So: Having further motivated our positing of the new ZFC, as an essentially modest and conservative move, we inquire, what further potential axioms might attract our attention?
The first thing that springs to anybody’s mind is “the existence of God”; but here I must be a spoilsport. Simply by its bare wording, it yields us almost nothing – no more than a positing of “the Great Pumpkin”. But then, almost everyone who subscribes to the statement means a heck of a lot more by it than they let on; that lot-more differs, from person to person; what it is, is inexplicit, at least until some War of Religion breaks out; and is mostly unconscious in each one of us. It’s about as useful as, in set theory, the proposition that “Sets exist”; or, in the Theory of Everyday Life, “Sh*t happens”. Both doubtless true, but not fruitful.
Thus: What conjecture might be worthy of being accorded provisional axiomatic status, by the Regressive Gambit (“By their fruits ye shall know them”)?
One we have already met.
(Axiom I) Der Herrgott ist rafiniert, aber boshaft ist er nicht.
Remarkably, this principle is tacitly assumed in enterprises as widely separated as the Theory of Everyday Life, and theoretical physics. Our Olympic judges are holding up a unanimous “10-out-of-10” on the score of Fruitfulness.
(A) In the former, it serves for such essential tasks as allowing us to accept, without qualms,
(i) the Existence of Other Minds; and for refuting that vile devil’s-spawn,
(ii) the falsity of the Myth of the Brains in a Vat, along with the ravings of Berkeley.
Note: The Man in the Street already accepts (i) without qualms, having fortunately been spared contact with the corrosive hyperscepticism of the tenured diabolists (excuse me – eliminative materialists); and has probably never heard of (ii), unless he has watched “The Matrix”. So who needs another axiom already? -- But here he simply joins the otherwise honorable ranks of those early mathematicians, who made implicit use of the Axiom of Choice in their proofs, without knowing it. (I) is indeed necessary for the deduction of (Ai) and (Aii) – trust me on this. If you deny it – well, if you lead the Unexamined Life, I can’t argue with you.
(B) In physics, it undergirds (though it does not establish) such covertly metaphysical principles as Uniformity, Symmetry, Analyzability …
Again: the Lad in the Classroom imagines, between toothsome chomps on his chewing-gum, that these are affordances of Science itself, along with the opposite Coulomb charge of electrons and protons, and the curious mating habits of the mantis; for he has met these ideas in no other setting. But again: They are prior principles guiding the practice of science, not something we discovered in a lab.
A merely autobiographical aside, of no philosophical import whatsoever:
I tend instinctively to accept (I), though on very little evidence. If pressed, I defend it – lest Reason flee her throne. But in the matter of physics, I tend to think its reach is less than one might suppose. Already the paradoxes of quantum theory seem to infirm it to some extent.
(I), then, though some deem it crucial, is not comparable to the seemingly intuitive Axiom of Choice, but is more like one of the ambitious axioms for Large Cardinals, with which, in recent years, theoreticians have attempted to trick out Set Theory.
And here is another one, a classic – an axiom with the gloves off, comparable in overvaulting ambition to the hypothesis of full-bore Measurable Cardinals, GCH, Surreal Numbers, you name it. Its implications are absolutely incredible (and I mean that, alas, in both senses of the word), so please be sitting down before it is stated. Ready? Here goes:
(Axiom II) God cares about each one of us, individually.
This proposition is central to the theology, and even moreso to the psychology, of almost every Christian congregation. It is in bad odor with most contemporary philosophers – but leave these aside. We have each, in our own lives, experiences that make us contemplate (II), and either nod -- or wring our hands.
I’ll neither publically doubt that proposition, nor defend it, not being privy to any arguments beyond those already long-since trotted out on either side, nor to any personal experiences that might change the game.. But let us note that, largely from within the ranks of scientists, there has arisen a conjecture intermediate in strength between (I) and (II) –
(Axiom AP) the Anthropic Principle
in any of its forms, concentrations, or dilutions.
Indeed, quite apart from its genesis, it is mainly scientists who pay attention to this hypothesis (and employ it in concrete calculations). To the average believer, who already accepts (II), (AP) is hardly news, since (II) entails it.