axioms not only fit together, 
but all extensions of set theory ‘considered in practice’ seem to be equiconsist -
ent with large cardinal axioms; their arithmetic consequences are compatible, 
and (eventually) so are their projective consequences, thus providing, for exam-
ple, a coherent theory of the reals (i.e. of V
ɷ+1
). This is illustrated by the axiom of 
determinacy (an assertion about infinite games and pivotal to current research 
in set theory) that turns out to be equiconsistent with the assertion that there are 
infinitely many Woodin cardinals.
93
 The axiom of determinacy (with extrinsic 
justifications) and ‘there exist Woodin cardinals’ (with intrinsic justifications) 
are proved to coincide. “This sort of convergence of conceptually distinct do -
90
 Woodin, “The Transfinite Universe”.
91
 Ibid.
92
 Hampkins, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”.
93
 Caicedo, reviewing Woodin’s “The Realm of the Infinite”.

379
a new hope for the symbolic, for the subject
mains is striking and unlikely to be an accident,” as Peter Koellner
94
 wrote in 
2006.
95
 Yet despite these successes, with the amount of mathematical effort and 
development which was required – which yielded the correct axioms for a cer -
tain part of the universe of sets, i.e. V
ɷ+1
 (in which the real numbers appear), and 
noting that this is just an infinitesimal fragment of the universe of sets V, then 
the prospects for understanding V itself to this same degree (or even just V
ɷ+2
 in 
which all sets of reals appear, and which would reveal whether the continuum 
hypothesis is true) seem daunting.
96
Probing the ‘whole’ universe of sets (the global argument)
Taking another track, the successes of the inner model program (from the doc -
trine of constructibility) were encouraging, and led to the realization that the 
large cardinal hierarchy is a very ‘robust’ notion.
97
 The inner model program is 
the detailed study of large cardinal axioms, by seeking generalizations of the 
definition of the axiom of constructibility (Gödel) which are compatible with 
these axioms (such as the axioms for measurable cardinals and beyond),
98
 i.e. 
by enlarging the constructible universe (denoted by L) to accommodate targeted 
large cardinal axioms. The axiom of constructibility, i.e. V = L, severely limits 
the nature of infinity, but the constructible universe L, does have the benefit 
of being pristine and analyzable (unlike the cumulative hierarchy of sets V), 
and enlargements of L, being L-like, offer the same advantage. Inner models, 
i.e. enlargements of L, are constructed using both constructible elements and 
extender elements as defining parameters, where an extender is derived from 
the ‘elementary embedding’ which is closely linked to the large cardinal under 
consideration.
99
 Large cardinals can be defined in terms of elementary embed -
dings, i.e. an embedding of V itself into some inner model M,
100
 and extenders 
are non-trivial fragments of the elementary embedding.
101
 More precisely, one 
94
 Peter Koellner is a philosophy professor specializing in mathematical logic at Harvard Uni-
versity.
95
 As quoted by Zhaokuan Hao, (Fudan University, China) in “Gödel’s Program and Ultimate 
L”, Presentation at the National University of Singapore, September 2017.
96
 Woodin, “Strong Axioms of Infinity and the Search for V”.
97
 Woodin, “The Transfinite Universe”.
98
 Ibid.
99
 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”.
100
 Where M is a transitive model of ZFC containing all the ordinals, and an elementary em-
bedding is a function (not the identity) between models which preserves truth. Ibid.
101
 Woodin, “The Transfinite Universe”.

380
norma m. hussey
has to require that the large cardinals of the inner models which constitute the 
solution have some form of ancestry in the large cardinals of V. As such, the 
extender models are constructed as refinements of V, which preserve enough 
extenders from V to witness that the large cardinal axiom holds.
102
By the year 2000, the inner model program had been unconditionally extended 
to the level of Woodin cardinals – which are limits of strong cardinals, and con-
ditionally extended to the level of superstrong cardinals.
103
 But the incremental 
nature of the program seems to be an absolutely fundamental aspect: each new 
construction of an enlargement meeting the challenge of a specific large cardi-
nal axiom comes with a theorem that no stronger large cardinal axiom can hold 
in that model. Since it’s very unlikely that there could ever be a strongest large 
cardinal axiom, this methodology seems unlikely by its very nature to yield an 
ultimate enlargement of the constructible universe, i.e. to ever succeed in pro -
viding the requisite axiom for clarifying the conception of the universe of sets.
104
The pursuits for a new axiom for set theory from below (analyzing the set-theo -
retic universe fragment by fragment) looked daunting, and from above (a form 
of extension of ZFC based on the whole universe) seemed hopeless.
105
 
Then something happened…
Circa 2010 the situation changed dramatically, with the “remarkable and com-
pletely unexpected” discovery within the inner model program, that at a specif -
ic, critical stage in the hierarchy of large cardinal axioms, there is a breakaway 
point:  at the level of one supercompact cardinal, the inner model construction 
must yield an enlargement which is compatible with all stronger large cardinal 
axioms, i.e. it must yield an ultimate enlargement of the inner model L.
106
 The 
requirement for enough suitable extenders from V to witness that the supercom-
pact cardinal holds, necessarily must generate ‘phantom’ witnesses (extenders) 
for all large cardinals stronger in the hierarchy.
107
 This is now formulated in the 
102
 Ibid.
103
 Woodin, “In Search of Ultimate-L”, p. 1.
104
 Woodin, “Strong Axioms of Infinity and the Search for V”.
105
 Rittberg, “How Woodin Changed his Mind”.
106
 Woodin, “Strong Axioms of Infinity and the Search for V”.
107
 Woodin, “The Transfinite Universe”.

381
a new hope for the symbolic, for the subject
‘universality theorem.’
108
 If such an inner model could be found, then it would 
accommodate all large cardinals (that are consistent with ZFC), which amounts 
to a paradigm shift in the whole conception of inner models.
109
 Is this not a 
mathematical event?
The entire framework for the inner model program was transformed from a pro -
gram for the incremental understanding of large cardinals into a program for 
perhaps understanding V itself, and it led to the isolation of a candidate for an 
axiom that V is an ultimate version of L.  W. Hugh Woodin is on the brink (in 
a decadal sense) of demonstrating a transcendent version of Gödel’s axiom of 
constructibility for the universe of sets; but where Gödel’s axiom is minimal (i.e. 
the mimimum possible universe of sets containing all ordinals), Woodin’s axi-
om is a maximal statement about the nature of the universe of sets. Everything 
now hinges on the fate of the ‘ultimate-L’ conjecture which, if true, provides an 
absolutely compelling case that V = ultimate-L.
110
 
Strategic-extender models
The various approaches to inner model theory have to be fundamentally al -
tered to provide the solution to the inner model problem for one supercompact 
cardinal.
111
 The ultimate-L program introduces a new class of model for V – a 
different, previously unknown class of extender model called the strategic-ex -
tender models, which can no longer be layered (as in the non-strategic case);
112
 
based on the strategic-extender models, one can formulate the axiom, V = ulti-
mate-L, without referring to the detailed fine-structure theory of these models, 
or even using the definition of the structures yielding the levels of these models. 
The point (here) is that, in the context of a proper class of Woodin cardinals, 
there are naturally defined approximations to ultimate-L and the collection is 
rich enough to make a definition of the axiom, V = ultimate-L, possible without 
specifying the detailed level-by-level definition of ultimate-L. Woodin notes that 
108
 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”.
109
 Rittberg, “How Woodin Changed his Mind”.
110
 Woodin, “Beyond the Age of Independence by Forcing,” Presentation at the Chinese Math-
ematical Logic Conference”, May 20, 2017.
111
 Woodin, “In Search of Ultimate-L”, p. 1.
112
 Ibid, p. 93.

382
norma m. hussey
this situation is analogous to being able to formulate the axiom, V = L, without 
specifying the definition of L.
113
Strategic-extender models were developed by generalizing on the earlier suc -
cess in understanding V
ɷ+1
 and the projective sets – which provided a good the -
ory of the reals (assuming the existence of Woodin cardinals). They make use of 
the class of all hereditarily ordinal definable (HOD) sets, i.e. the class containing 
all those sets which are definable from ordinal parameters.
114
 Originally defined 
by Gödel,
115
 Woodin gives an equivalent reformulation of the class HOD which 
highlights it as some sort of merge of the definitions of the cumulative hierarchy 
V and of the constructible universe L. For the power-set in L, the subset is logi-
cally definable from parameters in the set, whereas in HOD, the subset is defin-
able without parameters. He also notes here that there is a remarkable theorem 
of Vopenka which connects HOD and Cohen’s method of forcing,
116
 and which 
illustrates why Cohen’s method is so central in set theory (and for reasons other 
than simply establishing independence results).
117
Ultimate-L program
The program to prove the ultimate-L conjecture is a 4-stage project. Stage 1 has 
been completed (~2013) and it survived its first critical test; this was the con-
struction of an L-like weak extender model for the finite levels of supercompact -
ness. Then, “in an unexpected twist,” Woodin reported, “the remaining three 
stages collapsed into a single final stage, and subsequent analysis has provided 
the key clues as to how to proceed.”
118
 With many evidences leaning toward a 
positive answer for the ultimate-L conjecture, Woodin is quite confident of the 
outcome. However, as he wrote in March 2017, “given the series of unexpected 
events to date on this subject, an abundance of caution seems prudent here.”
119
 
113
 Ibid, pp. 94–95.
114
 Woodin, “The Transfinite Universe”.
115
 More precisely, the definition of the class HOD originates in remarks of Gödel at the Princ -
eton Bicentennial Conference in December, 1946. According to Woodin, the first detailed 
reference appears to be Ref [8] in Ibid.
116
 Theorem (Vopenka). For each set G ⊂ Ord, if G ∉ HOD, then HOD
G
 is a Cohen extension of 
HOD.
117
 Woodin, “Strong Axioms of Infinity”.
118
 Woodin, “Beyond the Age of Independence by Forcing”.
119
 Woodin, “In Search of Ultimate-L”, p. 3.

383
a new hope for the symbolic, for the subject
As of May 2017 there was a draft of a proof and, according to Woodin (at that 
time), “things look promising.”
120
In 2019, Woodin said that the ultimate-L conjecture reduces the entire post-Co -
hen debate on set theoretic truth to a single question which must have an an-
swer – true or false, i.e. it cannot be meaningless. Either the conjecture is true, 
in which case the axiom, V = ultimate-L, is very likely the key missing axiom for 
V . Or it is false, in which case the program to understand V , by generalizing the 
(local) success in understanding V
ɷ+1
 and the projective sets, fails.
121
The axiom V = Ultimate-L (conjectured)
· Large cardinal axioms can (and therefore do) exist in the universe given by 
ultimate-L. 
This is the key new feature which sets it apart from all previous generalizations 
of L. Ultimate L is mathematically analyzable – in this sense it is L-like – and 
yields a much deeper understanding of the large cardinal axioms; it identifies 
more precisely the transition for large cardinal axioms from the possible to the 
impossible (i.e. the demarcation of consistent and inconsistent sets), and pro -
vides a framework for a continuing evolution in the understanding of this tran-
sition.
122
 One obtains a new generation of inconsistency results for the large car -
dinal hierarchy (in the setting where the axiom of choice fails) which includes a 
mild strengthening of Reinhardt cardinals, and also Berkeley cardinals.
123
· The conjectured axiom affirms the ZFC axiomatic, and in particular the axi-
om of choice.
It recognizes all large cardinal axioms (both known, and those yet to be discov -
ered) consistent with ZFC, but will eliminate essentially all the large cardinal ax -
ioms known to contradict the axiom of choice.
124
 For example, the existence of 
120
 Woodin, “Beyond the Age of Independence by Forcing”.
121
 W. Hugh Woodin, “The Continuum Hypothesis”, Presentation at the University of Münster 
(WWU), Germany, June 20, 2019.
122
 Woodin, “The Transfinite Universe”.
123
 Woodin, “In Search of Ultimate-L”, p. 3.
124
 Ibid.

384
norma m. hussey
weak Reinhardt cardinals – which was already proven to contradict ‘choice’ by 
the well-known Kunen inconsistency (1971)
125
 – is actually refuted here.
126
 More 
fundamentally, it reveals deep connections between large cardinal axioms and 
proving instances of the axiom of choice.
127
 One also obtains, in the context of the 
axiom of choice, that what seem like natural generalizations of axioms of defin-
able determinacy, are false if sufficient large cardinals are assumed to exist.
128
· The continuum hypothesis is decided.
Surprisingly in fact, it holds,
129
 which runs counter to the intuitive expectations 
of notable Platonists, from Gödel, to Woodin himself, to Badiou. The type of mul -
tiplicity of the power-set is decided and with a minimum of excess; the contin-
uum is the second infinity. In fact the axiom renders Cohen’s method of forcing 
completely useless for establishing independence from the resulting conception 
of the universe of sets;
130
 it settles all questions about ‘small’ sets which have 
been shown to be independent by that method. This is because the axiom V = 
ultimate-L (or V = L, for that matter) strongly couples the width of the universe 
of sets V, to its height. Woodin explains that V has two dimensions, width and 
height; the height is a large cardinal issue, and the width is – how many reals 
do you have, i.e. how many sets of reals. V = ultimate-L binds the width to the 
height; if you change the width, you cannot recover the axiom without changing 
the height. Cohen’s method changes the width, but preserves the height of V, 
and that’s why it is useless – it cannot affect the axiom.
131
Woodin further conjectures that this axiom is true, i.e. that it will eventually be 
validated on the basis of accepted and compelling principles of infinity, exactly 
as the axiom of projective determinacy was validated,
132
 and the truth of which 
only became evident after a great deal of work, i.e. mathematical results proven 
125
 Kenneth Kunen, “Elementary embeddings and infinite infinitary combinatorics”, J. Sym -
bolic Logic 36 (1971), pp. 407–413. This is Ref [7] in Woodin, “The Transfinite Universe”.
126
 Woodin, “The Realm of the Infinite”.
127
 Woodin, “Strong Axioms of Infinity and the Search for V”.
128
 Woodin, “In Search of Ultimate-L”, p. 3.
129
 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”.
130
 Ibid.
131
 Woodin, “The Continuum Hypothesis Set Theory”, Presentation at the California Institute 
of Technology (Caltech), February 23, 2019.
132
 Woodin, “Strong Axioms of Infinity and the Search for V”.

385
a new hope for the symbolic, for the subject
by the set theoretic community. This demonstrated (for him) that the discov -
ery of mathematical truth is not a purely formal endeavour.
133
 Using the phrase 
“truth beyond our formal reach,” Woodin maintains that there’s a component 
in the evolution of our understanding of mathematics which is not formal, that 
there is mathematical knowledge which is not entirely based in proofs.
134
Some meta-ontological implications
Set theory’s contemporary indeterminacy – which renders the impossibility 
of regarding it as an absolute referent – stems from the independence of large 
cardinal axioms, and the undecidability of the continuum hypothesis. The 
non-intuitive nature of the axiom of choice, which is “without any known pro -
cedure,”
135
 doesn’t help either.
136
 There are many implications raised not just by 
the fact, but also by the manner of the resolution of these issues by the axiom  
V = ultimate-L, if indeed it is demonstrated to be true…
▸ The conjectured axiom, V = ultimate-L proposes to complete the system of 
axioms which define the universe V; it offers a coherent conception of the 
universe of sets, presenting an unambiguous concept of the transfinite uni-
verse. Addressing the questions of independence with respect to large cardi-
nal axioms and the continuum hypothesis, it resolves the contemporary cri-
sis of formalization. The resulting conception of V is maximal and univocal, 
i.e. not in the least indeterminate or relative; it thereby affirms the universe 
(or non-pluralist) orientation, and facilitates the adoption of set theory as an 
absolute ontological referent.
▸ The skeptical assertion that V is meaningless (i.e. a fiction) is undermined 
by the articulation of a coherent and ultimately complete (but not closed) 
set theory. V = ultimate-L provides a framework for understanding and ana-
133
 Rittberg, “How Woodin Changed his Mind”.
134
 Woodin, “The Continuum Hypothesis Set Theory”.
135
 Badiou, Being and Event, p. 228.
136
 The axiom of choice has long been subjected to criticism. It is a purely existential axiom 
in that it asserts the existence of a function of choice without providing any means of con-
structing it; it is impossible to specify any rule that might guide an infinite set of arbitrary 
choices. Yet it has been proved that it does not introduce any contradiction. Moreover, the 
axiom of choice is required to establish that every set can be well-ordered.

386
norma m. hussey
lyzing large cardinal axioms as true axioms about the universe of sets, and 
no longer as independent set-theoretical statements. Large cardinal axioms 
exist in the universe given by V = ultimate-L, which allows the number theo -
rist, for example, to account for new truths generated in the universe of num-
ber theory by their existence. Such examples serve to ground a refutation of 
the skeptic.
▸ The universe given by V = ultimate-L – unlike the constructible universe (giv -
en by V = L) – recognizes all large cardinals (both known, and those yet to 
be discovered) consistent with ZFC, but refuses an incoherent “prodigality of 
being.”
137
 The impossibility, heretofore, of determining consistency in large 
cardinals is surpassed, so that an axiomatic realized in the ultimate-L model 
transcends the fear of incoherence, the tension of uncertainty. As such, it 
offers a coherent way out of the constructible universe.
▸ V = ultimate-L, affirms the axiom of choice. In fact ‘choice’ emerges as a 
crucial axiom for determining formal consistency in large cardinal axioms, 
which therefore are no longer independent from the resulting conception of 
sets. For Badiou, the axiom of choice provides the precise concept of the be -
ing (as opposed to the act) of subjective intervention; he associates it with 
a ‘principle of infinite liberty.’
138
 But in the contemporary world, freedom is 
generally associated with a principle of independence. Perhaps freedom will 
yet come to be re-defined (positively) as the expression of choice within a co -
herent yet unrestricted axiomatic, rather than (negatively) as the exemption 
(independence) from any formalization.
▸ Woodin’s axiom affirms the truth of the continuum hypothesis. Woodin origi-
nally set out to falsify the continuum hypothesis; between 1999 and 2004, he 
presented three very similar yet different arguments against the continuum 
hypothesis, in each case, exploring the universe from below.
139
 But in the 
course of his endeavour to find the true axioms for V
ɷ+2
 – in which the contin-
uum hypothesis would have an answer – he was obliged to start a program 
in inner model theory which led to the completely unexpected discovery of 
137
 Badiou, Being and Event, p. 283.
138
 Hallward, “On the Development of Transfinite Set Theory”, p. 339.
139
 Rittberg, “How Woodin Changed his Mind”.

387
a new hope for the symbolic, for the subject
the breakaway point in the hierarchy of large cardinal axioms, and the sub -
sequent isolation of an axiom that V = ultimate-L. The truth of the continu -
um hypothesis is something demonstrated by this axiom, but the resulting 
model was not the original expectation or motivating goal.
Alain Badiou’s objection to the continuum hypothesis as true (ever since he was 
working on what was to become Being and Event) stems from a consideration 
that adopting it leads to a restriction of set theory’s axiomatic powers, and is 
therefore opposed to the principle of maximality.  Yet, as he says, there were rea-
sons for hesitating, since adopting it would introduce no contradiction, and the 
mere negation of the hypothesis doesn’t tell us what type of infinity the conti-
nuum really is either.
140
 Perhaps Badiou’s objection to the truth of the conti-
nuum hypothesis stems more particularly from an objection to the restrictions 
(minimality) of the constructible universe – which is the only situation (here -
tofore) which has demonstrated that truth. But Woodin’s axiom is a maximal 
axiom for V; it accommodates all genuine large cardinal axioms, including that 
of a measurable cardinal which therefore implies the existence of Silver’s 0
#
 and 
brings the doctrine of indiscernibles (generic) into the resulting conception of V . 
The key point here is that ultimate-L is not at all L, that in fact the inner model 
approach has to be altered – in a fundamental way – to provide the solution for 
one supercompact cardinal; ultimate-L is not simply an L enlarged to accom-
modate large cardinal axioms, but (having developed from a coherent theory of 
the reals) an L re-cast such that the resulting model can accommodate the reals 
which are overlooked (indiscernible) in L.
The normalization of the state – its minimum of excess (represented by the truth 
of the continuum hypothesis)
141
 – which was primitively achieved in the con-
structible universe, is accomplished here in the context of an axiomatic which is 
capable of challenging a much greater indeterminacy, that of consistency with 
respect to large cardinals; genuine large cardinals are admitted due to their (de -
monstrable) formal consistency and false large cardinals are refuted. It seems 
then that the state proves to be unduly excessive only in the absence of a coher -
ent formal framework; beyond the narrow constructible universe (which can’t 
admit even a measurable cardinal) and before an ultimate universe (which ad -
140
 Badiou, “Towards a New Thinking of the Absolute”, pp. 17–18.
141
 Badiou, Being and Event, p. 296.

388
norma m. hussey
mits all large cardinals consistent with ZFC), the specter of independent large 
cardinal axioms coincides with an excess of the state which is immeasurable. 
Once the formal framework has the capacity to identify and accommodate all 
genuine large cardinals, the state seems to settle into normalcy and lose its (po -
tentially) exorbitant power of oppression.
▸ The axiom completely negates the ramifications of Cohen’s method, leading 
Woodin to present a vista for set theory “beyond the age of independence by 
forcing.”
142
 
Badiou’s maxim that “there are only bodies and languages, except that there 
are truths,” is based on the existence of the pure theory of the multiple as an 
absolute ontological referent. In this respect, he writes “it is that theory, and not 
the general principle of the existence of truths, that it is a matter of defending 
here.”
143
 Set theory as an absolute referent implies a system of axioms fully iden-
tified such that no important, significant, useful property of sets will remain 
undecidable; any such theory has to “banish the specter of undecidability as 
demonstrated by Cohen’s method of forcing.”
144
It is noteworthy here that Alain Badiou recently remarked that he now accepts 
that his definition of universality – as genericity (due to Cohen) – in Being and 
Event is purely negative.
145
 He added that “genericity signifies, exclusively, that 
universality is not reducible to properties in the situation,” and that maybe it 
is impossible to solve the question of universality with a negative definition of 
this kind. He said that universality does not just consist of negative genericity, 
but also of something which has a form of relationship to an affirmative concept 
of absoluteness which he proposes in The Immanence of Truths.
146
 Badiou also 
notes that ‘elementary embedding’ somehow plays a similar role to ‘forcing,’
147
 
in the sense of as a truth procedure. He says that the absolute class V is embed -
142
 Woodin, “Beyond the Age of Independence by Forcing”.
143
 Badiou, “Towards a New Thinking of the Absolute”, p. 13.
144
 Woodin, “Strong Axioms of Infinity”.
145
 Badiou, Presentation at the Graduate Workshop on Being and Event, Columbia University, 
New York, October 24, 2017, subsequently published in Sometimes We Are Eternal, p. 58. 
146
 Badiou, referring to the English translation (yet unpublished) of L ’Immanence des Vérités.
147
 As reported by Berankova, in “The Immanence of Truths: the Absolute between the Singu -
lar and the Universal”.

389
a new hope for the symbolic, for the subject
ded in one of its ‘attributes’ (Spinozist notion), i.e. in the transitive subclass M, 
which is thereby transformed into a model of V. There is a relation between V 
and M but this is not the identity, i.e. V will remain different from the attribute 
M and contain sets that we cannot find in M. Thus, to say that a ‘truth’ is abso -
lute does not mean that a ‘truth procedure’ is absolute; rather, the absolute is 
embedded in its attribute.
Ultimate-L – with strategic-extender models and the class HOD – seems to em-
brace these truth procedural conditions. The subset in HOD is definable without 
parameters, and HOD is connected to Cohen’s method of forcing (Vopenka), so 
it seems that forcings are somehow already inscribed in ultimate-L giving a wid -
ening of V. But this widening is strongly coupled with a heightening of V to the 
level of a supercompact cardinal and (thus) beyond, yielding an inner model 
which is very ‘close’ to V .
Forcing, i.e. the Cohen symptom of a sort of legal insubordination to a restricted 
(conservative) formalization – which is effective and necessary to demonstrate 
the insufficiency of ZFC – does not itself propose a solution to the crisis (of ZFC) 
which it initiated, but the indiscernibles which it indicates must be incorporat -
ed in any such solution. It only becomes redundant as an intervention when the 
formalization is sufficient. This is also the case with the ‘attributes of the abso -
lute,’ i.e. large cardinals, which are indicated by the procedure of elementary 
embedding.
Alain Badiou has not written about ultimate-L, but then he generally does not 
write about what is only conjecture in set theory – he waits for results and their 
establishment among the set theoretical community. Woodin’s axiom however 
seems to present the possibility of a new framework for infinite thought which 
is universally consistent in the sense indicated by Badiou; it completes a system 
of axioms which define a model which is very ‘close’ to V with the capacity to ac -
commodate all of the reals, together with all large, demonstrably consistent ‘at -
tributes of the absolute’ (large cardinal axioms). It seems to embrace a principle 
of maximality,
148
 and it affirms a ‘principle of infinite liberty’ (axiom of choice).
148
 This is in the sense that “any intellectual entity, whose existence can be inferred without 
contradiction from the axioms that prescribe it, exists by that very fact,” Badiou, “Towards 
a New Theory of the Absolute”, p. 11.

390
norma m. hussey
Implications for the symbolic, for the subject
The argument presented in this paper is that humanity’s symbolic organization, 
or a symbolic, requires a credible ontological referent. The breakdown of the hi-
erarchical symbolic (of tradition) accompanied the ruin of the ontological sup -
position – of the being of the one – on which it relied. The modern revelation 
of the pure multiplicity of being as such cannot be undone. A new symbolic 
(if such is possible) has to recognize the non-being of the one, which means 
that it can only be non-hierarchical, which is to say, egalitarian. An egalitarian 
symbolic therefore requires a credible ontology of pure (infinite) multiplicity as 
referent; mathematical set theory is proposed (Badiou).
In Being and Event, Badiou wrote that metaontology “serves as an unconscious 
framework for every orientation in thought.”
149
 In this respect, the commonplace 
doubt that an ultimate completion of set theory can be articulated, corresponds 
to a general conviction of impossibility – with respect to an egalitarian symbolic –  
which has dominated since the 1980’s; such a conviction equates a symbolic 
solely with hierarchy, and resigns itself to (or lauds) the reign of the a-symbolic, 
(unless it attempts a retrogression to hierarchy).
But set theory – with the axiom V = ultimate-L (conjectured) – now affords the 
real possibility of a coherent, ultimately complete, and therefore credible, ontol -
ogy of multiplicity. As such it proposes the possibility of an egalitarian symbolic. 
This is not to suggest that an abstract formalization in itself constitutes a con-
crete new symbolic, but a coherent ontology of multiplicity is an a priori require -
ment, i.e. a condition of possibility. The articulation – even as conjecture – of 
a coherent, rational thinking of infinite multiplicities challenges the dominant 
conviction of impossibility with respect to an egalitarian symbolic;
150
 it re-opens 
the question of a symbolic for an age of multiplicity. It implies that the idea of an 
egalitarian symbolic may be more than just a utopian dream. 
149
 Badiou, Being and Event, p. 282.
150
 It is of course still quite possible that ultimate-L will fail, and that an optimal axiom (or 
otherwise concluding axioms) for V will eventually take a form other than that of an inner 
model. Nevertheless, the significance of the supercompact cardinal and the universality 
theorem cannot be ignored.

391
a new hope for the symbolic, for the subject
This can modify, or even flip, a subjectivity; a wishful orientation (unfounded 
desire), or even a hopeless one, has new reason to hope (desire combined with 
expectation). A condition of possibility (as opposed to impossibility) gives a 
subjective motivation to act (rather than hopelessly react) in the concrete world. 
Ontology may indicate how to proceed: such is Badiou’s endeavor, with the 
adoption of forcing (in Being and Event) and elementary embedding (in L ’Im -
manence des Vérités) as the being of truth procedures. It is by such “arduous 
and protracted procedures” that “the new pulls itself away from the old.”
151
 But 
the advance from doubt, or beyond conviction, to reasonable expectation that 
we can have access to a ‘place’ of thinking where such truth procedures can 
be decided, gives impetus to the subject to pursue them in the objective world, 
whenever chance arises, or the ‘event’ happens.
More speculatively (and this is an impressionistic response to accounts of the 
remarkable developments in set theory by Hugh Woodin and colleagues), ontol -
ogy might even suggest that recent phenomena such as the generation of mass 
testimony (‘target witnesses’) by activists in, for example, the ‘Me Too’ or the 
‘Black Lives Matter’ movements, to challenge the inherent male/white/power 
bias in the existing legal, social and economic systems, will have effect. Or that 
the advocation of ‘choice’ by activists, for instance for same-sex marriage or 
abortion, is valid. More tantalizingly, the deep awareness of the requirement 
for social and economic justice by contemporary (genuine) climate activists,
152
 
suggests the inadvertent generation of non-targeted or ‘phantom’ witnesses. All 
of these orientations have their correspondences in (speculative) ontology in the 
present moment.
Moreover, the development of the axiom V = ultimate-L demonstrates how 
the unexpected discovery (encapsulated here in the universality theorem) can 
change everything. It harnesses and combines the efforts of the ‘universists’ in 
the three camps of set theory, i.e. the inner model program, the generic and the 
large cardinal doctrines, using the resources of all three but privileging none. 
Those aspects of mathematical truth discovered separately in each of the three 
151
 Zachary Luke Frazer in the introduction to Badiou, The Concept of Model, re.press, Mel -
bourne 2007, p. xvi.
152
 This is captured by Naomi Klein in “This Changes Everything: Capitalism vs. the Climate” 
(Simon and Schuster, New York 2014) – again, an intriguing title.

392
norma m. hussey
doctrines are addressed and deployed, which suggests that in other domains 
perhaps those of a shared conviction, but opposing methods, might engage with 
each other’s work.
The subject and reversal of exile
Before devising the mathematical ontology, Alain Badiou wrote a theory of the 
subject (1982).
153
 While he subsequently revised his thinking,
154
 some of those 
teachings have resonance here; in particular, his comparison of the subjective 
figures in Greek tragedy according to Sophocles (in Antigone), and to Aeschylus 
(in the Oresteia):
 155
In Antigone the theme is ‘reversal’ by which the way of the new is barred; it is 
not governed by any new right. The crucial point is the retrogression towards the 
origin. This is native reversal, reversal from the formal (what is learned) to the 
originary native form. It leaves no way out for the subject (Antigone) – wandering 
below the unthinkable – other than death. This subjective figure must always pre -
vail in times of decadence and disarray.
Such a subjective figure seems appropriate for our times; it captures the gener -
al disorientation which prevails under the triumphant reign of the a-symbolic, 
with no prospect for any new right (a symbolic in an age of multiplicity) but only 
the retrogression to finitude by which the way of the new – actual infinity – is 
barred. This leaves no way out for the subject other than nihilism.
In the Oresteia the theme is the ‘rupture’ that allows for the advent of the new; 
it is a new coherence instituted by the interruption of the repetitive series that 
made up the whole previous social order. The key point is the interruption of the 
power of the origin. Orestes (the subject) demands a discussion based on facts; he 
stands firm and does not give in to the murderous seduction of the Erinyes. There 
is no return to order, and the subject is prepared to sustain exile, exile-without-re -
turn. Reversal of exile will not take place without the re-composition of a different 
153
 Badiou, Théorie du Sujet, Éditions du Seuil, Paris 1982.
154
 Badiou, Logics of Worlds, especially the “Formal Theory of the Subject”, pp. 43–89.
155
 Badiou, Theory of the Subject, Bloomsbury Academic, London and New York 2013, pp. 
161–168.

393
a new hope for the symbolic, for the subject
order – which is not simple. Neither will it take place without the structural an-
chorage of the native – indeed it is from the materialistic impasse of the latter that 
the practical existence of the former proceeds. 
Here the subjective figure seems to correspond to the universe orientation (or 
faithful subject) with respect to the situation of set theory; it captures the in-
terruption (the Cohen event) of the repetitive series (V = L) that made up the 
previous natural order. There is no return to order, and the subject is prepared 
to sustain exile without return (set theory’s indeterminacy).
After more than 50 years of roaming (sustained exile) – refusing the native rever -
sal of the skeptic, and overcoming the exile-without-return of the generic (due 
to Cohen)  – set theory now affords the possibility, not of a return to order, but 
the recomposition of a different order (V = ultimate-L). It is only the insistence 
of the interruption (forcing/generic) which, in the end, allows for the advent of 
a new right capable of completely recomposing the whole logic of the decision. 
Risking the hyperbole of understatement, this is not simple. Moreover, the re -
versal of exile will not take place without the structural anchorage of the native 
(“the anchoring of the ordinals in being”
156
); ultimate-L, as a new, consistent 
universe capable of accommodating all cardinals, is still structurally anchored 
in L which is the minimum possible universe of sets containing all ordinals, but 
with L re-shaped, re-composed. Indeed it is from the materialistic impasse of 
the latter (i.e. the constructible power-set is never the set of all subsets of the 
set), that the practical existence of the former proceeds (by generalizing the suc -
cess in understanding the reals, i.e. V
ɷ+1
). Finally – faithful to the conviction that 
“even though we have to return, and it is this return that makes the subject [set 
theory] – there can arise an enlightened overcoming of what no longer entails 
any return.”
157
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 Badiou, Number and Numbers, p. 83.
157
 Badiou, Theory of the Subject, p. 168.

394
norma m. hussey
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