Is the musical group that recorded the album ‘Revolver’ identical with the set {Harrison, Lennon, McCartney, Starr}?
If so, then how can an abstract object (in this case, a set) perform a physical act (record ‘Revolver’)?
If not, then how can we tell apart the band and the set?
Either you conclude that the act of recording the album was performed by the members of the set, not the set itself, or you conclude that sets are not entirely abstract. Personally, I would take the latter interpretation, but math does not deal in the question of what is and is not “abstract”.
As an aside, there’s also a mathematical object called a “group”. A group is a set together with an operation on that set (such as the integers and addition, or rotations of an object and composition of those rotations) that meets certain conditions: The operation must be closed (operating on any two members of the set gives you another member of the set), it must be associative (for instance, (a+b)+c = a+(b+c) ), it must have an "identity element that, combined with any other element, gives you that other element, and every element must have an inverse, an element that when combined with it gives the identity. Though of course, the Beatles are not a “group” in this mathematical sense.
Er, this is a joke, right? The appearance of the terms ‘set’ and ‘group’ seems to be a bit of a giveaway… 
Since my count 22 other musicians provided music or background vocals on Revolver, the answer is hell, no, not even close, because 26 does not match 4.
It seems to me that the concept of a ‘set’ is pretty abstract. It developed in the late 19th and early 20th century as a way to try to formalise mathematics as a branch of logic.
Seemed quite intuitive for a while, until Russel’s paradox appeared. There’s a famous case where Gottlob Frege was about to publish a definitive book on set theory, when Russel’s letter appeared.
Frege’s response is classic: “Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.”
It’s considered a great example of scientific honesty!
There are things in mathematics called sets, and groups, but I am not sure what is your philosophical point? It is true, and not particularly mind-bending, that “physical” entities such as people are fuzzy— you say that is I in the room playing that instrument? What about my clothes? Gut bacteria? What am I? None of that is a mathematical problem, merely philosophical.
Better not to publish your set theory that is obviously inconsistent than to publish it and have to retract it afterwards. Hardly anything could have been more fortunate! (Unfortunate to have fucked up in the first place, but them is the breaks.)
Probably still worth publishing the book, so that, once set theory was put on a rigorous footing that excludes Russell’s paradox, subsequent mathematicians could go through and figure out how much of the work is salvageable (probably most of it).
My favorite legend about Russell and sets of people goes something like, he was at a dinner where someone challenged him to prove that if 2 = 1, then he is the Pope (ex falso sequitur quodlibet). Without hesitating, Russell replied, “The Pope and I are two people. If 1 = 2, then we are one. Therefore, I am the Pope.”
(I didn’t realize ‘group’ had a specific meaning in math (though I shouldn’t be surprised, given that ‘almost all’ does) otherwise I wouldn’t’ve used it.)
Unlike you, on first pass I’m inclined toward the first of the two options you give. There probably is value in seeing Harrison, Lennon, McCartney, and Starr as The Beatles and {Harrison, Lennon, McCartney, Starr} as another thing entirely.
There are many different theories of sets, but in let’s say ZFC, the only terms are sets. The elements of any set are also sets. So maybe from that point of view it starts to become nonsensical to consider whether the Beatles form a set.
I thought sets were what groups played 
And in the Beatles, as a group: which one is the identity element, and which the inverse?
Assuming Paul is not dead, of course…
Funny that Paul who was supposed to be dead has a good chance of being the last survivor.
A dog - Canis familiaris - is an animal, a physical thing. A contemptible person - home sapiens - can be referred to as a dog. A person’s feet can be called dogs. A bad movie can be a dog. Dog down the hatch means fasten it. Closely following or stalking another is dogging, and non-physical rumors can dog a physical person. A physical person can be characterized as a hot dog, who could be eating a hot dog without any accusation of cannibalism.
OTOH, axes, class, digit, factor, flip, graph, origin, power, and prime are all common terms that have specific or special definitions in math.
Words are extremely malleable. They have denotations and connotations. They can describe abstractions and physical items. They can behave as nouns, verbs, adjectives, and adverbs. “Set” has over 400 dictionary definitions. You shouldn’t let them slop over one another. That causes mental confusion like asking whether a mathematical set can perform a physical act. Words can do anything for love but not that.
I recall there was for a while in the late 70s and early 80s, a fad called “the new math”.
The idea that we should replace teaching standard arithmetic etc with “Understanding the basics”.
So kids were supposed to “understand” things in fundamental terms, not just ‘rote learning’. In practice, since most math teachers didn’t really understand the idea itself, this mostly seemed to consist of introducing the phrase “the set of” meaninglessly into every sentence. And pointless drill in using bases other than 10, which is something only computer engineers really need.
Tom Lehrer parodied it well.
And even after “new math” faded away, the “look for the four in the tens place” method of subtraction lived on, to the point where anyone under 40 (50?) likely has no idea what Lehrer is on about when he describes the “three from two is nine, carry the one” method.
Also known, to anyone under age 55 or so, as “math”. Most of the people these days who complain about “new math” and “why can’t we just do it the way we did it when I was in school” did do it that way when they were in school. They just don’t remember any of it, and are proud of not remembering it.
More like the early 1960s to the mid 1970s - the song making fun of “New Math” came out in 1965. In the late 1970s, the processes of what had been “New Math” were the standard, and people had just learned to call anything they didn’t like “The New Math” (like people talk today about “millennials” when they mean people in their 20s (because millenials now are in their mid-40s))