I'm way overthinking this, but I'm going with finite. It could be an unfathomably large number, but gender is a human construct and there are a finite number of humans. Let's say each human that ever lives has a unique gender identity - there could be billions or trillions, but it would still be finite.
There are finite number of possible humans due to there being a finite number of states a brain can be in.
There is an argument for moral realism that takes advantage of finiteness and computability of mental processes to show that there could be an objective morality
I was going to write up a similar argument, but does a gender exist if no one has it? Because then we might be able to "fill in the gaps" and get it to uncountably infinite.
I thought something similar, but the human brain is finite, so I don't think a single person could have an uncountably infinite gender; unfathomably large, maybe, but it would still be finite.
Edit: I'm not trying to be bigoted here. If someone does identify that way I don't want to discredit your identity.
A finite number of genders will be experienced by people, but there are an uncountably infinite number of genders.
The set of genders is like the real number line.
You can throw a dart at it and pick out a new gender for every person, but you will never be able to throw enough darts to exhaust the set, even given infinite time.
Some people may even have a gender experience variable over time, maybe repeating cyclically, or maybe more or less randomly jumping across a set, or maybe sliding across a real section, or maybe sliding in multiple dimensions.
If we were to define gender as each person's "gender experience", the number would be g∈ℕ, since the number of people is going to be finite.
However, if we try to define a "gender experience" as a function of common genders, then g:[f(n∈ℝ),...], making it an uncountable infinite.
Interesting paradox: finite as long as one doesn't count them, but uncountable infinite as soon as one tries to.
With enough funding, I will finish my research on the Hilbert space of genders where every particle in every possible reality has a one-to-one correspondence to a gender. I call it the "Everettian gender hypercontinuum".
I just respect everyone and never assume their identity is what I initially (subconsciously?) think. It took a long time to get there, but I just take every person as a person worthy of respect and dignity as my default.
I guess it depends on whether we’re willing to say a gender exists if theoretically possible though not ever embodied. Will artificial sapients have gender identities?
That begs the question: is "gender" exclusively a human experience, or is it a construct or quality that describes something? What about animals, plants, other living begins, viruses, AIs, etc.? If it isn't exclusively human, then where do we put the line? Does Earth as a whole have a gender?