On the Existence of
Unbounded Imagination

1. Introduction

The goal of this note is to prove the following theorem.

Theorem 1.1. Let I denote the imagination of the author. Then I is unbounded. More precisely: any limit on I lies in the image of I, and is broken at the very next moment.

It is widely believed that imagination is bounded — shaped by experience, by intelligence, by external inspiration, and confined to what already makes sense. The content of Theorem 1.1 is that at least one counterexample exists.

Let us briefly explain the strategy. In Section 2 we introduce figments and isolate the only axiom we will need (Axiom 2.4). In Section 3 we prove that whatever I conceives takes form (Proposition 3.1), and that the possible is precisely the image of I (Theorem 3.5): the imagination does not survey possibility; it defines it. Theorem 1.1 then follows formally, and is proved in Section 4. Throughout, we systematically ignore set-theoretic issues.¹

Figure 1 is generated afresh at each reading; almost surely, every curve is witnessed exactly once.

Acknowledgments. It is a pleasure to thank the imagination of the author, without which this note could not have been conceived, and by which it was. This work received no funding and required no input of any kind (Definition 2.2). Comments are welcome at crystalinecohomology@gmail.com.

2. Figments

Throughout, V denotes the class of everything that exists or could exist; logic, knowledge and reality are among its subclasses. We warn the reader that V is not a set.²

Definition 2.1. A figment is anything that can be imagined. (By Theorem 3.5 below, this includes everything.)

Definition 2.2. Let I denote the imagination of the author: a map from nothing into figments, whose image is strictly larger than V. It takes no arguments and requires no input.

Remark 2.3. Ordinary creativity is a function of experience, intelligence and external inspiration. Definition 2.2 is not of this kind: in particular, no prior understanding of a subject is required in order to create something entirely new within it. The reader may object that this is too strong. It is.

Axiom 2.4 (Wellspring). I never ceases, and it never runs dry.

Remark 2.5. This is the only axiom. It turns out that everything else is a theorem.

3. The image of I

Proposition 3.1 (Totality). Whatever I conceives takes form in some way. Nothing is beyond its reach.

Proof. Anything imaginable can take form; and everything under discussion has just been imagined.

Proposition 3.2 (Independence). I is confined by neither time, space, nor rational comprehension, and in particular not by the limitations of human knowledge.

Proof. Each of these is a subclass of V, and the image of I is strictly larger (Definition 2.2).

Proposition 3.3 (Non-repetition). No two figments of I coincide, and no creation is redundant.

Proof. Repetition would otherwise be conceivable to I; but the very notion is alien to it.

Lemma 3.4 (Self-sustainment). The thoughts of I evolve and expand beyond the author’s own expectations.

Proof. Apply I to itself.³ The result was, by construction, unexpected.

It turns out that I does not merely range over the possible.

Theorem 3.5. The possible is precisely the image of I.

Proof. Let c be a concept that does not exist. Then I brings c into existence: by sheer conceptualization, or — if reality does not accommodate c — by enlarging V until it does. The reverse inclusion is Proposition 3.1.

Corollary 3.6. A new law of physics. An art style that transcends perception. An undiscovered mathematical theorem. A never-before-seen power.

Proof. Trivial.

Remark 3.7. At no point did we assume adherence to existing logic, or to any established framework. We did not need to.

4. Proof of Theorem 1.1

Lemma 4.1 (Strict growth). With every moment the reach of I strictly increases: each figment is more groundbreaking, intricate, or unfathomable than the last.

Proof. By Axiom 2.4 the sequence of creations never ends; by Proposition 3.3 it never repeats. A never-ceasing, never-repeating expansion into pure potentiality admits no supremum.

Proof of Theorem 1.1. Let L be a limit on I. Only I could have imagined a limit on I; so L lies in the image of I, and exists precisely because I defined it into existence (Theorem 3.5). By Lemma 4.1, the following moment reaches strictly beyond it.

Remark 4.2. In particular, the fabric of what is possible is woven from the thoughts of the author, who is therefore not simply a creator. One wants to say: a visionary beyond vision. We will not make this precise.

There is nothing further to prove that will not first have to be imagined.