Why has Russell's definition of numbers using equivalence classes been finally abandoned? ( If it has actually been abandoned).

Solution 1:

The question might be a better fit for HSM.se but, until it's there, my answer won't focus on historical details so much as mathematical motives.

(1) numbers were higher-order properties, not of things , but of sets

Numbers are lots of things. Is the example above worth taking as a definition, axiom or theorem? You can try each approach, but we try to leave as much complicated machinery as possible to the later theorem-proving stage.

(2) Numbers are defined as equivalence classes

Which, after $0$, are "proper classes". I won't be terribly specific about that, because the details vary by your choice of set theory. But since we can't have a set of all sets that aren't elements of themselves, we have to say some collections of sets you can imagine aren't sets, and we typically say, ironically enough given the original motive for set theory, that sets are distinguished from proper classes in that they can be elements of classes.

Eventually, we want to define integers as equivalence classes of ordered pairs of integers with the same difference between coordinates, e.g. $-3$ is the set of $(n+3,\,n)$ for non-negative integers $n$. But $(a,\,b):=\{\{a\},\,\{a,\,b\}\}$ requires $a,\,b$ to be elements of things, i.e. sets, so they can't be the enormous equivalence classes proposed in (2).

(3) To identify each number class we would need a "standard" in each class. (4) But the use requires us to admit the existence of the elements of these standards. (5) We choose as standards sets whose elements exist "at minimal cost". (6) We finally abandon the definition of numbers as equivalence classes (with a special element as standard) and define directly each number by its "standard".

A few points:

• If you think about it, (3) immediately allows us to jump to (6) and thereby obviate (2), regardless of whether you make the observations in (4), (5).
• Defining $0:=\{\},\,Sn:=n\cup\{n\}$ and putting these into a thing called $\omega$ with no further elements, and claiming $\omega$ is a set, is something we already do in just about every interesting set theory's axiom of infinity (although I imagine some prefer a slightly different formulation). We don't do that because we're trying to solve the problem Russell was thinking about; we do it because a lot of interesting mathematics requires infinities. And that one axiom lets us skip all of (1)-(5) and never do any "philosophy" at all.

(7) We finally put this set in order using the successor function

Oh dear, I seem to have gotten ahead of myself. ;)

Finally, let's note that none of this lets us decide what the equivalent to (1)-(7) would be for infinite sets' sizes. What is the representative set equinumerous to $\Bbb N$, for example, or to $\Bbb C$? Roughly speaking, it would go like this:

• (1)/(2) would proceed as before;
• For (3)-(6)'s choice of cardinals, see here. Long story short, the details vary by the set theory used (and to an extent the model thereof), but that link gives the gist of it;
• (7)'s a bit trickier, and in some set theories you can't even order all the set sizes!

Solution 2:

The main (unique?) motivation has zero relation with your (4). The definition of numbers as equivalence classes has a very big technical problem: the equivalence classes themselves are "too big", namely, proper classes.

Solution 3:

The problem is not that the original definition requires the existence of the elements of the standards (Thumb, Index etc.) If we have a reasonable Set Theory, we can always find a set with five elements.

The problem is that the equivalence class so defined is a proper Class, not a Set; and the aim is to construct as much mathematics as possible using Sets only, as constructed using the Axioms that we allow ourselves.

So we define $5$ iteratively as $$0=\emptyset$$ $$1=\{0\}$$ $$2=\{0,1\}$$ $$3=\{0,1,2\}$$ $$4=\{0,1,2,3\}$$ $$5=\{0,1,2,3,4\}$$

which are all well-defined Sets.