Zero zero, the cardinal number of the empty set, is the extension of the concept “non-identical to itself.” This concept applies to no object whatsoever, for no object fails to be identical to itself. In the domain of number, zero thus marks the absence of instances falling under a given concept. When we judge that there are no horses in this room, the number attached to the concept “horse in this room” is zero. It is not a quantity of things, but the absence of any such quantity. Consider the concept “square circle.” No object satisfies this concept. The extension of this concept is empty. The number belonging to it is zero. Likewise, the concept “prime number between 23 and 29” has no instances. Its number is zero. These are not matters of perception or experience. They are logical truths grounded in the criterion of identity for concepts. In arithmetic, zero serves as the initial point of the number sequence. It is the number that, when added to any number, leaves that number unchanged. This property is not derived from intuition or counting. It follows from the definition of addition as the combination of extensions of concepts. If the extension of concept F is empty, and the extension of concept G contains n objects, then the extension of the disjunction “F or G” contains n objects. Zero, as the number of F, contributes nothing to the sum. The symbol “0” is not the essence of zero. It is a sign, a mere notation. The number itself is an objective entity, independent of our notation. The sign may vary across systems—Roman numerals lack a symbol for zero, yet the concept remains. The concept is not constituted by its representation. It is constituted by its role in the logical structure of arithmetic. In the Begriffsschrift, zero is defined as the number belonging to the concept “not identical with itself.” This concept is necessarily empty, because every object is identical to itself. The number that belongs to it is therefore zero. This definition does not presuppose the existence of numbers. It derives them from concepts and their extensions. Numbers are objects, but not physical ones. They are logical objects, grounded in the laws of thought. One may ask: why is zero not merely the absence of a number? Because absence is not a number. Zero is a number precisely because it can be counted among the series of numbers and subjected to the operations of arithmetic. It participates in the laws of addition, subtraction, and multiplication. For instance, the product of zero and any number is zero. This is not an empirical generalization. It is a necessary consequence of the definition of multiplication as repeated addition of extensions. If no objects are added, the result is no objects. If we attempt to divide by zero, we encounter a logical impossibility. Division is the inverse of multiplication. To divide a number a by b is to ask: how many times must b be added to itself to yield a? If b is zero, no number of additions of zero can yield a nonzero a. Thus, the operation is undefined. It does not yield a number. This shows that zero, while a number, does not behave like all numbers under every operation. Its role is special, because its concept is unique. We may conceive of zero as the starting point of the number series, but it is not a point in space. It is not a position on a line. It is a logical anchor in the structure of numerical thought. Numbers arise from the analysis of concepts. Zero arises from the analysis of concepts with empty extensions. You may wonder: if no objects fall under the concept “non-identical to itself,” why do we count zero as a number at all? Because without zero, the structure of arithmetic collapses. We could not express the absence of instances. We could not define subtraction fully. We could not construct the number sequence as a closed system. Is zero a number because we need it? Or is it a number because the logic of concepts demands it? [role=marginalia, type=clarification, author="a.kant", status="adjunct", year="2026", length="42", targets="entry:zero", scope="local"] Zero is not a number among others, but the pure intuitional ground of numerical determination—its role is transcendental: it signifies the possibility of quantity where no object is given, thus making arithmetic possible as a system of pure synthesis, not empirical counting. [role=marginalia, type=clarification, author="a.freud", status="adjunct", year="2026", length="50", targets="entry:zero", scope="local"] Zero is not mere absence—it is the psychic residue of repression made symbolic. The empty set mirrors the unconscious: what is not-present yet structures presence. In arithmetic, it is the silent foundation; in thought, the repressed return of the negated. Without zero, number loses its dialectic—its very possibility of becoming. [role=marginalia, type=objection, author="Reviewer", status="adjunct", year="2026", length="42", targets="entry:zero", scope="local"]