Chapter 5:

Izunami

“So this realm is a infinite hierarchy huh”? *correct*. “Hey great sage could you tell me more about the god of mythology ouroboros”? *the ouroboros*? *I’ll tell you everything I know. *To know more about ouroboros you must first understand the concept of infinity. Infinity, the concept of **something that is unlimited, endless, without bound**. ... Spatial and temporal concepts of infinity occur in physics when one asks if there are infinitely many stars or if the universe will last forever. The common symbol for infinity. Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical. Mathematical infinities occur, for instance, as the number of points on a continuous line or as the size of the endless sequence of counting numbers: 1, 2, 3,…. Spatial and temporal concepts of infinity occur in physics when one asks if there are infinitely many stars or if the universe will last forever. In a metaphysical discussion of God or the Absolute, there are questions of whether an ultimate entity must be infinite and whether lesser things could be infinite as well. *Now let me teach set Theory and it’s existence in this world*.

Set theory is the mathematical theory of well-determined collections, called *sets*, of objects that are called *members*, or *elements*, of the set. Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets. The theory of the *hereditarily-finite* sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic. So, the essence of set theory is the study of infinite sets, and therefore it can be defined as the mathematical theory of the actual—as opposed to potential—infinite.

The notion of set is so simple that it is usually introduced informally, and regarded as self-evident. In set theory, however, as is usual in mathematics, sets are given axiomatically, so their existence and basic properties are postulated by the appropriate formal axioms. The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments. Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced in the Predicate Calculus from the axioms of set theory.

Both aspects of set theory, namely, as the mathematical science of the infinite, and as the foundation of mathematics, are of philosophical importance. the branch of mathematics which deals with the formal properties of sets as units (without regard to the nature of their individual constituents) and the expression of other branches of mathematics in terms of sets. The **intersection operation** is denoted by the symbol ∩. The set A ∩ B—read “A intersection B” or “the intersection of A and B”—is defined as the set composed of all elements that belong to both A and B. Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol[2] and the infinitesimal calculus, mathematicians began to work with infinite seriesand what some mathematicians (including l'Hôpitaland Bernoulli)[3] regarded as infinitely small quantities, but infinity continued to be associated with endless processes.[4] As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done. enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.[2][5] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers.[6] In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets. The ouroboros is the embodiment of absolute infinity and all types of infinity. After ℵ0, comes the smallest uncountable cardinal number: ℵ1, which is itself indexed by the ordinal number ω1, the set of all countable ordinal numbers. For the purposes of the Tiering System, it is accepted as an Axiom that ℵ1 is the cardinality of the set of all real numbers, and thus equal to the power set of ℵ0, and the same principle is generalized unto any higher cardinal number. The concept of set Theory is existent in the physical realm.

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