“An Axiom, you know, is a thing that you accept without contradiction. For instance, if I were to say ‘Here we are!’ that would be accepted without any contradiction, and it’s a nice sort of remark to begin a conversation with. So it would be an Axiom. Or again, supposing I were to say, ‘Here we are not!’, that would be—”
“—a fib!” cried Bruno.
“that would be accepted, if people were civil”, continued the Professor; “so it would be another Axiom.”
“It might be an Axledum”, Bruno said: “but it wouldn’t be true!”
—Lewis Carroll, Sylvie and Bruno Concluded
To get to the “good stuff” in math, you almost always need some set theory. Zermelo-Fraenkel set theory (ZF), plus the axiom of choice (AC; ZF+AC=ZFC) has become the standard first-order axiom system for set theory.
Before diving into the details, some generalities on axiom systems. Nowadays we’re pretty chill about them; you can take any collection you like (hopefully consistent) for a theory, and then you can start writing your thesis. Not, perhaps, an interesting thesis, but at any rate Bruno won’t complain that your axioms aren’t true!
For the Greeks, the axioms and postulates were true, in some sense. Idealized, sure, but descriptive of reality. This tie began to fray with the discovery of non-Euclidean geometries. Algebraic axiom systems, like those for groups and for fields, appear by the end of the 19th century.
For roughly two thousand years after Euclid, most math developed without axioms. Take calculus as an example. You have the rules of calculus, but you don’t see anything like the Euclidean treatment of geometry. This remained true even as people subjected its foundations to stricter and stricter scrutiny. Mathematical intuition reigned supreme.
Hilbert’s Grundlagen der Geometrie (Foundations of Geometry, 1899) pushed towards a more formalist attitude. A celebrated quote of his, from years earlier, sums it up nicely:
One must be able to say at all times, instead of points, lines, and planes: tables, chairs, and beer mugs.
At times Cantor seemed to endorse this perspective:
Mathematics is entirely free in its development, and its concepts are only bound by the necessity of being consistent, and being related to the concepts introduced previously by means of precise definitions.
—Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of sets)
But he held strong opinions on what’s true in mathematics:
I entertain no doubts as to the truths of the tranfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.
—Letter from Cantor to Jeiler, quoted in Dauben (p.147).
On the other hand, he referred to the “Cholera-Bacillus of infinitesimals”, and called them “nothing but paper numbers!” (Dauben, p.131). The Continuum Hypothesis was for him a question of fact.
Two other themes run through this period: mathematics as a mental activity, and as logic.
Recall that Boole titled his famous treatise An Investigation of the Laws of Thought: on Which are Founded the Mathematical Theories of Logic and Probabilities. Cantor’s definition of “set” in his last major work reads
By a set we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought.
Here is the first sentence of Dedekind’s Was sind und sollen die Zahlen?: “In what follows I understand by thing every object of our thought.” His proof of the existence of an infinite set relies on this ontology:
Theorem: There exist infinite systems.
Proof: My own realm of thoughts, i.e., the totality S of all things which can be objects of my thought, is infinite. For if s signifies an element of S, then the thought s′, that s can be an element of my thought, is itself an element of S…
[Dedekind then appeals to his definition of infinite as having a bijection with a proper subset.]
Frege severely criticized this injection of psychology into mathematics. Cantor’s “proof” of the Well-Ordering Theorem suffers from it, as it consists of successively choosing elements of the set to be well-ordered. If we take this literally, then the choices must take place at an increasing sequence of times t1<t2<…. This limits us to ordinals that are “realizable in ℝ’’, and thus to countable ordinals (see post 4). Yet Cantor claimed that every set can be well-ordered, in particular ℝ.
This is why Zermelo was at pains to say in his second proof of the Well-Ordering Theorem, “…the ‘general principle of choice’ can be reduced to the following axiom, whose purely objective character is immediately evident.” (My emphasis.)
Both Frege and Russell held that the truths of mathematics are logical facts. Thus we find debates on whether Zermelo’s axiom of choice is logically valid. Not surprising, historically. Aristotle’s logic dealt with propositions. From “proposition” we obtain “propositional function”, that is, a proposition with a free variable, like “x is mortal”. It becomes a proposition if we assign a value to the variable (“Socrates is mortal”), or quantify over it (“All men are mortal”). The class of all things satisfying a propositional function went by the name, “extension of a concept”.
Zooming out from these specifics, logic and mathematics both lay claim to necessary truth. This is elaborated in Kantian philosophy. Kant classified mathematical facts as synthetic a priori: necessary truths that go beyond analytic truth, which are true by definition. Poincaré classified the Axiom of Choice as a synthetic a priori judgment, just like the principle of induction.
The rise of formal logic and axiomatic set theory resulted in a sharply drawn boundary between logic and set theory. We have the axioms and rules of inference of first-order logic; then we have the axioms of ZFC or similar systems, which are particular first-order theories. Things weren’t so clear at the dawn of the 20th century.
Diagonal Argument 