Zermelo & Philosophy
The Importance Of Being Ernst
The idea of a set is to capture a little bit of the idea of “is an instance of”. The relation “is an instance of” is usually denoted by ε, from the Greek copula εἰμί.
So anyway, in 1898, there were two conceptions of set floating around.
The older one came out of the discussion of George Boole’s work as perfected by Peirce (though there were important precursors, especially Leibniz).
The younger concept is found in the pathbreaking work of Georg Cantor. I will give a brief fake history to motivate his work.
The late 18th & early 19th century mathematicians had developed two general methods for dealing with the linear partial differential equations (pdes) coming out of the theories of potentials and heat: Fourier’s Method and Green’s method. Both work in the same way: a special solution is found in a tractable case (a fundamental for Fourier, an impulse for Green) and complicated solutions are built by linearly combining these solutions with a weight that depends on the particular situation. One can explore the nature of these solutions - how and when the combinations converges on the true solution, whether the answer depends on the approximation technique etc. - by looking at them geometrically or arithmetically. The geometric approached reached something like perfection after Riemann. In retrospect (for instance as the below quote by Cohen argues), Cantor was lead inevitability to set theory by looking at the arithmetic side of linear pde theory.
The essential disagreement between Cantor’s approach and that of the post-Booleans was that the post-Booleans considered a fixed ‘domain of discourse’ whereas Cantor had an essentially incompleteable universe. The post-Booleans started with two special distinct objects called Top/True/Everything and Bottom/False/Nothing and worked into the middle. Cantor always1 started from a given universe and built up new objects by proscribed methods.
This was more than just a philosophical difference. Cantor’s famous power set and diagonal construction is capable of creating non-elements of claimed sets of all sets. Early formalizations of set theory struggled to be expressive enough to allow Cantor’s constructions and not so expressive to allow pathological objects in the same way novice programmers struggle to write loops which always work.
This meant that in German philosophy in the early 20th century an enormous amount of attention was being paid to developing syntaxes which rigorously distinguish between the noun everything and the quantifier every thing. And if you know your history if philosophy you will immediately see where I am going with this.
That’s right, it was exactly this issue that caused Carnap to break with Heidegger. Initially quite sympathetic to Heideggerian language as a replacement for metaphysics, Carnap later came to the conclusion that Heidegger’s Being And Time era language is not capable of explaining modern science, which means - in Carnap’s opinion - the language doesn’t live up to its stated goal of replacing Aristotelean metaphysics with an umwelt which takes seriously that species have origins and spacetime is relative.
Okay that Virgil Abloh installation has nothing to do with set theory, I just had enough with the dour black and white of 19th century guys.
Just doing what I want finally brings me to the reason for this post: given the importance of set theory for the Carnap-Heidegger debate, can we find for general philosophical import in Zermelo’s set theory axioms themselves?
Unfortunately, doing this will take a lot of typing so I’m only doing a set up today. First, as a preamble, my question means that I will be looking at the axioms slightly differently than usual. I want to highlight concepts of general philosophical interest, not questions about independence, model theory, etc..
Specifically, I want to be able to refer to the axioms as a whole so I will start with a basic axiom list, following Zermelo’s 1908 paper as translated in Ebbinghaus’s 2010 collection. I am giving loose intuitive definitions to be used later when I am trying to find some philosophical meat. Also, I don’t promise 9 more posts. If I think some axioms have the same philosophical connections, then I’ll combine.
Extensionality: Each set is determined entirely by its memebers.
Elementary sets: There is a set with no members. For each pair of objects (not necessarily distinct), there is a set with only those objects (or that object) as members.
Seperation: One can form subsets by propositions.
Power Set: For each set, there is a set which is all and only the subsets of that set
Union: For a given set, there is a set which is all and only the elements of the elements of the given set.
Choice: Given a set of disjoint nonempty sets, there is a set with one element from each of the disjoint nonempty sets
Infinity: There is a special set that contains the empty set and whose image under the successor function is a subset of the special set.
Two axioms were later added by Zermelo but first published by Frankel and Von Neumann:
Replacement: Given a function whose domain is a set, its range is also a set.
Foundation: Every set is either empty or contains an element distinct from the set.
So that this post isn’t too bare, I will give at least a bit of a philosophical look at one axiom. I will go with extensionality, because it is first.
It’s often proposed, for instance by Quine, that the appeal of working in sets is exactly extensionality. That said, there are some mathematically and philosophically trivial ways of working without extension. For instance, one could take ordered sets as the basic concept. But more philosophically and mathematically interesting is distinguishing by intension.
If you are a computer scientist, then you don’t believe in extension. Take the following functions:
Trombleysort: given a list, check if the list is ordered. If so, then quit. Otherwise, set up and solve a random boolean satisfiability problem. If there is a satisfactory answer to that problem, then perform one bubblesort operation. Otherwise, randomly permute the entire list. Send this list back to the start.
All three of these functions have exactly the same domain (finite ordered tuples with a strict total orders, or lists for short) and the same range (lists whose elements agree with the total order). For the right choice of order, each list is mapped onto the same ordered list for all three. These are the same function. In fact, extensionally, there is only one sorting function.
As concepts, Heapsort, Bubblesort and Trombleysort have the same members, the same ordered pairs of lists and ordered lists. But they aren’t themselves members of the same concepts. Heapsort and Bubblesort are deterministic algorithms and Trombleysort is not. Heapsort is an efficient algorithm, Bubblesort and Trombleysort are not.
What extensionality adds is that sets that have the same members are also members of the same sets. This makes sets more tractable than more philosophically obvious collections like concepts.
Goong forward, in more modern foundations the selling point is almost2 always more intension than in Zermelo Set Theory. For instance, in Martin-Löf’s type theory propositions have to have witnesses which make them true. For example, “971 is a composite number” requires if not a factor, then a method of finding a factor and a guarantee that the method works. That this is impossible is the meaning of falsity.
Martin-Löf’s very explicit inspiration is his fashion icon’s Aristotle: “To say of what is not, that it is”. Aristotle’s logic requires all knowledge to begin with the particular and build up inductively. That this disagrees radically with contemporary mathematical practice has been noted since Schopenhauer. Schopenhauer’s critique was picked up by LEJ Brouwer who built it into a positive system.
But this story is well known. What I want to call attention to is something I first saw pointed out by Hans-Deiter Ebbinghaus. The Aristotle/Schopenhauer/Martin-Löf theory models learning as fully analyzed experience.
There is another model of learning, Plato’s theory of learning as remembering. Plato’s theory is much closer to Euclid’s Elements, in which the definitions, postulates, common notions and propositions have a hazy relationship with one another. It’s not clear, for instance, that the Pythagorean “Theorem” is posterior to the Parallel “Postulate” (they are logically equivalent with the other axioms).
I think these two different views of learning arise from very different dialectical understandings of what philosophy is. Is philosophical language trying to negate ordinary language to become more precise? Then allowing “common notions” and discovering their implicit depth in the manner of Plato and Zermelo is what is called for. Is philosophical language trying to be a backgroundless correct language for description? Then only fully analyzed axioms are called for, in the manner of Aristotle and Martin-Löf.
Well, anyway, I think any phiosophy essay is allowed to end on a Plato/Aristotle contrast. I look forward to writing more on this in the future.
It is sometimes wrongly supposed that Cantor believed in so-called ‘naive set theory’, where there each proposition has a set as its extension. This can be disproven by a casual reading of his papers.
The sole exception as far as I know is Quine’s New Foundations.