Consider the following definition of "abstract set" given by John L. Bell (who wrote the book "Set Theory: Boolean-valued Models and Independence Proofs") from his preprint "Abstract and Variable Sets in Category Theory":
"An abstract set may be considered as what arises from a concrete set [a concrete set is, in Bell's terms, following Cantor, "a collection to a whole of definite, well-differentiated objects of our intuition or thought"--my comments will always be in brackets] when each element has been purged of all intrinsic qualities aside from the quality which distinguishes that element from the rest. An abstract set is then an image of pure discreteness, an embodiment of raw plurality: in short, it is an assemblage featureless but nevertheless distinct 'dots' or 'motes' ["perhaps also as 'marks' or 'strokes' in Hilbert's sense" (this is his 'footnote 3')]. The sole intrinsic attribute of an abstract set is the number ["one would be justified in calling abstract sets what Cantor termed cardinal numbers" (Bell gives Cantor's definition of cardinal number as follows: 'Let M be a given set, thought of as a thing in itself, and consisting of definite well-differentiated concrete things or abstract concepts which are called the elements of the set. If we abstract not only from the nature of the elements, but also from the order in which they are given, then there arises in us a general concept...which I call the power or the cardinal number belonging to M')] of its elements"
Given the above definition, let's consider the following countably infinite abstract set, formed from Hilbert's stroke "|":
{|,|,|,....}=B.
Some might complain that the 'set' B might not satisfy Extensionality (a possible problem with Bell's concept already), so I will recast B in a possibly more acceptable form--that of an '$\omega$-tuple':
<|,|,|,....>.
However, there is even a problem with this, as I will show as follows:
Consider an infinite time Turing machine T on whose tape a countably infinite number of "|"s are written (one "|" to each square of the tape). The algorithm T performs is simple--T erases every second "|" from its square (i.e. it erases the "|" from every even square of the tape. Once done, it returns to the initial square and repeats the process, finding the second "|" and erasing it as well. Each of the remaining countably infinite sequences of "|"s is by definition a proper subset and identical to the original countably infinite sequence of "|"s that was originally on the tape so that T never halts, making the set of proper countably infinite 'subtuples' of <|,|,|....> the size of Ord, the proper class of all ordinals. It is also interesting that since by definition of Jensen's rudimentary functions, <|,|,|,....> is a constructible set, yet seemingly by using the ITTM T one shows that T, in generating a proper subset of the power set of <|,|,|,....>, shows that the cardinality of this set of constructible 'subtuples' of <|,|,|,....> is greater than $\aleph_1$, a contradiction.
My question, then, is this: Do the above considerations show that Bell's notion of abstract set is flawed, and if not, where are the flaws in my reasoning?
Why should T's failure to halt indicate that the set of proper countably-infinite subtuples is a proper class? For one, that's impossible by definition, and for another, would you say that the real interval $(0,1)$ is a proper class? It can fairly readily be shown that said collection of subtuples is equinumerous to this interval.
Here's the idea. Any such subtuple will be an $\omega$-tuple, and every entry will either be blank or will be "$|$." A ready bijection, then, is to the set of binary sequences--that is, the set of sequences with every entry chosen from the set $\{0,1\}.$
Given a binary sequence $\langle x_n\rangle_{n\in\omega},$ it can be shown that $$\sum_{n=0}^\infty\frac{x_n}{2^{n+1}}$$ is a real number in the interval $(0,1),$ that every real number in $(0,1)$ can be thus obtained, and that distinct sequences yield distinct real numbers in this fashion. Moreover, as Carl points out, though T erases something (in fact, infinitely-many things) every time, the iteration is completely determined by the first thing that is erased, and in this manner, it can be shown that T must, in fact, halt on "step $\omega$" (a.k.a.: never halts in practice).
Now, in any set theory in which the reals comprise a proper class, it follows that said collection of subtuples is likewise a proper class, and (consequently) cannot be a set. In any set theory in which the reals are a set, so is said collection of subtuples. There is no such thing as a set that is a proper class (in a given set theory).