I have serious doubts about any construction that strips away all distinguishing feature from entities until they are bare units, which are still supposed to be distinct (which is what you need for them to form a set, as if they are not distinct the set would have only one element and not two). Such a construction would have Leibniz turning in his grave, which makes me think that that cannot be how Cantor did the construction.
Kit Fine might like what he calls Cantor's definition/construction, but it smacks of nonsense to me, von Neumann's construction seems far less problematic.
It does seem to be a poorly defined concept as given there. How are we to know that the two units you get from Fido and Felix are the same as the two units you get from topsquark and Plato, or from a beer and a bumblebee? It seems to me that the question "what is two?" is now given the answer "two is the two-ness of any two things", which is at least circular, and completely fails to define the concept anyway. It's what you think it is - regardless of whether it is the same as what anybody else thinks it is.
Cantor's definition of numbers (at least as presented in the video) seems an awful lot like the categorical definition of ordinal numbers. They start exactly the same: the ordinal 0 is the empty category, with no objects, and no arrows. It's the "diagram that isn't a diagram".
The ordinal 1 is more interesting: it consists of a single object, and a single arrow, the identity arrow on that object (Identity arrows are often "not drawn", because of their triviality). Conceptually, this is the "sense" conveyed by the "=" sign. A thing is, what it is.
The ordinal 2 is even more interesting: it consists of two objects, each with their own identity arrow, and exactly one non-identity arrow $\bullet \to \bullet$. This gives us a way of *distinguishing* one object from another, in the sense conveyed by the sign "$\neq$".
The ordinal 3 is a "commutative triangle" (the diagram that displays what we mean by "composition"), together with the relevant identity arrows.
It's harder to explain with words what larger ordinals look like, but essentially, we have an arrow $a \to b$ if and only if $a < b$. Each "old" ordinal is a strict sub-diagram of the "newer" ones.
To get "cardinals", we just erase the (non-identity) arrows, so we have just the collection of objects. If we identify these objects with "units", we get the Cantorian view.
Of course, this sort of "begs the question", what are objects, and what are arrows? Well, the objects are very similar to the Cantorian "units", they could be lots of things. In the "classical" view, they are the usual:
$0 = \{\ \}$
$1 = \{0\}$
$2 = \{0,1\}$
but there's nothing that says they might not be "sheep" or "dollars" or "elephants" or whatever. They're like empty hangers in your closet, you put on them the particular labels that make sense in a particular context.
Arrows are similarly "vague", they could be anything that can be "composed" in some fashion (in the "classical sense" of ordinal numbers they represent "set-inclusion", which is composable because set-inclusion is transitive).
What's interesting about this set-up, is it transcends the "language" barrier-the properties of "first, second, third, etc." are expressed visually in a way that does not depend on words.
There is a fourth view not expressed in the videos, but one I daresay is important, which I will attempt to explain:
Natural numbers are the smallest infinite set for which induction works. This does not say what numbers ARE, it says what they DO. Formally, there are some additional requirements (to create an arithmetic) which have been codified as "the Peano axioms". So any kind of system which obeys the Peano axioms counts as "numbers". This is a subtle view, and one hotly contested by those that feel that numbers "exist" in some concrete sense, but it is one adopted by most people (whether they realize it or not) who actually work with numbers (a sort of: "Who cares what they are, as long as we get agreeing answers when we calculate with them" approach). This approach assumes that the "ultimate reality" of "what numbers are" is INTERNAL, and what we communicate (with others) are "the consequences of numerical property". Accordingly, "what numbers are" may be as mysterious as "is the green I see, the green you see?" (an example of qualia).
I'm sticking with Von Neumann's definition.
First, Fine's objection to Von Neumann's definition strikes me as empty. To have cardinality 2 is to be bijective with the set which defines 2, so what's his problem? It seems like his problem is that "twoness" isn't internalized in a set of cardinality 2 - but so what? The fact that you decide if a certain property (like having cardinality 2) applies to something (a set of cardinality 2) by appealing to - elsewhere - that which defines the property (bijective with Von Neumann's definitional set) is the case with EVERYTHING, so his complaint seems to me to be nothing more than special pleading, because... and I don't even know the "because". I can't fully make out what it is that he finds so objectionable. Appealing to a definition to decide if a property pertains to a thing isn't some manner of ontological crime by failing to respect a thing's intrinsic properties. Rather, it's the way all properties are attributed, always. Definitionally correct property attribution changes nothing - neither adding nor subtracting anything - about the thing's intrinsic nature. There's simply never an issue of the use of definitions somehow failing to respect a thing's intrinsic nature - but he seems to think that the whole numbers should somehow be treated differently. Why?
Second, Cantor's approach strikes me as directly self-contradictory."Deprive these two objects of all of their individuating features, beyond their being distinct from one another." But that's absurd. Being "distinct" from another thing isn't a feature of a thing, but rather is a relationship between two things... and, worse, it's a relationship that's decided EXACTLY by observing that there is at least one individuating feature which the two things do not share.
Therefore I hear the translation of Cantor's approach as: "Take distinct things, and remove all of their distinguishing characteristics - and then distinguish them." It's pure nonsense, a self-contradiction, a non sequitur.
Some interesting, and adroit observations. Here's my objection to von Neumann's definition:
We can form just as easily (more easily, in fact, since we do not need the union operator) the sets:
$\emptyset, \{\emptyset\}, \{\{\emptyset\}\},\dots$ etc.
and these sets, too, can serve as a definition of natural numbers (these are "Zermelo's natural numbers"). So which is it? How can we claim that either set (of our respective sets) is more deserving of the title "THE natural numbers"?
I humbly submit, therefore, that the "essence" of natural numbers is some property that these two *models* SHARE. In other words, they are not just isomorphic sets, but isomorphic recursively-defined sets defined by isomorphic recursions. This is a STRONGER statement than a mere cardinality equivalence, or even an order-preserving bijection.
In other words, the von Neumann naturals are just ONE way to construct a "natural number object", something that behaves the way our intuitive process of counting leads us to believe numbers ought to behave. The intuition is a priori, the von Neumann construction is an "after-the fact" justification of SOUNDNESS.
Your last statement:
actually hits upon what I feel is a fundamental flaw in human thought: I like to characterize this flaw as "everything we tell ourselves is a lie." We are constantly, on various levels, conflating equality with equivalence. This is why I think some people have trouble with very abstract subjects, they can only relax their notions of the "identity" of things so far. Truth, in truth, is unknowable, we are left with vague approximations that will have to serve, as time is short, and life moves on whether we attempt to comprehend it or not.Therefore I hear the translation of Cantor's approach as: "Take distinct things, and remove all of their distinguishing characteristics - and then distinguish them." It's pure nonsense, a self-contradiction, a non sequitur.
The "Cantorian Model" presented here seems to be the most universal at first. I have thought extensively about what I was really dealing with when studying math. This video seems to present a model that was fairly parallel with my own.
Numbers do not exist; they are abstract. Numbers are used to symbolize a real-world occurence. We use numbers to describe phenomena. Therefore, 2 really means 2 distinct object. I suppose in this sense I was thinking along the lines of the Freige-Russel Theory. The key word here is distinct. The Cantorian Model fails to classify any two objects as distinct, however, this is why we use numbers: to quantify a distinct thing or group of things. A mon avis, the Cantorian Model is unnecassarily abstract. The Freige-Russel Model is much more practical, as it reflects our motive for using numbers - eventual application to real world scenarios.