The Independence of the Continuum Hypothesis

I've just had the good fortune to discover that both of Paul Cohen's papers establishing the independence of the Continuum Hypothesis (CH) from Zermelo-Fraenkel set theory (ZFC) are freely available online, with Part I to be found here, and Part II here. It's odd that attempting to find both papers by the obvious route - going to the PNAS website and looking through the archives - only leads one to JSTOR, when in fact PubMed apparently carries PNAS archives dating all the way back to January 1954.

Why is Cohen's work on the Continuum Hypothesis important, you might wonder? To this, my response is that it has rather interesting implications for a long-cherished belief amongst mathematicians going by the label "mathematical platonism", the essence of which is that mathematical concepts have an independent existence of their own in some abstract universe outside our mental lives, with their truth being a purely objective matter which we are duty driven to accept; what is more, every mathematical statement is necessarily either definitively true or false. The existence of two equally consistent and yet contradictory systems of mathematics in the guise of ZFC+CH and ZFC+(~CH) is obviously impossible to reconcile with such a position, and unlike the old difficulties with Euclid's parallel postulate, which one can at least come to terms with by thinking of flatness as a mere local approximation of a curved manifold, there is no possibility of regarding one system as a limiting case of the other, nor even the possibility of appealing to the greater psychological intuitiveness of one of the two alternatives, as is usually done with the axiom of choice. Who has ever laid eyes on an actual infinity, to know what is "intuitively" true in such a realm?

My conviction is that the only truly tenable position for one to hold at this juncture is that of the formalist who regards mathematics as a game of symbol manipulation, in which we are free to choose whatever starting rules we please, just as long as they're consistent with each other; from this position, neither the axiom of choice nor the independence of the continuum hypothesis present any difficulties whatsoever. Mathematical platonism, like the assumption that space is Euclidean or that there is such a thing as absolute time, is nothing more than a false belief which has been engendered in us by the selective pressures which shaped our minds; there are no pure ideas lying out there in some abstract realm just waiting for some thinker to come along and discover their timeless essence, only brains which have evolved to accommodate a universe in which certain regularities occur, and it is from these regularities that notions like perfect circles and the real number line are born. Ours is a purely physical universe, and even our very ideas are part of that physical reality we inhabit.

PS: This Slashdot post also has interesting things to say about the whole matter. Also see these articles by W. Hugh Woodin in the Notices of the AMS.

Comments

[clark kluggermann is a troll]

["clark kluggermann"'s mother dropped him on his head as a child, which is why he's now a cowardly little retard who thinks he can't be identified when trolling.]

[dof]

Strange, I always had the intuition CH was "true" because i found GCH so compelling.

Every element from a countable set can be represented by a finite sequence of symbols from a finite alphabet.

Every element from the continuum can be represented by a countable sequence of symbols from a finite alphabet.

It seems a bit weird to "allow" aleph-1 to be different from c, as we will still need a countable sequence to represent an arbirary element from a set with cardinality aleph-1, which just happens to be the same as for c.

[Abiola Lapite]

"Strange, I always had the intuition CH was "true" because i found GCH so compelling."

What's interesting is that as the Slashdot commenter points out, there are equally compelling reasons to believe it's false! This only goes to show just how much a matter of taste this all is, as neither logic nor intuition is a worthwhile guide here.

[dsquared]

This isn't true. It's only a counterexample to particular forms of Platonism, and not to the more popular versions believed by, for example, Roger Penrose or Kurt Goedel. (Quite a while since I last did this, so from memory)

It's true that this result is not consistent with a simple form of Platonism about numbers (this would be a "naive Platonism", in that it's an assertion about the Platonic existence of particular mathematical entities). If there were such entities as numbers, then they would either form a continuum or not, and there would be a fact of the matter[1].

However, there are other closed Platonisms which would not be affected my this. In particular, to cite an important example in the philosophy of mathematics, WVO Quine is a closed Platonist about sets. It's perfectly well open to him to say that there are two or more kinds of sets (ZFC+CH sets and ZFC+~CH sets), and to go on believing in sets because he regards them as metaphysically necessary entities for other reasons.

Furthermore, it doesn't bite at all against forms of Platonism which don't tie themselves to specific claims about mathematical entities, but just maintain that there is a world of mathematical truths, and that mathematics is the business of discovering these truths[2]. This was the actual view of Goedel, who believed that we made these discoveries empirically, through a species of categorical intuition. Cohen at one point appeared to endorse this view (though he later moved away from it), so it's unlikely that his work is "Obviously" inconsistent with it.

In other words, Cohen on the continuum doesn't destroy Platonism any more than Goedel's Incompleteness Theorem destroyed Formalism (although it did put paid to Hilbert's formalist project).

[1]I'm assuming that the conclusion that there was a fact of the matter as to whether the CH was true, but that this fact of the matter was unknowable, would be sufficiently unattractive that nobody would hold it, but it would certainly be a consistent view for even a naive Platonist to hold.
[2]Note that the formulation Abiola uses above - that every mathematical statement is definitely true or false - is much too strong. Neither form of Platonism described above commits one to this much stronger view, and I'm not aware of anyone seriously having maintained a "closed" Platonism which did, other than as a strawman. If one believes that 1+1=2 is true, then one is to that extent a Platonist, without being committed to anything about set theory (this, IIRC was the actual view of Kronecker).

[Julian Elson]

I find the continuum hypothesis too narrow for my tastes. I want the cardinality of power sets of countable sets to be less than the cardinality of R... or something.

[Abiola Lapite]

"If there were such entities as numbers, then they would either form a continuum or not, and there would be a fact of the matter."

The very existence of an infinity of numbers is predicated on certain axioms that are not in and of themselves god-given, as no one has ever seen or perceived an infinity of anything; as such, to speak of a "fact of the matter" in this regard is unacceptable. That constructivists like Brouwer and his followers are out there shows that it is perfectly possible to engage in mathematical research without the use of axioms that force us to accept the "reality" of a continuum; all we have to do is renounce the notion of an actual infinite, which in any case has no basis in nature.

"but just maintain that there is a world of mathematical truths, and that mathematics is the business of discovering these truths[2]. This was the actual view of Goedel, who believed that we made these discoveries empirically, through a species of categorical intuition"

1 - He never made clear why exactly we could trust in this "intuition" of his. At best he could say "the mathematics produced is useful in the real world", but that does nothing to establish that "intuition" is some sort of infallible guide to knowledge; the same "intuition" told men for 2500 years that the parallel postulate was "obviously" universally true. Remember Kant's claims about it being an example of synthetic a priori knowledge?

2 - Gödel's claims about the nature of mathematical truth are much stronger than you let on, and in fact what you portray as my caricature of Platonism is indeed what he believed. The following is a Gödel quote from a paper by Charles Parsons*,

["It seems to me that the assumption of such objects [classes and concepts] is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions"]

and in the same paper we are told the following:

["In 1964 the question of the “objective existence of the objects ofmathematical intuition” is said (parenthetically) to be “an exact replica of the question of the objective existence of the outer world”"]

and yet further on

["it is reasonable to ask whether Gödel is a realist by one criterion suggested by the work of Dummett, according to which realism admits truths that are “recognition-transcendent”, that is obtain whether or not it is even in principle possible for humans to know them. In the sphere of mathematics, an obstacle to this view for Gödel is his confidence in reason; he expresses in places the Hilbertian conviction of the solvability in principle of every mathematical problem."]

In other words, Gödel was a hardcore Platonist of the sort you claim I throw up as a strawman.

"It's perfectly well open to him to say that there are two or more kinds of sets (ZFC+CH sets and ZFC+~CH sets)"

This is incoherent unless you're talking from a formalist viewpoint. A platonist would insist that there is a *single* universe of sets outside of which no sets exist, even if we don't know what the correct axiomatization of it may be; as such, claims about ZFC+CH sets and ZFC+~CH "kinds" of sets existing side by side make no sense within such a framework. We aren't talking about mere alternate geometries here, but conflicting claims about the nature of a single heirarchy of numbers.

As for your claims about Roger Penrose, plenty of others would differ; see for example Daniel Dennett, or even the following discussion of Penrose's claims by Edward Nelson:

http://www.math.princeton.edu/~nelson/papers/tokyo.pdf

"Neither form of Platonism described above commits one to this much stronger view, and I'm not aware of anyone seriously having maintained a "closed" Platonism which did, other than as a strawman."

Then maybe you ought to read The Mathematical Experience, or even take another look at what Gödel himself had to say on the matter (see above). Straw man indeed ...

To the extent that certain mathematical notions seem objectively true and universal to the human mind, it is only because said mind evolved under a physical regime that was consistent with a particular axiomatic system, which has become implicitly embedded in our thinking. Once we start talking of ideas "existing" without thinkers, we're heading off into mysticism-land, and Gödel's own adherence to Platonism veered off in just such a direction.

*Charles Parsons, "Platonism and Mathematical Intuition in Kurt Gödel's Thought", Bulletin of Symbolic Logic, Volume 1, Number 1, 1995.

[dof]

Anyway, another way to phrase why I find GCH so compelling is from an information-theoretic point of view.

Another way to write

aleph(i+1) = 2^aleph(i)

is

log aleph(i+1) = aleph(i)

and the log operator here has the meaning: how many symbols from a finite alphabet ("bits" for base 2) would be needed to represent a generic element from a set of this cardinality?

But maybe all this means is that if you take -GCH, you just "happen" to have different cardinals with the same logarithm. I sounds messy, but I guess that was true for Gödel's theorem in his days.

[dsquared]

In or out of order:

"The very existence of an infinity of numbers is predicated on certain axioms that are not in and of themselves god-given"

This is begging the question. A Platonist about numbers would not accept that their existence is "predicated on certain axioms".

"He never made clear why exactly we could trust in this "intuition" of his"

I'm sorry; could you be clear here about when you'er making your own argument and when you're saying something that you believe to be implied by Paul Cohen's work on the continuum hypothesis? This seems to me to be clearly in the first category.

"In other words, Gödel was a hardcore Platonist of the sort you claim I throw up as a strawman"

No he wasn't. The claim I described as a strawman was the claim that Platonism commits you to a belief that every mathematical statement is definitely true or false. "Solvability in principle of every mathematical problem" is a much weaker claim; you will, without fail, go down every wrong path there is to go down in philosophy of mathematics unless you keep clear about this.

"A platonist would insist that there is a *single* universe of sets outside of which no sets exist"

No. That would be a closed Platonism. An open Platonism would simply make the existence claim that there exist sets.

"As for your claims about Roger Penrose, plenty of others would differ"

My one claim about Penrose was that he is a Platonist, and nobody disagrees with this; your linked reference actually says that he is. It's true that lots of people disagree with Platonism, but I didn't sign up for a defence of Platonism (I'm a Wittgensteinian constructivist myself). I objected to your simplistic assertion that the independence of the continuum hypothesis rendered Platonism untenable.

"Once we start talking of ideas "existing" without thinkers"

Numbers and sets are not ideas; I thought I had dealt with this claim of yours already. It is not at all absurd to assume that if three rocks exist, then a set of three rocks exists.

[Delmore Macnamara]

It seems rather a long step from the indubitable "both CH and ~CH are consistent with ZFC" to the doubtful "there is no fact of the matter about CH" & thence to the rather postmodern "mathematical reality has no objective existence".

As I am quite sure you know, both Godel & Cohen hoped to find some additional sensible axiom that, added to ZFC, would settle CH. The fact that none has yet been found is no knockdown argument against its existence, particularly since only forty years have passed since Cohen proved the consistency of ~CH with ZFC. Of course Projective Determinacy would settle CH but its truth is hardly self-evident. I seem to recall that V=L would settle CH too, but I may be wrong about that & working set-theorists seem convinced - for reasons. I am too stupid to understand - that V=L is false

If Friedman's recent suggested project to prove that all equivalences of CH necessarily require 'a lot of quantifiers'(i.e. more than the four for the axiom schemes & ZF axioms & five for AxC)in ZFC were to have a positive result, I suppose that might be suggestive that, since no "simple" axiom settled CH, there is indeed no fact of the matter about it, but even then I don't think it would be decisive. Perhaps a "complicated" axiom can nonetheless be self-evident. As with God, I think the best approach to the existence of mathematical reality is strict agnosticism. I am doubtful that mathematical machinery can often settle philosophical problems (altho' wasn't one of the original motivations of Frege's Grundlagen & Grundgesetze to show that the axioms of arithmetic were not synthetic a priori?)

What do you think of Friedman's project to produce statments of "core mathematics" that require additional set-theoretic axioms to settle? As I understand it, this is a strategem intended to bring the Frege-Godel-Cohen line of FOM back into the mathematical limelight. Would that be a good thing?

[Delmore Macnamara]

For those interested in FOM & particularly Friedman's projects, the archives of the FOM list are highly to be recommeded: see
http://64.233.183.104/search?q=cache:-9pQuyM_7fkJ:www.cs.nyu.edu/mailman/listinfo/fom+fom&hl=en

It is a closed list & you have to have a serious interest & some competence in mathematical logic to join, which is why I haven't! But the archives are freely accesible to us knowlessmen.

PS my use of the word "hoped" in the previous post may have given the impression I thought Cohen was dead. AFAIK he is still alive; certainly he was on a Panel on CH at the recent Atlanta meeting of the AMS.

[Abiola Lapite]

"This is begging the question. A Platonist about numbers would not accept that their existence is "predicated on certain axioms"."

Oh really? Then what is the claim that they exist based on? It can't be empirical evidence - no one's ever seen a disembodied "1" floating about - so is it divine revelation?

"No he wasn't. The claim I described as a strawman was the claim that Platonism commits you to a belief that every mathematical statement is definitely true or false. "Solvability in principle of every mathematical problem" is a much weaker claim; you will, without fail, go down every wrong path there is to go down in philosophy of mathematics unless you keep clear about this."

This argument doesn't hold the slightest water. It's easy enough to reformulate the claim that the continuum hypothesis is true as a problem: "does there exist a cardinal lying between aleph_0 and c?" There you go.

"My one claim about Penrose was that he is a Platonist"

No, you said the following, remember?

"It's only a counterexample to particular forms of Platonism, and not to the more popular versions believed by, for example, Roger Penrose or Kurt Goedel."

So you did in fact say something about the nature of Penrose's platonism, and that was just what I was addressing. Penrose is yet another believer in the notion you insist on calling a strawman.

"Numbers and sets are not ideas; I thought I had dealt with this claim of yours already."

No, you haven't. What are they if not ideas? In what other sense can they be said to "exist?"

"It is not at all absurd to assume that if three rocks exist, then a set of three rocks exists."

Then why isn't it possible to categorically say that "there exists a cardinal lying between aleph_0 and c"? Surely both aleph_0 and c "exist", so it ought to be a simple yes or no matter whether something lies between them in terms of cardinality.

[dsquared]

"It's easy enough to reformulate the claim that the continuum hypothesis is true as a problem: "does there exist a cardinal lying between aleph_0 and c?" There you go."

And it's easy enough for a non-straw "open" platonist to say

"This question is ill-formed. The sets defined by ZFC+CH are continuous but the sets defined by ZFC+~CH are not. However, it is a *fact* rather than a formalism that this is the case, and my Platonism consists in making this second assertion rather than in being committed to the existence of any particular entity."

or for a non-straw "closed" platonist to say:

"We don't know at this point. The presence or absence of such a cardinal would be consistent with ZFC axioms, but ZFC is not the last word on set theory".

or indeed, to say:

"No, assuming you mean "does there exist?" in a Platonic sense, because infinite sets are not among the entities I admit as having an independent Platonic existence. The natural numbers were created by God; all else is the work of man".

You might or might not find any of these views convincing, but that's not the issue here. The issue is that all three of them are consistent positions to take and none of them are inconsistent with the independence of the continuum hypothesis.

and onward ...

"No, you said the following, remember?"

I did, and it's increasingly looking like I'm right. The paper you linked to doesn't give any particular reason to believe the author thinks that Cohen's proof is a counterexample to either Penrose's or Goedel's forms of Platonism, (this is not surprising, because it isn't). You apparently have your own arguments against Platonism, and you can find lots more in the literature because it is a very controversial position (for example, I don't believe it) but I think you should do me the courtesy of allowing me to choose whether I want to get involved in this separate debate, rather than pretending that it's part of one I'm already involved in.

Please, I repeat, distinguish between arguments of your own and those that you believe to be implied by the independence of the continuum hypothesis.

"What are [numbers and sets] if not ideas?"

They are numbers and sets. They differ from "ideas of numbers and sets" in more or less the same way in which dogs and cats differ from ideas of dogs and cats, which is similar to (though distinct from) the way in which a set differs from its members. The distinction between a thing and a reference to a thing is a pretty deep and important part of philosophical logic which (as I have mentioned) you will go down every possible wrong path if you don't keep clear.

"Surely both aleph_0 and c "exist", so it ought to be a simple yes or no matter whether something lies between them in terms of cardinality."

No Platonist is committed to either your premis or the entailment you assume. The Platonism which I (I think, mistakenly) attributed to Kronecker would not say that mathematical entities other than the natural numbers existed. And a belief that aleph_0 and c both existed (to which no Platonist is necessarily committed, "surely" or otherwise) does not entail belief that the question of the continuum hypothesis is simple (or indeed that it is "yes or no"; see above).