Continuum hypothesis, state of the art

Continuum problem has worried mathematicians since the time of the creator of set theory, Cantor. The great mathematician Hilbert put it first in his famous list. In a sense, it is considered solved – only many do not consider it a solution, and it still occupies the minds of philosophers and mathematicians.

On cardinalities of sets

Let me remind you that the “size” of infinite sets cannot be determined by recalculating the elements. Instead, the concept of “cardinality of a set” is used, which is determined on the basis of mutual mappings of elements. For example, any infinite set of integers (for example, the set of prime numbers) can be counted by assigning numbers to them. All such sets have cardinality aleph-zero (sorry, the lowercase formulas got corrupted on Habré and won’t be fixed quickly, as the support says)

\aleph_0

The set of real numbers, however, cannot be counted, which is proven very elegantly by the diagonal method. The power of the set of points of any finite and infinite figures on the plane, in volume, etc. is also equivalent to the set of real numbers and is denoted as

\aleph_c

Since the set of formulas (finite lines in a finite alphabet) is countable, most real numbers have no representation, they are a faceless mass, so faceless that despite the fact that there are still a continuum of ‘faceless’ numbers, none of them can be given as an example, showing it explicitly. Such faceless elements are called indiscernibles (I note that there is no Russian article on this subject on the wiki, and I also don’t know an analogue of the Russian word).

Despite the fact that from such a set we cannot explicitly present a single element, there is no problem for the axiom of choice to do this. Proofs that begin with the words “take an arbitrary element of the set” work great if there is axiom of choice, and that is why this axiom so often gives rise to monsters – for example, Banach-Tarski paradox (I note that the very cutting of the sphere in this paradox is indiscernible).

We have so far become acquainted with two infinite powers: countable and continuum. It turns out that there is a standard way to produce more and more powers: the powerset operation – getting all subsets of this set. It is denoted as

powerset(X) = 2^x

– indeed, for finite sets the number of elements increases in this way. It is reliably known that

2^{\aleph_0} = \aleph_c

Continuum hypothesis

The first question that arises, and it actually arose with Cantor, is there a set with cardinality between countable infinity and continuum? That is

\aleph_0 < ???  < 2^{\aleph_0} = \aleph_c

Oddly enough, this turned out to be a very difficult problem, and it is not for nothing that it was included in Hilbert’s list of problems at number 1. First, many years later, Gödel proved that the existence of such a set cannot be proven in standard set theory. A quarter of a century later, Cohen proved that this statement cannot be refuted. So this statement is independent from set theory, which immediately gives rise to two versions of set theory – where such a set exists and where it does not.

If such a set exists, then it is indiscerniblebecause if it were not so, then we could prove its existence simply by giving an example of its construction.

You should have believed me right away that there are two such versions. There are more of them. First, there is the ‘neutral’ version, which says nothing about the existence of sets of intermediate cardinality. Secondly, there is a version with the continuum hypothesis – it is categorical, there are no such sets. But if we allow the existence of sets of intermediate power, then we can axiomatically state that there are exactly one, two, three, 188338 such sets, any number between 17 and 83, any finite number, any finite prime number, etc.

It is interesting that the number of such sets with intermediate cardinality cannot be infinite for rather complex reasons. But any finite number is welcome! Any finite subset of integers is admissible, which gives a countable number of variants of the theory.

There is a stronger version of the continuum hypothesis (CH) – generalized continuum hypothesis (GCH). She claims that not only

2^{\aleph_0} = \aleph_1 = \aleph_c

but that this is true for any power:

2^{\aleph_\lambda} = \aleph_{\lambda+1}

GCH stronger CH and it also follows from it A.C. (axiom of choice).

Philosophical aspect

So, is the problem solved? (that’s how it’s marked in Hilbert’s list)? Formalist from mathematics (this is not a name-calling) this solution will completely satisfy. For a formalist, we are free to specify any axioms and get different results with different axioms.

For platonist But such an answer is unsatisfactory, since mathematics is a reflection of the highest reality of the universe, and in fact there is a solution – yes or no. We must simply find suitable axioms where the solution will be.

Of course, you can ask – so what is the problem? Accept the continuum hypothesis (CH) that there are no sets of intermediate cardinality, here is the solution. But CH – this is a crutch, declaring the desired result an axiom. It’s as if in a buggy function that sometimes outputs NaN, we would check for this value and output, for example, 0.0, without fixing the bug itself.

But the search for such an axiom is a very difficult thing, because the axiom should be simple and self-evident – much simpler than itself CH.

Gödel was a Platonist, and wanted to find a solution, so he began a search towards higher powers that go beyond

\aleph_0, \aleph_1, ..., \aleph_\omega = \aleph_{\aleph_0}, ..., \aleph_{\aleph_{\aleph_...}}, ...

and much further, for unattainable powers. The search direction turned out to be very fruitful. One might expect that set theory can be “bent” in any direction by adding different axioms. But, apparently, there is still reality behind the formulas, and the Platonists are right – all the new axioms were strung on top of each other, forming an almost correct thread, going into infinity towards the absolute universe.

On one of the levels that separated small big power (small large cardinals) from large high power (large large cardinals) I almost found a solution to the problem.

Universes

Few scientists can boast of the existence of a universe named after him. I know only three such scientists: Von Neumann, Gräthendieck and Gödel. Moreover, Gödel has two universes. The first has nothing to do with mathematics – it is Gödel’s rotating universewhere everything revolves around everything.

It is an exact solution of general relativity for special conditions. Not only does everything revolve around everything, but there is also a time cycle passing through any point in space-time. It was created as a counterexample to Mach’s principle. But we digress.

Von Neumann Universe

Von Neumann Universe

The first mathematical universe – This is von Neumann’s V. It consists of levels; when adding a new one, we take the set of all subsets. V without an index means the entire universe, that is, the class of all sets (let me remind you that the set of all sets is not a set in all classical set theories. The exception is New Foundations)

Let’s skip Grothendieck’s Universe. Gödel came up with universe L, which also unfolds in stages – but using the principle of definability by formulas. Each next level is assembled from the material of the previous one using possible formulas – combinations of sets obtained at previous levels. The second name of this universe is constructive universe.

What’s the difference between L And V? V says nothing about the properties and quantity of elements and the ability to determine them. IN L but, by construction, it cannot be indiscernibles. Thus

V = L + indiscernibles

If we accept that V=Lthen many problems are immediately solved, for example, CH And GCH – after all, in L No indiscernibles which means both problems are solved automatically.

But alas, as it turned out, V=L castrates the power hierarchy by killing big big power completely, and this is not interesting. Although the border V=L and lies very, very high, above everything, where you can climb with the help of endless

\aleph_{\aleph_{\aleph_...}}

above unattainable powers, above hyper-hyper-unattainable and much higher still. But this is not enough.

Even Gödel himself did not believe that V=L. Alas, the hope turned out to be false. It’s a pity, because L has a number of convenient properties. For example, all sets in it are defined by one or more formulas. We can choose the shortest one and sort all the sets in a unique order. And there are also a countable number of sets in it, since the set of all formulas is countable.

I’m sorry, what?

Skolem’s paradox

What happens? We talked about uncountable sets, went higher and higher, obtained sets of monstrous power, and now it turns out that there are a countable number of all sets?

Yes, in L This is true. This is called Skolem’s paradox. First-order theories fatally fail to control the power of their models. If a theory has an infinite model, then there are models any infinite power.

A model is a set of “chips” that satisfies all the formulas of the theory.

model theory

model theory

But let’s understand the paradox. So, there is a set of continuum that says “Ya countless“We say (in L): come on, show who is in you, like list() in Python. The system thought about it and generated an endless string of formulas – definitions of the sets that are included in it. But there are countless of them! How can this be?

Formally, the paradox is apparent, since the statement that the set is uncountable exists inside theories, and we count the chips outward. These are two different levels of reality. But you are unlikely to be completely satisfied with this answer.

Then you can consider that the definable elements in L – these are “reference” points that are drawn on the curves. There are many more points in total, but these reference points are enough for us.

With real numbers everything is the same – we are always dealing with a countable quantity and usually this is enough if the remaining points (indiscernibles) behave quite peacefully. But this is not always the case; the axiom of choice allows us to construct a subset of real numbers that cannot be measured. Many monsters hide in the shadows among indiscernibleswhere our gaze does not penetrate.

This is why neither Gödel nor most other mathematicians believe in V=Lbecause this hypothesis makes the world too primitive.

Later studies

William Woodin back in the 90s I tried to come closer to solving the problem of the continuum hypothesis. First he began to operate endless logic – logic, where formulas can be of infinite length, adapting it to set theory. He further put forward a plausible hypothesis which, if accepted, solves CH in a negative but definite way:

2^{\aleph_0} = \aleph_2

that is, between the countable power and the continuum power there is exactly one power.

Unfortunately, neither infinite logic nor the hypothesis itself satisfy the principle that an axiom must not be at least more complex than the statement it proves. The continuum hypothesis itself is not difficult to understand, but I understand what Woodin wrote only at the very top. Later, Woodin himself moved away from his hypothesis and went in a different direction.

He tried to expand like this Lto V=L did not truncate the universe at such a low level, but contained almost all large, large powers. To make it clearer, I will give a picture of the structure of the universes:

The black cone is the universe V. For finite sets it coincides with Lbut she is already lower V. It breaks down at the axiom level V=L, killing interesting big big powers. Woodin tries to create an expanded universe Ultimate-L, which is much higher. Border V=L separates “small” large powers from “large” large powers.

Above the red line are “crazy big big” powers. They are so great that the axiom of choice no longer works there A.C. (and therefore GCHhence, CH there is definitely false!). But Woodin’s latest research hints that it’s not just A.C., but also its simpler and much more obvious variants. That is, monsters live there and, perhaps, this whole structure is contradictory.

Woodin hopes that he Ultimate-L will decide CH in a positive way (that is, from changing his opinion)

Why all this?

Note that almost all mathematics does not rise above the first two infinite powers – countable and continuum. With a stretch, when we talk about subsets of real numbers we rise to the level of the second aleph. Even the first unusual (singular) power

\aleph_\omega = \aleph_{\aleph_0}

appears nowhere in ordinary mathematics.

I know of only three cases where the existence of higher powers influenced something outside of set theory:

  • It was recently discovered that some aspects of machine learning depend on the continuum hypothesis

  • some properties of very fast-growing functions are proved using the axioms of the existence of high powers – example

  • unlimited growth period laver tables is proven using an axiom of one of the highest powers – however, I think that the function does not grow fast enough to exceed the limit of provable statements in formal arithmetic – perhaps there is a simple proof that does not use set theory.

So what does all this affect? Woodin said in one of his lectures that there are two options: either someday all this mathematics will be needed in the Theory of Everything, or not. Agree, the first option would be interesting – you can imagine that, for example, a coefficient equal to 1 if the continuum hypothesis is true and 0 otherwise made its way into the formula for the ratio of the mass of the up quark to the down quark.

But it may also be that this whole kitchen has nothing to do with physics. Woodin believes that this is also an optimistic option! Because it means that we the people have seen something (Woodin is obviously a Platonist) that lies outside our universe. On my own behalf, I will add that this is very similar to the Mathematical Universe Hypothesis (MUH), created by Max Tegmark.

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