Types of infinities and brain stem

    This article is a continuation of an article about huge numbers . But now we will go even further - in the infinity of infinity.

    For this we need ZFC - the theory of sets Zermelo, Frenkel + Choice. Choice is the axiom of choice, the most controversial axiom of set theory. She deserves a separate article. It is assumed that you know what the "power" of the set is. If not, then google, for sure this is stated better than I can. Here I will only remind some

    Known facts

    • The power of a set of integers is denoted by $ \ aleph_0 $. This is the first infinite power; such sets are called countable.
    • The power of any infinite subset of integers is simple, even, etc. - also countable.
    • The set of rational numbers, that is, the fractions p / q, is also countable; they can be passed by a snake.
    • For any power, there is a powerset operation - the set of all subsets that creates more power than the original one. Sometimes this operation is referred to as raising a two to a power, i.e.$ powerset (\ aleph_0) = 2 ^ {\ aleph_0} = \ aleph_c $. powerset from the calculated power is the power of the continuum.
    • Continuum power is possessed by: finite and infinite segments, planar and volumetric figures, and even n-dimensional spaces as a whole
    • For ordinary math, the following power, $ powerset (\ aleph_c) $ practically not needed, usually all work happens with countable sets and continuum power sets


    Little known facts

    In ZFC, not all collections of elements can be sets. There are collections so wide that it is impossible to allow them to be sets; paradoxes arise. In particular, the “ set of all sets ” is not a set. However, there are set theories where such sets are allowed.

    Further. Set Theory ... What Objects? Numbers? An apple? Oranges? Oddly enough, ZFC does not need any objects. Take the empty set {} and agree that it means 0. 1 denote by {{}} the deuce as {{{}}} and so on. {5,2} is {{{{{{{}}}}}}, {{{}}}}. Using integers, we can create real ones, and collections of real ones can create any shapes.

    So set theory is ... how to say ... hollow theory. This theory is about nothing. More precisely, about how you cannest (i.e., nest in each other) braces.

    The only operation defined in set theory is$ \ in $- a symbol of belonging. But what about unification, exclusion, equality, etc.? These are all macros, for example:

    $ (A = B) \ equiv \ forall x ((x \ in A) = (x \ in B)) $

    That is, in translation into Russian, two sets are considered identical when, when testing any element for belonging to them, we will get the same results. The

    sets are not ordered, but this can be fixed: let the ordered pair (p, v) be {{p} , {p, v}}. Inelegant from the point of view of the programmer, but enough for a mathematician. Now the set of all param-value pairs sets a function, which is now also set! Et voila! all mathematical analysis, which works at the level of second-order languages , since it speaks not of the existence of numbers , but of the existence of functions , collapses into a first-order language!

    Thus, set theory is a poor theory without objects and with one relation icon, which has absolutely monstrous power - without any new assumptions, it generates from itself formal arithmetic, real numbers, analysis, geometry and much more. This is a kind of TOE mathematics.

    Continuum Hypothesis - CH

    Is there power between $ \ aleph_0 $ and $ 2 ^ {\ aleph_0} $? Cantor could not solve this problem; the “king of mathematicians” Hilbert praised its importance, but only later it was proved that this hypothesis can neither be proved nor disproved. She is independent of ZFC.

    This means that you can create two different maths: one with ZFC + CH, the other with ZFC + (not CH). In fact, even more than two. Suppose we reject CH, that is, we will believe that between$ \ aleph_0 $ and $ 2 ^ {\ aleph_0} $there is still power. How many can there be? One, two? Godel believed that only one. But, as it turned out, the assumption that there are 2, 17, 19393493 of them does not lead to contradictions. Any number, but not infinite!

    When in formal arithmetic we come across an unprovable statement, for certain reasons we know that, nevertheless, this statement, although not provable, is actually either true or false. In set theory this does not work, we really get different mathematicians. How to relate to this? There are three philosophical approaches:

    Formalism: why, in fact, be surprised? We set the rules of the game of symbols, different rules - a different result. No need to look for a problem where it does not exist

    Platonism:But how then to explain that completely different theories, for example ZFC and New Foundations, built on completely different principles, almost always give the same result? Does this mean that behind the formulas is some kind of reality that we are studying? This view was held, for example, by Godel

    Multiverse: We can have many axiomatics, sometimes giving the same result, sometimes not. We must perceive the picture as a whole - if color is associated with different systems of axioms, then the colored tree of effects is mathematics. If something is true everywhere - it is white, but there are also colored branches.

    Higher and higher.

    In the future, for simplicity, we will accept the continuum hypothesis, i.e. $ powerset (\ aleph_0) = \ aleph_1 = \ aleph_c $- it is very comfortable. In fact, we will also accept the stronger axiom, the generalized continuum hypothesis that between x and powerset (x) there are never intermediate powers. Now we iterate the powerset and everything is simple:

    $ \ aleph_0 \ rightarrow \ aleph_1 \ rightarrow \ aleph_2 \ rightarrow ... \ rightarrow \ aleph_ {36463634} \ rightarrow $

    How far can we go? After an infinite number of iterations, we get to$ \ aleph_ \ omega $- infinite power in order! By the way, its existence was not obvious to Cantor. But a second! After all, the powerset function is always defined, therefore$ \ aleph_ \ omega $ can't be the last!

    $ \ aleph_ \ omega \ rightarrow \ aleph _ {\ omega + 1} \ rightarrow \ aleph _ {\ omega + 2} \ rightarrow ... $

    To obtain $ \ aleph _ {\ omega + 3} $it is necessary to repeat powerset infinity and three more times . Have you already started to demolish the roof? It's only the beginning. Because again, having iterated powerset an infinite number of times, we get to$ \ aleph _ {\ omega + \ omega} = \ aleph _ {\ omega2} $, after which, naturally, $ \ aleph _ {\ omega2 + 1} $

    Having reached infinity an infinite number of times , we obtain the index$ \ omega ^ 2 $. How do you like this power, for example:$ \ aleph _ {\ omega ^ 3 + \ omega ^ 2 + \ omega4 + 48745} $? While we iterated powerset over the list of ordinals, here are the initial ordinals:


    but there are many, many more. So we will skip it all right away and do it

    Big step right away

    Attention! What is written below may be dangerous for your brain! We iterated powerset a countable number of times, but do not we wave to the continuum ? Honestly, I myself am a little bit sausage from the fact that the cycle can be performed a continuum of times, but set theory requires existence

    $ \ aleph _ {\ aleph_1} $

    Next we will go faster:

    $ \ aleph _ {\ aleph_1} \ rightarrow \ aleph _ {\ aleph_2} ... \ rightarrow \ aleph _ {\ aleph_ \ omega} = \ aleph _ {\ aleph_ \ aleph} \ rightarrow $

    The last Alef has an index of zero, but the local latex does not allow it to be put - there are too many levels. But the main thing is that you understood, no matter how much new monstrous power we would create, we can say - yeah, this is just a repeater , and put this whole construction in the form of an index to the new Aleph. Now the capacities are growing like a snowball, we can not be stopped, the pyramid of Alephs is higher and we can create any power ... Or not?

    Unreachable power

    What if there is power so big $ \ theta $that no matter how we try to reach it “from below”, building structures from Alephs, we will not achieve it? It turns out that the existence of such power is independent of ZFC. You can accept its existence or not.

    I hear the whisper of Occam's razor ... No, no. Mathematicians adhere to the opposite principle, which is called ontological maximalism - let everything that is possible exist. But there are at least two more reasons why I want to accept this hypothesis.

    • Firstly, this is not the first unattainable power that we know. First ... this is the familiar counting power. Oddly enough, it has all the properties unattainable - it's just not customary to call it that:
    • There is no way to get infinite power “from below” - neither adding elements a finite number of times, nor iterating powerset () a finite number of times, using finite sets for seeding, you will not get infinity. To get infinity, you must already have it somewhere.
    • The existence of infinite power is introduced by a special axiom - the axiom of infinity. Without it, the existence of infinite power is unprovable.

    Second: if we reject the axiom of infinity, we get FinSet, a simple toy set theory with finite sets. Let's write down all these sets (the so-called theory model )

    {{{}}, {}}
    {{{{}}}, {{}}}
    {{{{ {}}}, {{}}}
    {{{{}}}, {{}}, {}}

    And we get ... an infinite set of finite sets ... That is, the modelThe theory of finite sets is infinite, and plays the role of the “set of all sets” in it. Maybe this will help to understand why the theory cannot talk about the “set of all sets” - such a set always exists as a model outside the theory and has other properties than sets inside. You cannot add the infinite to the theory of finite sets.

    And yes$ \ theta $it is the “multitude of all multitudes" of ZFC theory. In this video, at the end it is very beautifully said about unattainable power, but we have to go on.

    Even further.

    Of course, we can go further by iterating $ powerset (\ theta) $. After going through all the steps described above, building huge repeater towers, we again run into an unattainable cardinal (but now we do not need new axioms, with the axiom of the existence of unattainable power that we just added, this has become provable). And again and again.

    $ \ theta_0 \ rightarrow \ theta_1 \ rightarrow \ theta_2 \ rightarrow ... \ rightarrow \ theta_ {36463634} \ rightarrow $

    Note that now the arrow does not make sense to us like executing the Powerset () function, but GetNextInaccessible (). Otherwise, everything looks very similar, we have:

    $ \ theta _ {\ theta_1} \ rightarrow \ theta _ {\ theta_2} ... \ rightarrow \ theta _ {\ theta_ \ omega} \ rightarrow \ theta _ {\ theta_ \ theta} \ rightarrow $

    Now then we will definitely achieve anything ... Or not?

    Hierarchy of large capacities.

    Yes, with GetNextInaccessible we run into hyper-unattainable power. Its existence requires one more axiom. There are hyper-hyper-unreachable powers. Etc. But there are other ways to determine power , not only through unattainability:

    As a rule, behind each link there is a whole endless hierarchy with an arbitrary number of prefixes of hyper- and repeaters. However, the total number of formulas that determine unattainable cardinals is not that big - because the number of formulas is countable !!! Therefore, sooner or later they will end. Where they end, a red line is drawn. Everything below this line is defined more unsteadily, albeit formally.

    The red line itself marks the end of Gödel’s universe (but do not forget that Gödel created TWO different universes) - the universe of sets constructed from below using formulas. Capacities above the red line are called hmm, "small", and below - large: The

    main idea in them is that the universe of sets becomes so large that it begins to repeat itself in different senses. Each line, as always, requires a separate axiom, and several. And more interestingly, all this is not as useless as you might think. For example, the strongest axiom (rank-into-rank), in the very bottom line, is needed to prove the fact about the tablets .

    Below is a survey, the last choice is decrypted here .

    Only registered users can participate in the survey. Please come in.

    Which point of view is closer to you:

    • 25.8% Formalism 72
    • 10.4% Platonism 29
    • 19% Multiverse 53
    • 44.6% Complex-difficult! 124

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