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I'm trying to understand the evolution of the concept of number since Frege/ Russell and to see the "big picture".

What are the main motivations explaining the change from Russell's definition using equivalence classes ( in "Introduction to mathematical philosophy") and the current definition of (natural numbers) using the successor function?


The "stages" I can see are the following. Would you please assess the reasons I have imagined to explain (to myself) the passage from one stage to another?

(1) Frege / Russell recognized that numbers were higher-order properties, not properties of things , but of sets

(2) Numbers are defined as equivalence classes, using the relation of "the set X is equinumerous to set Y" (iff there exist at least one bijection from X to Y)

(3) To identify each number (that is each class) we would need a "standard" in each class. For example, one could use{ Thumb, Index, Middle finger, Ring finger, Pinky finger} as a representative of the sets having 5 elements. In that case, one would say:

the number 5 is the set of all X such that there exists a bijection from X to the set { Thumb, Index, Middle finger, Ring finger, Pinky finger }

and

X has 5 as cardinal number iff X belongs to the set 5

(4) But the use of these representatives requires us to admit the existence of the elements of these standards. Furthermore, it obliges us to admit that the existence of numbers depends on contingent facts of the world, that is, the existence of these elements belonging to our " standards".

(5) So to get rid of these existential presupposions, we decide to chose as standards sets whose elements exist "at minimal cost". As standard for the set "zero", we use { } (as we did before. But as standard for the set 1, we now use

{ 0 } (that certainly exists if 0 = { } exists.

and as standard for the set 2, we use { 0, 1} , etc. In this way, our construction becomes independent of the existence of concrete things in the world.

(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". So instead of saying that "2 is the set of sets that can be put in $1-1$ correspondance with the standard $\{ 0,1\}$", we simply say that

the number $2$ is (by definition) the set $\{ 0,1\}$.

(7) We finally put this set in order using the successor function ( $S($number $x)$ is by definition the union of number $x$ and of $\{x\}$) which "generates" an infinite series of numbers "out of" the null set.

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    $\begingroup$ That's a decent philosophical reasoning - although I can't speak to its historical accuracy, not being versed in the history myself - but never underestimate the role of sheer pragmatics: proper classes are weird and talking about sets of cardinal (or ordinal) numbers is important, so it is useful for the "number of elements" (and similar notions) of a set to also be a set rather than a class. It just ultimately reduces the "overhead cost" for the arguments we want to make. Again, I'm not versed in the history (hence this isn't an answer), but I suspect this did play a major role in the shift. $\endgroup$ – Noah Schweber Apr 7 at 14:26
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    $\begingroup$ I think this approach, as Noah points out, fell out of favor when it became very obvious that unrestricted comprehensions are problematic, and classes are in general annoying to work with. So, it's a lot easier simply denoting a set to represent a given cardinality, and forcing any set with the same cardinality to share a bijection, rather than being a member of some class, that we don't even know much about. $\endgroup$ – Don Thousand Apr 7 at 14:29
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    $\begingroup$ (6) "sounds" a little bit different IMO... The original proposal of Frege and Russell was also to solve the philosophical problem of "what numbers really are" (assuming that the question is meaningful...). The current set theory construction aims at defining inside the "universe" of sets a structure that has exactly all the properties of the natural number. From a mathematical point of view this is enough, but form the point of view of Frege and Russell it is quite doubtful to asserts that numbers are conjured out of the empty set. $\endgroup$ – Mauro ALLEGRANZA Apr 7 at 18:38
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    $\begingroup$ See Paul Benacerraf, What numbers could not be. $\endgroup$ – Mauro ALLEGRANZA Apr 7 at 18:44
  • $\begingroup$ I don't see this as an "either/or" situation. I am of the opinion that the natural numbers are exactly what we learn as children when we first learn to count. That they constitute an infinite set takes a bit more doing. All formalization of the number systems is just a way to keep mathematicians honest and/or busy. $\endgroup$ – Steven Thomas Hatton May 27 at 20:13
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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!
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    $\begingroup$ @ J. G - Thanks for this clear and detailed answer. $\endgroup$ – Eleonore Saint James Apr 7 at 15:06
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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.

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    $\begingroup$ I don't know why this got downvoted. It cuts to the heart of the question. $\endgroup$ – TonyK Apr 7 at 14:56
  • $\begingroup$ @ TonyK @ Martin Bias - What was downvoted? Personnaly, I upvote your both answers. $\endgroup$ – Eleonore Saint James Apr 7 at 15:00
  • $\begingroup$ @EleonoreSaintJames, my answer as two upvotes (you, TonyK) and a downvote. The other upvote of TonyK is also mine. $\endgroup$ – Martín-Blas Pérez Pinilla Apr 7 at 15:02
  • $\begingroup$ @EleonoreSaintJames: Once you reach a certain reputation, you can click on the vote counter to see the number of upvotes and downvotes. This answer currently has two upvotes (from you and me!) and one downvote. $\endgroup$ – TonyK Apr 7 at 15:03
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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.

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