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Alexandre Freiburghaus Mazuir
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Queen-troduction

Honestly, what would you say is more obscure ? Set theory or queer theory ? In an attempt to turn two minuses into a plus (because same-sign operations are perfectly well-defined), I will attempt to use both Russel’s paradox and its answers, as well as modern understandings of gender and sexuality, to explain one another.

Defining sets-uality

But what could be a common denominator between sexuality and sets ? Well, but a crucial one : the terminology is emergent, and emergent only. That is to say that the way both sets and sexuality get constructed is bottom-up instead of top-down. One of the most common critiques made of queer lexicon is that it would be simply “creating more boxes”, and would therefore not truly respond to anyone’s need of individuality. Each letter added to the LGBTQ acronym would be one more “option” available to take, with the looming threat of needing to add a few billions of them to the list, or at least as many as there are people, if we truly wanted everyone to feel personally included. This view, however, comes from a mistaken understanding of what the terminology (and a fortiori sexuality itself) is, one where the label is what matters most, despite this going against typical human experience1. Indeed, it is precisely the paradigm of a top-down approach to sexuality. To better see why, let us first take a detour to set theory.

Russel’s, more like Ru-slays

Russel’s paradox is one of if not the most foundational result of set theory, and, as it turns out, it shows exactly that sets too cannot follow from a top-down approach, where the name dictates what the set is made of. Suppose ad absurdum that sets originated from their name, or prescriptive qualifiers, then consider the set predicated as “the set of all sets that do not contain themselves”. This is, for Russel, a paradoxical entity. Indeed :

  • If the set contained itself, then that would mean it should not be part of the set of all sets that do not contain themselves, and therefore should not contain itself.
  • If the set did not contain itself, then that would mean it should be part of the set of all sets that contain themselves, and therefore should contain itself.

Both cases therefore end up in contradiction, rendering the nature of the object uncertain, and often even undesirable. One of the standard changes made to set theory to avoid such paradoxes is that sets should instead be viewed as a collection of preexisting things, and that their names merely describe what they contain, without any imposed restrictions on the content itself2. This is often referred to as the “arbitrary” school of thought, one that severs the “necessary” ties between a set and the concept/name therein associated. To sum it all up, the descriptive name of a set follows from the nature of its elements, rather than the elements being chosen depending on what the description of the set is : this is a bottom-up approach, rather than a top-down one.

“Bottom” and “Top” are technical terms here

And said approach can also be applied to sexuality. Let us view one’s sexual orientation as the descriptive name of a set of attractions (that is, some element of the powerset of all people, namely those the person is attracted to3). Then, the name itself given to the sexuality would only describe what can be found in it, not imposing anything, and with no necessity for it to be a unique identifier (multiple sets could be described in a similar way - a same sexuality label - without the need to contain the same elements, as it is merely a meta-description, and not a relevant part of the set’s identity).

Furthermore, and on that note, notice also that this approach allows us to explain how two people with the same sexual orientation may not have the same type. Their sets can be described using the same name, with no need for the sets themselves to be identical. If sexuality instead prescribed who you found attractive, in a top-down approach, then said distinction could not be made, as two people with the same label should necessarily like the same people4. This issue could obviously be circumvented (keeping a top-down view) by fine-tuning the name of the sets, similarly to indexing them as “the heterosexuality of Alice” and “the heterosexuality of Berth”, thus making them slightly different. However, ironically, this would be much closer to the problem of an abundance of “boxes” that we mentioned in earlier, the one we precisely wanted to avoid, as the labels themselves would matter (rather than simply being used for discussion purposes), and for we would then probably need one for every type (so, arguably, one for every person).

This model also perfectly accounts for the essential fluidity of sexuality. It may seem absurd, keeping a top-down approach, to say that someone’s sexuality could change, as that would require some massive ontological shift, completely moving from one property to the next, but looking at it from a bottom-up perspective, and using some dynamic definition of sets, that change all of a sudden seems much more mundane : people regularly find new people attractive, and lose interest in some they might have previously had a crush on, adding or removing elements of the set. This simple element-driven evolution of their attraction set would then perfectly explain the need for the set itself, their sexuality, to be renamed once enough of its contents has changed.

In short, the bottom-up approach allows us both to give its proper importance to an individual’s own type and identity, as well as to avoid paradoxical sets. But could we find an equivalent to Russel’s worries in gender ?

Paradoxy would make for a great drag queen name

If everything we have said until now is true of sexuality, there are great parallels to be made between the domains of sets and gender too. Rather than a collection of attractions, the “gender-set” can instead be viewed as a collection of socially gender-valenced interests and/or experiences5, only then being given a label. As such, one can wonder whether this arbitrary-influenced vision of gender can be, much like in set theory, motivated through some paradox. Let us consider a few candidates :

The agender agenda

On a personal note, I, the author, am non-binary, and more precisely agender. This is simply the label most fit to my collection of experiences, and thus it does not make any claim about my relation to gender by itself. However, agenderism can be hard to wrap one’s head around, to the point that I often feel weird mentioning it to people with no proper explanation. And this struggle people might feel they often perceive as some sort of paradox. What makes agenderism seemingly paradoxical is the (understandable) misunderstanding that agenderism is the total absence of gender. Consider the following two interpretations :

  • If I identify as agender, then I have a gender (“agender”), so I cannot identify as agender.
  • If I identify as agender, then my collection of gender-experiences is empty, so there is nothing for me to identify as.

In reality, (2) is not truly paradoxical, as it relies on the ambiguity between an empty set and there being no set at all. Yes, agenderism on its own is weird, the same way talking about the empty set as a set is weird (if sets are collections of things, how can a “collection” of one element be a set, let alone one that contains nothing ?) but the empty set nevertheless exists and could then be considered a gender-set, which therefore could be labeled. However, there is another mistake at play here, in both (1) and (2), as mentioned before : that of assuming agenderism is the absence of gender experience. In reality, though it vastly depends on individuals6, agenderism most often correlates to a rejection of the gender divide. But, crucially, there still are some experiences that cause agender people to label themselves that way (said rejection, for example), implying the existence of a (non-empty) gender-set. As such, though it might appear at first paradoxical7, agenderism is actually not problematic to the top-down framework. We might then want to find some other example.

Disaffirming gender care

How about we defined a gender more skin fit to Russel’s paradox ? Introducing : ungenderism. Let us define ungenderism as “the gender identity for those not identifying with ungenderism”. Now, obviously, this is quite similar to “the set of all sets that do not contain themselves”, and that was the whole point. Observe that, as expected, we get a seemingly 1-to-1 mapping between this gender and Russel’s paradox as defined earlier :

  • If I identify as ungender, then I have the experience of not identifying as ungender, so I do not identify as ungender.
  • If I do not identify as ungender, then I cannot not8 have the experience of identifying as ungender, so I do identify as ungender.

With ungenderism, it appears we do run into the same exact problems as with Russel’s paradox : there is no stable state, neither being ungender nor not being ungender. The possibility for us to define ungenderism in the top-down view is a sufficient reason to claim the top-down framework is unfit to understanding gender9, the same way that Russel’s argument was enough to reject (naïve) top-down views of set-theory. We have therefore established, and solidified, a link between the two fields.

It’s giving c***-clusion

To conclude, you will rarely ever hear about set-theory in political debates. And that is a shame. On the other hand, gender identity is a subject many people talk about without having given it much thought, for to most it is somewhat alien yet present in the zeitgeist. However, no matter how alien it may feel, this is the reality many people are living in. A reality based on personal experience rather than the words we use to explain it. A reality also too often suppressed by people questionning themselves, spending more time thinking whether they are “doing it right ?”, fitting in the right box, rather than “doing what feels right ?” and making their own. Every experience is valid. Every label you may give it is, if you deemed it so, the right one. Every class is not a set. It is from collection and community that labels and acceptance both eventually emerge.

Be gay, do set theory.

Footnotes

  1. In many ways, this is also similar to the prescriptive/descriptive divide in natural language, where a prescriptive view (here top-down) is enforced by institutions, despite the descriptive (bottom-up) model being used by close to every linguist. Words find their meaning in discussions, not dictionaries. ↩

  2. Note that, in ZFC, it is still possible to arrange sets based on a certain property using the “specification” axiom schema, which could then be seen as top-down. Instead, we argue that the needs for the description to specifically be a formula in Lϵ\mathcal{L}_\epsilon and for the constructed set itself to be some subset of an already existing set both show weakness of that interpretation. We will thus ignore this view for the rest of the essay. Nevertheless, this shows that labels still are a contentious discussion, even in set theory. ↩

  3. We simplify a lot here, for instance ignoring degrees of attraction between people of someone’s type. Attraction usually fluctuates between individuals, which would require some other framework to properly model. ↩

  4. And, also problematically, they should logically be attracted to everyone matching that labels’ prerequisite, which is obviously not what happens for most people. ↩

  5. In all fairness, it would be completely reasonable to also include life experiences in the construction of sexuality, as a way to also account for queer identity in general, but in the interest of time we will skip over that part, and leave that problem for other gays to solve. ↩

  6. Otherwise we would be back in a top-down view of gender. ↩

  7. One might also notice that the “paradox” here is merely a rejection of agender identity. The strength of Russel’s paradox crucially is that it goes both ways (pun intended). Therefore, agenderism is also not a true paradox for there still is one consistent direction (assuming you are not agender does not imply you are). ↩

  8. Yes, this paradox relies on double negation, but what can I say, we already mentioned same sign operations were fine, so might as well say same polarity ones are too. ↩

  9. Remember that the paradox breaks in the bottom-up approach, as the “then” statements do not necessarily follow. ↩