Undecidability

Undecidability, as Lacan deploys it across Seminars XV and XXIV, is not a merely technical result borrowed from mathematical logic but a structural consequence of the subject's constitutive relation to the not-all and to objet petit a. The "Gödel-style" undecidability Lacan invokes names the formal property of any sufficiently rich logical system that contains propositions which can be neither proved nor disproved from within the system's own axioms. Lacan's move is to argue that this mathematical limit is not an accidental feature of formal systems but indexes something essential about the subject's relation to the universal quantifier ("all"): because the logical function of the "all" is grounded not in an achieved totality but in the cause effected by objet petit a — the remainder that no symbolic system can re-absorb — any field of knowledge structured around the subject will always be capable of generating the undecidable. Undecidability is thus the logical signature of the not-all: the same structural impossibility that prevents the feminine position from being totalized (no exception closes the series) also prevents any subject-indexed proposition-field from being complete.

In Seminar XXIV, Lacan sharpens this by triangulating undecidability with the Real and the sinthome. The Real is defined as what does not cease not to be written — pure impossibility — and it is precisely at this register of the impossible that undecidability lives. If S1 never fully represents the subject for S2, then there will always be a proposition — a truth about the subject — that outstrips the signifying chain available to represent it. Psychoanalysis, Lacan argues, produces only a "semblance" of truth, not truth itself, because the very apparatus of its address (the signifier, the Other, transference) is structurally incomplete. Undecidability is thus not a gap to be closed by further analysis but the permanent, constitutive remainder left by the subject's entry into language.

In jacques-lacan-seminar-15 and jacques-lacan-seminar-15-1, undecidability appears at the intersection of three canonical concepts: objet petit a, the not-all, and the splitting of the subject ($). The argument is that the logical possibility of always generating undecidable propositions in any sufficiently structured field is not an external mathematical curiosity imported into psychoanalysis, but is internally generated by the structure of the not-all — the same logical operator that prevents the symbolic order from achieving totality also prevents any discourse indexed to the desiring subject from being complete. Objet petit a is the engine of this non-closure: as the irreducible remainder that the Other cannot re-absorb (see the canonical definition above), it is what ensures the "all" can never fully ground itself. Undecidability is thus positioned as an extension and specification of the not-all, giving it a formal (logical) face, and a re-application of objet petit a's structural role as cause-of-desire to the domain of propositional logic.

In jacques-lacan-seminar-24, the concept migrates toward the register of the Real and the sinthome. Here undecidability is explicitly linked to the Gödel result and to the impossibility definitive of the Real ("what does not cease not to be written"). This positions undecidability as a bridge concept between the formulas of sexuation (not-all, universality) from the earlier seminars and the late Borromean topology: if the big Other is constitutively incomplete — formalised through Gödel's theorems as Lacan notes in his account of the Other's structural castration — then the undecidable proposition is precisely what occupies the hole in the Other, the place of S(Ⱥ). The concept thus sits at the convergence of the incompleteness of the symbolic order, the not-all of feminine sexuation, and the Real as impossibility, functioning as a logical name for what the sinthome must knot together without resolving.

Seminar XV · The Psychoanalytic Act (alt. translation)Jacques Lacan · 1967 (p.155)

once we are in this field, we will always be capable of making the undecidable emerge in it... this privilege of the function of quantification depends on what is involved in the essence of the all and its relation to the presence of the little o-object.

The quote is theoretically loaded because it performs Lacan's key displacement: the "privilege of the function of quantification" — the logical capacity to universalize — is shown to depend not on formal completeness but on "the essence of the all and its relation to the presence of the little o-object," thereby grounding undecidability not in the limits of axiom systems but in the structural non-closure effected by objet petit a as the cause that prevents any "all" from closing.