However, the situation changes if there are uncountably infinitely many people...

@Qiaochy Yuan although one wonders how there happened to be so many people :)

In the same spirit, there is no probability distribution on $\{1,2,\ldots\}$ that assigns $\frac{1}{n}$ to the set of multiples of $n$ (one could argue that such a probability would be fair because $1/n$ of the integers are multiple of $n$). Its non-existence is a corollary of Borell-Cantelli's lemma using that $\{p \Bbb N\}_p$ would be independent events, where $p$ runs through the set of prime numbers.

@Malady The "Aleph-null child per family" policy.

Can't the "no probability distribution" problem be addressed by converting this problem to a limit? Isn't that how we usually deal with infinities in math?

@Barmar Not sure what you ask. But considering the uniform probability distribution on $\{1,\ldots,n\}$ and letting $n \to \infty$ does not yield a probability distribution on $\Bbb N$.

@Didier Doesn't it yield that the probability approaches 0 as n approaches infinity?

@Barmar As explained in this precise answer, there is no probability on $\Bbb N$ such that $\Bbb P(k) = \lim_{n \to \infty} 1/n = 0$ for all $k$. The allegedly "probability limit" obtained won't be a probability measure.

@QiaochuYuan , you mention So, there's no such thing as choosing a random element of a countably infinite set uniformly at random which is surely true. But isn't it further the case that So, there's no such thing as choosing an element of a countably infinite set. How could you? I guess, you could choose from a partial set of the set but that's just saying you can't do it. I don't see how you can choose 1 item from a countably infinite set. Of course, I may be wrong.

@Fattie If you give me your favourite countably infinite set. I'm sure I can pick one of its elements.

@Barmar well, but this is what I'm saying (note my answer). Say we're mathematicians discussing the philosophical construct "infinite sets". We're then discussing the idea "huh, here's a thought, from these philosophical constructs, what could it mean to 'choose' one item, let's discuss some ideas on that and pick the idea we like, and we'll call those B-F infinite sets and then we can work with those rules." Similarly if you're going to say "given that choosing an item is meaningless, but we're making up a definition, let's also define the probability of choosing one, hey, i suggest we use

.. the limit concept, which is kind of congruent to other similar uses with abstract concepts, so lets do that" and I say "huh that's a nice idea, let's define that probability in B-F infinite sets just as you said, let's write a book about that and deduce things about B-F infinite sets". That's all FANTASTIC but .... as the OP asks "If the question is indeed valid...". There is a two word answer to that "It's not.".

@didier, I'm sorry I'm not sure if you're "agreeing with me humorously" or "pointing out that I'm wrong humorously". Take any countably infinite set; take me what set you took, now show me how you chose one randomly - you cannot. Next, show me how you chose any one item (ie not randomly, you can choose). The only way to do so is to take a finite subset, and choose one from that. Which is simply equivalent to saying "oh obviously you cannot choose one from an infinite set".

I imagine with some very special countably infinite sets you can choose a subset (for example, possibly "ordered" sets speaking loosely) but that evidently has nothing to do with what the OP is saying. (And again it's totally impossible to randomly choose one from such an infinite set; this is just a side discussion about being about to choose one at all.)

@Fattie I'm not agreeing nor disagreeing. It seems that you are more concerned about the philosophical debate about the meaning or essence of mathematics, and that we are actually not talking about the same things. I, (and a lot of people around the world, and in particular on this website) consider mathematics as a language, in which it is absolutely possible to pick and element in an infinite set, the latter being a totally valid, well established notion in this language. I can't validate nor invalidate your point: we do not talk about the same things.

@Fattie As for the random part: I was not talking about randomly picking an element. I was just talking about picking an element. For instance, $\Bbb Q[X]$ is infinite countable, and I'm sure you can pick one of its elements, even though not randomly. But I do not want to enter a philosophical debate: it appears to me that the philosophical / ontological nature of the OP was an excuse to ask the following question: is there a uniform probability distribution on the set of integers?

Re, "the situation changes if there are uncountably infinitely many people." How can that make sense? There are uncountably many real numbers, and that's intimately related to the fact that there is no "successor" to any real number, there are no gaps between them. Real numbers are a continuum. What would it mean for "people" to be a continuum? Last time I tried counting people, they seemed to be very countable.

@Didier, Did you mean to say that you can't pick an element randomly, or did you mean to say that you can't pick an element randomly with uniform probability over the whole set? Picking an element randomly is easy: First, employ some simple, well-defined procedure to choose two of the elements. Then, toss a coin.

@SolomonSlow I just did not talk about random at all

@Didier, My mistake, I thought that when you said, "...even though not randomly," that it somehow had something to do with randomness.

@Didier exactly, what you say here: I (and [others]) consider mathematics as a language, in which it is absolutely possible to pick and element in an infinite set, the latter being a totally valid, well established notion in this language. (1) Precisely what I said at length to Barmar. (2) Exactly as you say the ascii tokens "infinite set" have (certain) "well established notions" in the certain set of humans, yourself and others. (3) OPs actual question is "Is [blah] an invalid premise to begin with?, answer "Yes, it's utterly invalid and totally, completely meaningless", with addend

-um being "In conventional established mathematics, those words have useful and well established meanings and you may be interested in some results from that, but, reminder that the answer to the question in your headline is unequivocally 'Yes, it's utterly invalid and totally, completely meaningless'".

@Fattie Then we just do not agree, and none of us will convince the other. At least, I won't try to convince you. I'm fine with this. \\ To Solomon : I used the word randomly in a late comment, replying to a previous question. Looking closely to my comments, you'll see that I never used it before I was asked to do so. \\ This discussion is becoming completely off-topic, and is on the edge of being a spam conversation, so I'll stop it here. Have a nice day/evening depending on your timezone

@Didier as you said re the question ".. was an excuse to ask the following question.." yes, you're totally right, it could be OP was asking a "mathematical" question, using one of the usual (totally idiotic) analogies like "in a hotel with an infinite number of rooms .. etc". You're right, that may be the case regarding the question.

@Didier lol yes that's enough of a comment chain, cheers :) it is 101°F here ATM so, such is life.