Abstract
Sets: where the intentional genesis of the mathematical concept of set is discussed.
Que serions nous, sans le secours de ce qui n’existe pas?
Paul Valéry
A version of this chapter has been published before as da Silva 2013.
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Notes
- 1.
As I argue below, a set-constituting intentional subject must act in certain ways for sets to come into existence – this much is required by the concept of set. But the concept itself does not prejudge the constituting powers of the subject. However, as I also argue here, to consider the concept in full generality, as an a priori conceptual science requires, we must give the set-constituting subject maximal freedom consistent with the idea of a set-constituting agent. The existence of a set a (for example, and empty set, an infinite set or the choice set in general) can then be established axiomatically provided the existence of a accords better with the acts of a very liberal set-constituting subject than the non-existence of a.
- 2.
Remember, I take the concept of ego as Husserl understood it; in few words, “an intentional center of sense giving” (Moran and Cohen 2012, p. 90). One can think of the ego as an “intentional consciousness” generically and abstractly considered, which can materialize in an individual or a community of individuals, in a moment of time or throughout history. As a “center of sense giving” the ego creates its own “worlds”, inhabited by “intentional objects” or beings-for-the-ego that have the sense the ego endows them with. The “transcendental history” I plan to tell here is the chronicle of the acts, or experiences, that go into the constitution of realms of sets, mathematical and empirical, as intentional objects, and their correspondent theories.
- 3.
- 4.
See Chap 8.
- 5.
- 6.
“It is not the being of the world in its unquestioned evidence that is primary, and it does not suffice to ask simply what belongs to it objectively; on the contrary, what is primary in itself is subjectivity, which pre-gives naïvely the sense of the world, and then rationalizes it or, which is the same, objectifies it” (Crisis §14).
- 7.
See Chap. 3.
- 8.
An objectively complete domain of being, recall, is a domain where every possible state-of-affairs is determinately either a fact or not a fact, in which case the complementary state-of-affairs is a fact.
- 9.
See, for instance, Maddy 1980.
- 10.
See Shoenfield 1967, chapter 9.
- 11.
For Husserl empirical sets are objectualities of the understanding. For this reason, they belong together with states-of-affairs to the predicative level of involvement of the ego with empirical reality.
- 12.
See Husserl 2001 for the precise meaning of these terms. Although the set of eggs in the basket is a physical object occupying the same space of the eggs, and existing as long as they exist, the set cannot be identified with the eggs in the basket for there is an irreducible intentional element that goes into making the set. The eggs must be taken as a collection, which, on its turn, be seen as a single object (Maddy’s “realist” approach to empirical sets is tangled in confusion for not seeing this).
- 13.
It is not obvious that this set is empty; we could as well say it does not exist, that the constituting intention is frustrated. Many mathematicians and philosophers (including Husserl himself), at the time these questions were first being discussed, did not accept the existence of empty sets. I deal with this issue below.
- 14.
The question, then, whether there is an empty set will depend on whether, first, there is an empty collection and, second, whether this collection can be unified.
- 15.
A priori possibility of existence is, for Husserl, as far as mathematics is concerned, existence: “all mathematical propositions of existence have this modified sense […] not simply a ‘there is’ but rather: it is possible a priori that there is. […] All existential judgments of mathematics, as a priori existential judgments, are in truth judgments of existence about possibilities […]” [EJ § 96].
- 16.
In the following quotes, Hermann Weyl calls our attention to the transition from actuality to possibility, and further, to the absolute positing of the possible, beyond the possibility of actual intuition, that characterizes theoretical sciences: “[...] the transition to theoretical cognition proper: the transition from the a posteriori description of the actually given to the a priori construction of the possible” and further, the conversion of “the possible [...] into transcendental and absolute being, in its totality naturally inaccessible to our intuition” [Weyl 2009a, 69–70]. Also: “It is typical of the mathematizing sciences (in contradistinction to the descriptive ones) that they pass from the classification of the given examples [...] to the ideal, constructive generation of the possible” [Weyl 2009a, 56–7]. Phenomenological analysis, such as the one I carry out here, aims at revealing the “hypothetical” nature of the absoluteness “naively” (i.e. uncritically) attached to scientific domains.
- 17.
The mathematical theory of empirical sets, although never actually developed, bears similarities with physical geometry, which also involves abstraction and idealizations. Physical geometry does not consider the actual spatial experiences of a human being in particular, but the experiences in principle available to a human being in general, which is why physical space is endowed with properties (like unboundedness and continuity) that are not actually experienceable by real human beings. Room must be made, however, for all experiences that are possible in principle. Like physical geometry, empirical set theory can be abstracted and generalized into a purely mathematical theory. Empirical set theory never actually saw the light of day probably because scientists never saw any use for it.
- 18.
I believe, however, that allowing the empirical ego to perform infinite tasks stresses too much the notion of an empirical ego.
- 19.
Weyl 1994.
- 20.
About the notion of a priori Husserl says: “a priori […] means by reason of their validity, preceding all factuality, all determinations arising from experience. Every actuality given in experience, and judged by the thinking founded on experience, is subject, insofar as the correctness of such judgments is concerned, to the unconditional norm that it must first comply with the a priori ‘conditions of possible experience’ and the possible thinking of such experience: that is, with the conditions of its pure possibility, its representability and positability as the objectivity of a uniform identical sense.” [EJ § 90]
- 21.
The empirical infinite is the limit of a series of finites.
- 22.
As I have already stressed, Frege did not understand this abstractive act.
- 23.
Contentual mathematical theories are in this sense also formal, given that its objects are forms. Since set theory is in this sense formal, Husserl places it within formal ontology, a domain of formal logic.
- 24.
Well-ordered sequences of arbitrary length are formal generalizations of sequences of finite length that enumerate the acts of the temporal ego. These more general well-orderings enumerate the acts of an ideal set-constituting agent that can always perform an ever new act after no matter how many it has already performed, even if “how many” means transfinitely many, no matter which transfinite power.
- 25.
Potter 2004, p. 38.
- 26.
V0 can even be empty, in which case the first object appears only at level V1 and is the empty set (supposing it exists).
- 27.
Ordinals are abstract aspects of order-equivalent well-ordered sets made into objects. This is a phenomenologically complex process in which particular abstract aspects of members of a family of order-isomorphic well-ordered sets is by ideation made into a universal that is instantiated in any member of the family. In set theory, a set is chosen to represent this entity. Since the sequence of all the levels of the hierarchy of sets that precede a given level is a well-ordered set, for the set-constituting agent can, I will suppose, keep track of acts it has already performed and consider them collectively, each level of the hierarchy can be labeled by the ordinal representing the sequence preceding it.
- 28.
Therefore, the notion of existence at work here is neither the realist (not even the modal realist) nor the constructivist. But it is not the nominalist denial of existence either.
- 29.
A sort of converse seems also to be true, as indicated above; any sequence of acts the ego has already performed can be gathered in a well-ordered set, for the ego can keep track of its acts and take them collectively.
- 30.
It is part of “traditional wisdom” that only set theoretical realism can give us the universe of sets in its full splendor, for, so the view goes, once an ego comes into the picture, no matter in what shape or form, the universe of sets must to some extent be trimmed (for example, Gödel’s partial universe L). I do not see how realism can have this power independently of some suspicious metaphysical principle of plenitude of reality, like Potter’s “if a set can exist, it does exist” or Maddy’s “maximize”. The principle of plenitude I subscribe to here, however, is an empty tautology: if a set can exist, then it does exist, for “to exist” only means “can in principle exist”.
- 31.
See Husserl 1970.
- 32.
Although the denomination may mislead one into thinking that an experience of frustration is a psychological experience, this is not how Husserl understands it. As an intentional experience (for example, of not-A), frustration discloses an objective impossibility, namely, that a certain intention (A) cannot be fulfilled intuitively, that any attempt at bringing something (that-A) to consciousness faces unsurmountable objective restrictions.
- 33.
Are these assertions devoid of meaning or just plainly false? Considering only that besides having elements sets can also be elements, sets can appear at both sides of ∈, and assertions of the type x∈x are apparently meaningful. However, considering that set-constituting acts presuppose the (joint) availability of the elements of the set, assertions of this type obviously lack sense: a set is not available if it is not yet constituted as a collectable object.
- 34.
Of course, in formal set theories, in which the relation of “belonging” is only formally equivalent to belonging proper, and “sets” can be anything whatsoever, “sets” may very well be conceived as “belonging” to themselves.
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da Silva, J.J. (2017). Sets. In: Mathematics and Its Applications. Synthese Library, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-319-63073-1_5
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