ALEXIS KARPOUZOS
INFINITY AND THE MIND Mathematics and Philosophy of set theory The Reflection Principle
What is the Ω (Omega) (1) ? Ω is the absolute infinity that is not limited from any constraint. Equivalently, the Ω is what human beings are thinking when they speak for the notion of infinity. The nature of infinity does not allow us to rationally, objectively, and completely understand the Absolute. Therefore, the Absolute can be comprehended only subjectively. In return, this leads to an identification of our shelf with the Absolute, which is equivalent with abandonment of our personal identity, and to a deconstruction of our egoistic Ego. To this point, a brief reference on the Reflection Principle will be provided, the way it is described in Set Theory. In particular, according to this principle, any property that characterizes the absolute set of Cantor V (the Absolute Infinite, the class of all sets) characterizes also any set. The equivalent philosophical version of the principle could be: “Any property that can characterize the Absolute is also a property of a smaller entity. The motivation behind the Reflection Principle has its roots on the idea that the Absolute has to be totally inconceivable. Therefore, if there exists a property A that it is only a property of the Absolute, then we could conceive it as the “unique thing with the property A”. But, the Reflection Principle does not allow such a thing. In particular, it states that any time that someone strongly thinks a very powerful property A, then the first thing that he can find that is characterized by this property, will not be the Absolute, but some smaller rational thought that just reflects this part of the Absolute which is expressed by the property A.
……..The Invisible The Reflection Principle points out the necessary vertical order: finite < transfinite < infinite. Those who study set theory see V as a fan that opens and expands above. The several levels of this fan are called individual universes or Va. Therefore, at the base of the cone (V) there is the empty set (V0). Moving upwards, the level V6 includes almost googol^200 elements, that is 10^100^200 since a googol equals 10^100.(2). The set Vω contains any finite figure that a human being can conceive. In the set Vω+ ω are objectively included all the standard mathematic theories. Finally, the set Vθ is the universe of the classic Set Theory. Apart from the above, there exists ρ, κ, λ until finally we reach Va. Nowadays; those who are occupied with Set Theory are occupied with a universe of a size (Va) more or less. Upwards and downwards, like a pick of a cloud, stands proudly the absolute Ω, the Absolute Infinite. From the above, arises that the complexity of any set x can be specified by an ordinal number that is called class of x. In general, the set Va is the set that includes the sum of all the sets with a class smaller than a, and it is the eminent irrational.
The Exuberant Principle (3) The question that arises is: What is the relation of the physical universe U with the universe V? Everyday experience opposes to any suspicion that U is very large. In traditional (classical) philosophy, there exists the plethoric principle or Exuberant Principle that suggests that the physical universe is equally rich with the set- theoretical universe of genuine platonic ideas. To the degree that only physical frame can be encoded into a set, it is expected that V can be as large as U or even larger. The Exuberant Principle requires that U is equally large or larger than V leading to the conclusion that the sets U and V are equally large.
……..The Cyclical principle of the Non-Transitivity…
It was previously stated that the vertical order of the infinite sets and the linear order of finite-transfinite-Absolute infinite awes. At least up to a point, both the finite and the transfinite are comprehensible notions, on the contrary the Absolute infinite is (or has to be) entirely incomprehensible and unspeakable. But the exuberant Principle suggests that the sets U and V are equally large. If it is so then Ω is larger/stronger than V, and V is equal to U. Therefore, according to the standard transitivity assumption, U is smaller and weaker than Ω. This is the simple vertical order from upwards to downwards. Though, what can be said if the above can be seen as a non-transitive relation? Meaning that, they could be understood by the unconventional perspective of a cyclical order, an non-transitive relation that does not follow common sense. A note on the game rock-scissors-paper combined with the discovery of the nontransitive dice can be helpful. That would mean that, according to the nontransitive relation, V is equally large and strong as Ω.
…The Problem of One/Multiple… This may be the way to solve the Problem of One/Multiple, which occupies both the philosophical and the Set Theory world. In short terms, can it be implied that all the different absolutes, i.e. God, Truth, Beauty, Class of All Sets, Noosphere, Good etc., are different facets of the unique and ultimate One? (4) Indeed this problem is analogous to Set Theory. In Set Theory there is a distinction between two different absolutes: a) The infinite that can be represented with Ω and b) the universe that can be represented with V. On the one hand, the Ω can be considered as the class of all ordinals, on the other hand the V is the class of all sets. Since any ordinal can be represented as a set, then in a simpler level, V can be considered larger than Ω. Though, the desire is to identify all the absolutes. Therefore, it can be conjectured that any set is also represented by an ordinal and thus according with the approach of cardinality this results to V= Ω. That is that the Absolute Infinite is as equally large as the Universe. The statement that Ω=V means that there exists one-to-one correspondence between the class of ordinal numbers and the class of sets. But since such a correspondence is by itself a proper class it is difficult to be sure it exists. (5)
…and the Axiom of Global Choice (6) If the infinites are considered to be of an absolute and vertical order, then indeed they cannot be assigned one-to-one, because otherwise there would be a proper class set. But, if they are considered as a dynamic and a nontransitive relation then this assignment can be done with a non-proper class outcome. Careful though! This can be achieved only through Conscience, which participates to this unification as a Global Physical Constant. This is the way that the assumption of the Set Theory that such a correspondence exists can be verified. This is the only way that the Axiom of Global choice can be verified. The aforementioned Axiom is an outcome from the stronger Axiom of Limitation of Size (7) which in a way is related with continuum hypothesis problem of Cantor.(8)
………………………………………………………………………………………….. 1. Omega is called the non imaginary Absolute Infinite. 2. A googol is the large number 10^100. 3. This Axiom suggests that the physical universe is equally rich with the settheoretical universe of the platonic Ideas. 4. Noosphere . The same way the physical objects exist and move in a physical space that is called three-dimensional visible universe, the same way the thoughts and consciences exist and move in the multi-dimensional mental space-time that is called Noosphere. 5. In Set Theory the term “class” declares a collection or a manifold of any kind. A class either it can or it cannot be unified in a set. If it cannot then it is called a proper class. Therefore V is a proper class that cannot be considered as a unity. 6. The hypothesis that there exists one-to-one correspondence between the class of ordinal numbers and the class of the sets. 7. In class theories, the axiom of limitation of size says that for any class C, if and only if it can be mapped onto the class V. 8. This hypothesis states that there is no set whose cardinality is strictly between that of the integers and the real numbers.