Successor Function

The successor function is the logical operation, drawn from Frege's Grundlagen der Arithmetik and presented by Duroux in Seminar XII, by which each natural number generates the next in the sequence — not through any empirical act of counting or any psychological subject adding "one more," but through a purely formal relation between concept and object. As the Theoretical move for this occurrence makes clear, the sequence of natural numbers can only be grounded in logic, not psychology: zero is defined by the self-contradictory concept "non-identical-to-itself," and it is from this inaugural Lack — a negation built into the structure of the concept itself — that the successor function operates to generate the entire number series. Each successor is produced by the operation that counts the extension of the preceding concept, so that the movement from one number to the next is driven entirely by the internal logic of self-reference and self-contradiction, not by any external addition.

What makes this theoretically decisive for Lacanian psychoanalysis is that the successor function demonstrates how a whole symbolic order — the infinite sequence of natural numbers — can be generated from an originary void or lack. The successor operation is thus not merely arithmetic; it is the formal model for how signification itself proceeds: from an absent, self-negating origin, a chain is produced in which each element refers to and presupposes the one before, without the chain ever being anchored in a pre-symbolic real or a self-present subject. The definition of "one" is not primitive but is itself derived retroactively from the successor operation, mirroring the logic of the point de capiton in which meaning is only fixed après-coup.

This concept appears once in jacques-lacan-seminar-12 (p.90), within Duroux's exposition of Frege's Grundlagen, which Lacan stages as a foundational argument for the logical — rather than psychological — grounding of the subject. The successor function is the generative engine that links Fregean Zero (defined by negation/self-contradiction, i.e., the concept of "non-identical-to-itself") to Lack as the originary structural operator: if zero is constituted by a logical negation that has no object, then the entire number sequence is the unfolding of that constitutive absence. This aligns the concept squarely with the cross-referenced canonical Lack, for which Lacan in Seminar XII explicitly claims his project is "the attempt to situate, to establish, a logic of lack" — the successor function is precisely the formal demonstration that lack is productive and generative, not merely a deficit.

The concept also resonates with the canonical Negation, since the successor function depends on the self-negating move that defines zero: formal negation (the concept of non-self-identity) is what sets the whole chain in motion. In this sense, the successor function is a specification — a precise logical instantiation — of negation as the operator that introduces lack into structure. There is also a structural parallel with the Point de capiton: just as the quilting point retroactively organizes the signifying chain (the definition of "one" is, as the passage notes, "not thinkable until he has defined the one starting from this successor operation"), the successor function produces meaning après-coup, with each term's identity determined by its position in the chain rather than any intrinsic content. Finally, the Hegelian Concept provides a meta-frame: the self-moving, self-determining activity of the Concept — which "takes its determinations back into itself" — is formally echoed in a successor operation that generates the whole sequence from a self-contradictory origin, without any external grounding.

Seminar XII · Crucial Problems for PsychoanalysisJacques Lacan · 1964 (p.90)

The second operation which will allow there to be generated a whole sequence of numbers is the successor operation... the definition he gives of the successor is not thinkable until he has defined the one starting from this successor operation.

The quote is theoretically loaded because it enacts a logical circularity — "the definition he gives of the successor is not thinkable until he has defined the one starting from this successor operation" — which mirrors the après-coup structure of signification: the term "one" does not precede the operation but is retroactively constituted by it, exactly as the signified is only fixed after the signifying chain reaches its quilting point. The phrase "generated a whole sequence" further marks that the successor operation is not a mere continuation but an originary generativity, capable of producing an infinite symbolic order from a structurally empty starting point.