# Why such stark contrast between the approach to the continuum hypothesis in set theory and the approach to the parallel postulate in geometry?

In geometry, each of the 3 versions of the parallel postulate (the Euclidean, Hyperbolic and Elliptic) can be used in conjunction with the first 4 Euclid's axioms to form the axiomatic basis of each of 3 respective self-consistent formal systems: Euclidean geometry, Hyperbolic geometry and Elliptic geometry.

In set theory, either the continuum hypothesis (CH) or its negation can be used in conjunction with the ZFC axioms to forms the axiomatic basis of each of 2 respective self-consistent formal systems: (ZFC + CH) and (ZFC + ¬CH).

In geometry, no scholar would even think of arguing that the Euclidean version of the parallel postulate is either its only "true" version or a "truer" version than the Hyperbolic and Elliptic, and that therefore Euclidean geometry is either the only "true" geometry or a "truer" geometry than the Hyperbolic and Elliptic.

Why, in contrast, do set theory scholars argue about whether CH or ¬CH is "true"? Why don't they approach CH and ¬CH just as geometry scholars approach the 3 versions of the parallel postulate and study the respective consequent self-consistent formal systems?

Taking as an example this question in mathoverflow, it would be unthinkable to ask geometry scholars "What is the general opinion on the parallel postulate?"

I am aware that Hamkins 2011 introduced and argued for a multiverse view in set theory, which is clearly consistent with Mark Balaguer's "plenitudinous Platonism" (*) position in philosophy of mathematics, as argued explicitely in Fuchino 2012. What I find remarkable is that said view was proposed that late in the development of set theory and that it seems to be still a minority position.

(*) Which can just be "plenitudinous fictionalism", as Balaguer himself is agnostic between Platonism and fictionalism, the important notion being "plenitudinous".

• This was a very educational essay. I wonder what kind of answer do you expect – Yuriy S Jan 26 '18 at 23:32
• Thank you. I am interested in whatever answer set theory scholars can give on the issue, without expecting a particular kind of answer beforehand. – Johannes Jan 26 '18 at 23:39
• Re "In geometry, no scholar would even think of arguing that the Euclidean version of the parallel postulate is either its only "true" version or a "truer" version than the Hyperbolic and Elliptic", prior to the discovery of general relativity, I think you would have found quite a lot of people arguing that Euclidean geometry is the "true" version, even if the other versions are perfectly valid to study mathematically. – Eric Wofsey Jan 26 '18 at 23:39
• I'm only speculating, but I think that Hamkins' (and some others') multi-verse notion in mathematics could only have seemed reasonable after the "multi-verse" interpretation of quantum physics, which itself was delayed due to the long popularity of the "Copenhagen interpretation" of Bohr. Also, all the usual non-Euclidean geometries do have models inside Euclidean space, so "consistency" of Euclidean implies that of the others... Think of all the "imbedding" theorems of Nash, et al. – paul garrett Jan 26 '18 at 23:40
• Paul Garret, we should not confuse the many-worlds (MW) interpretation of QM with the multiverse hypothesis in cosmology. Each branching world in MW has exactly the same physical laws and constants than all the others. The only difference between world_1 and world_2 is that in one the cat is alive and in the other it's dead. So, the MW interpretation has nothing to do with this issue. – Johannes Jan 26 '18 at 23:57

The premise of your question seems to be that mathematicians are generally agnostic about the truth of the parallel postulate, whereas there's a live controversy about whether the continuum hypothesis is true or not. I think you've got that almost exactly backwards.

In geometry, you're right that everybody knows that there are structures that satisfy the Euclidean axioms and other structures that satisfy the axioms of hyperbolic geometry. And there's a general agreement that both kinds of structures exist equally well, in whichever sense the speaker thinks mathematical structures "exist" at all.

However everybody still agrees that Euclidean geometry is the one we're usually interested in. It's the one that, by tacit agreement, everyone will understand a statement about lines and circles to refer to unless it is uttered in a context that explicitly refers to hyperbolic geometry.

This is the closest thing to universal mainstream agreement that Euclidean geometry is the real one I can imagine, short of making a claim about the shape of the physical world we live in. And if we go the latter route and ask the physicists, the general theory of relativity tells us that neither axiomatic system describes the real world exactly (and plenty of actual astronomical observations back that up).

(By the way, note that we can describe and work with the Euclidean space and plane entirely without reference to the classical axiomatic description of it -- namely as $\mathbb R^2$ and $\mathbb R^3$ with appropriate algebraic definitions of what we mean by a line, circle, etc.)

What about set theory, then? Rather than an active controversy, it looks to me like a vast majority of contemporary set theorists are content with saying that CH is undecidable from the axioms and that's the end of the line as far as philosophy goes. I don't see anyone wasting much breath (or ink, or bytes) on seriously arguing that CH is necessarily true, or necessarily false in some philosophical sense.

Paul Cohen, who proved that CH is not provable in ZFC, did opine at the time that he felt the continuum hypothesis was "obviously false". But that was 50 years ago, and very few of his successors seem to share his assessment, or even be interested in having one of their own.

In general set theorists are pretty wary of suggesting that set theory even has a "intended interpretation" that will give an absolute truth value to the CH -- especially compared with the ease with which everyone agrees that the Euclidean space/plane is the intended interpretation of geometry. It is generally acknowledged that the intuition that set theory speaks about some Platonic universe of actually existing sets is useful and beneficial (not least because without it there seems to be scant reason to care), but it is also generally acknowledged that arguments that rest on it tell more about the speaker's thought patterns than they tell about any kind of objective truth.

We know -- at least assuming that ZF is consistent, and otherwise why bother at all? -- that both ZFC+CH and ZFC+!CH have models, but in either case the models we get in that way are definitely distinct from whichever intended interpretation we might believe in.

So "set theory scholars argue about whether CH or ¬CH is true" is about as far from what I see at it could possibly be.

(JDH's multiverse concept seems to me like it's not so much an attempt to propose a definite answer to the questions you envisage, as it's a plan to give "no, we really, really don't need to waste time arguing these questions" a backing that feels more intellectually weighty than simply pointing to our decades of failures to get something useful out of philosophical musings in this area).

• I'm not sure I agree; there certainly seems to be a lot of discussion among set theorists (notably Hugh Woodin, for instance) as to what the 'real' set theory is and which way it decides CH. (Is it the Omega hypothesis? Is it 'Ultimate L'? Is it throwing out choice entirely in favor of countable choice plus AD? Maybe it's something else still?) – Steven Stadnicki Feb 9 '18 at 23:28
• @StevenStadnicki: What is "AD"? Cursory searches returned nothing... – Nemo Feb 10 '18 at 15:15
• @Nemo The axiom of determinacy; en.wikipedia.org/wiki/Axiom_of_determinacy (it's on much lighter footing than the others, though; for the most part I don't know that anyone considers this 'the' model of the universe rather than a toy model that's equiconsistent with other interesting things.) – Steven Stadnicki Feb 10 '18 at 17:30
• HM, thank you for your answer. One comment: even with the latest cosmological observations of null global curvature (Planck 2015), since an absolute value of curvature < 0.0001 cannot be distinguished from zero, the universe may still be finite and have spherical geometry, a subset of elliptic geometry (or be infinite and have hyperbolic geometry). In that case, spherical (or hyperbolic) geometry would be "the real one" at the largest scale. – Johannes Feb 11 '18 at 19:23
• @Johannes: But holding "at the largest scale" is not enough for the theory to be an exact description of physical space. At non-cosmological scales it is well established that physical space does not have uniform curvature, and therefore neither axiomatic geometry describes reality exactly. – Henning Makholm Feb 12 '18 at 1:05