Translator’s note
This paper, unpublished on any known mathematical outlet at the time, dates back to March 1946, and was found in a parcel in the Beth Library, in the ILLC common room, in the Autumn of 2025. Most plausibly, the author unsuccessfully attempted publication multiple times, culminating in a request to E.W. Beth himself. We publish it here, for the first time in almost eighty years, as we believe it is very well worth the attention of the open-minded logicians of the twenty-first century. Below we print what we hope to be the most faithful rendering possible of the original German. Much gratitude is owed to Max Imilian Wehmeier for his help in the translation process.
Foreword
My Dearest Evereth,
I hope this letter finds you well. I am sending you what I believe to be a legitimate, and indeed nothing short of groundbreaking, alternative foundation of Mathematics, namely Mennerl Set Theory with the Axiom of Bad Choice (MBC). You are my last hope for this work to be published: thus far, any ambition in this regard has been thwarted by the viciousness of Zermelo and his followers, an avid sect who keeps sabotaging my endeavours in order to retain cultural hegemony in mathematical foundations.
I believe your renowned wisdom will leave no room for doubt that this short notice of mine has the potential of shaking Mathematics from its foundations for the years to come. Indeed, I claim that MBC constitutes a more natural foundation for Mathematics than its more optimistic counterpart ZFC, since if Mathematics is to be an adequate modelling tool for the empirical world, it cannot but be able to account for the existence of universal bad luck, and the utmost chaos of the states of affairs human beings find themselves involved in.
I hope to hear your opinions soon,
Sincerely yours
E. Mennerl, Göttingen 1946
The Axiom of Bad Choice and Its Equivalents
In the present paper we outline a novel axiomatisation of Set Theory, Mennerl Set Theory with Bad Choice (MBC), which, as you may guess, is named after the author, who rightfully deserves recognition for such a contribution. Intuitively, the Axiom of Bad Choice states that for any collection of non-empty disjoint sets, there exists a function that picks exactly those elements that we do not want, and sends them to a thoroughly useless set we know nothing about.
We present a new suitable axiomatisation of Set Theory that includes the Axiom of Bad Choice (henceforth ABC), and prove its equivalence with the Disordering Principle. We conclude by outlining its most widely applicable corollary, namely the Murphy Law.
For this purpose, I add a “desirability predicate” $D$ to the language signature of $\cal{L_{\in}}$. This will allow the expanded language $\cal{L}_D$ to distinguish whether we care about a certain set, or better, if the existence of a certain set is in any way beneficial for our purposes. Notice how powerful and natural the addition of such predicate is: why would one not want to state whether a given set is desirable? Only one who has a political interest in preserving the depreccable Zermelian status quo. Given that in $MBC$ we hold that everything in our theory is a set (that is, we do not accept urelemente), by adopting it as foundational paradigm we will be able to state wheher or not the existence of any mathematical object is desirable. I believe that, out of respect for the reader’s intelligence, the advantages yielded by such expansion of the language need no further explanation.
Indeed, we are now equipped to formulate the Axiom of Bad Choice:
$$ \forall X(\forall y(y\in X)\rightarrow y \neq \varnothing)\rightarrow\exists f(\neg Df \land f:X \rightarrow\bigcup X \text{ s.t. }\forall y\in X(\neg Dy \land f(y)\in y)) $$The desirability predicate $D$ allows us to express the intuitive idea I hinted at at the beginning of the paper: for any collection of non-empty disjoint sets, there exists a non-desirable function from $X$ to its union $\bigcup X$, which selects a non-desirable element from any set, thus implying the existence of a non-desirable set. We call this function a disruptor and its range Jinx. For the reader whose senses are offuscated by their unwillingness to taking me seriously, I should like to remind them that no other than Kurt Gödel came up with a similar principle, which he called “universal bad luck”. Here is a quote, from Ueber die Ausmaße des Unglücks in der Mengenlehre nach Zermelo-Fraenkel und verwandter Systeme (Gödel (1931)):
“Bad luck is pervasive in all mathematising, and it would be foolish for any mathematical project not to acknowledge its existence. A day will come in which this stance, far from controversial, will be nothing short of a truism for the working mathematician”
If you do not believe me, I hope you will at least believe Gödel. Who, if not him, can speak of bad luck, after having discovered that the project of finding a way to prove all arithmetical truths is doomed by incompleteness? This is my appeal: if you accept incompleteness, you should also wholeartedly endorse bad luck. If you are honest enough to do so, its first ever formalisation and embedding into the beating heart of mathematics will come to you as a breath of fresh air. Furthermore, I will show it is equivalent to an uncontroversial principle, namely the Disordering Principle. Let us begin by rehearsing some preliminary definitions:
Definition (Chaotification). Let any $R \subseteq W \times W$ for a given domain $W$. Then, let $C(R)$ be the least subset $S$ of $W \times W$ containing $R$, such that $\neg DS$. We call this the chaotification of $R$.
Definition (Disordering of an Ordered Set). Let $(P, R)$ be any ordered set. Then, $(P, C(R))$ is the disordering of $(P, R)$
Which allow us to finally define the following:
Disordering Principle. Any well-ordered set can be disordered
Theorem.
$$ ABC \iff \text{Disordering Principle} $$
Proof. The “if” direction is trivial - indeed, I claim that not immedialtely seeing why this is the case denotes a lack of understanding such that no objections can be taken seriously. Hence, the proof of this entailment is left as an exercise to the reader. We are therefore only left to show the “$\Leftarrow$” direction. For this, it is crucial to notice that the disruptor can be seen as the chaotification of an arbitrary $f$. Therefore, consider $X$ a collection of non-empty disjoint sets. Then, by Pairing, Union, Replacement and Powerset, we know $X \times X$ is a set. Therefore, for any $f \subseteq X \times X$, we can define its chaotification $C(f)$, thus obtaining the desired disruptor function. $\square$
In $MBC$ we can easily prove the Murphy Conjecture, first proposed in 1799 by the enlightened British matemathician Lord Percival Murphy in his seminal (Murphy (1799)), which now fully deserves its upgrade to the name of Murphy Law. Murphy stated that
“for any mathematical structure with a given property, there must be one that has utterly monstrous and undesirable features. If this could be proven, mathematics will finally be able to be a universal language not only for Science, but for any endeavour of humankind.”
Now, take any set of mathematical structures, and let $\varphi$ define a subset $S$ defined by replacement on such set. Now consider $S^*$ a partitioning of $S$ such that all its elements are disjoint. By Bad Choice, we obtain a jinx set of undesirable elements, i.e., the set of Murphy’s “utterly monstrous and undesirable” structures, constituting a chaotification of the property $\varphi$.
Having proven the Murphy Law, as per Lord Murphy’s prophecy, nothing can stop Mathematics from providing a universal foundation of human endeavours. The last objection I will address is that of the Brouwerians, who will reject my axiom on grounds of its non-constructive nature. Fools! To answer their scepticism I will only highlight the ridicolous optimism of constructive mathematics: do they really believe that mathematics can be fully directioned by the human mind? They speak as if no calamity ever happened to them. To someone who is so blind to refuse to acknowledge the sovreignity of bad luck, I cannot but spitefully turn my back against. For the few enlightened individuals willing to recognise the uncontrovertibility of my results, on the other hand, wisdom and liberation awaits.
Bibliography
Gödel, Kurt. 1931. “Über Die Ausmaße Des Unglücks in Der Mengenlehre Nach Zermelo-Fraenkel Und Verwandter Systeme.” Monatshefte Für Mathematik Und Physik 39:173–98.
Murphy, Percival. 1799. “On Mishaps, Misdeeds and Misunderstandings: An Omnicomprehensive Theory of Unfortunate Events.” Transactions of The Royal Society 12:68–328.