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Infinity
“ | Only two things are infinite, the universe and human stupidity, and I'm not sure about the former. | „ |
~ Albert Einstein |
“ | To infinity and beyond! | „ |
~ Buzz Lightyear |
Introduction
Infinity is that which is boundless, endless, or larger than any number. It is often denoted by the infinity symbol.
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[ and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. 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. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes. 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. 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 mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the Universe is infinite is an open question.
Infinity Contradictions
Normally in fiction there will be statements of something being infinite in size but directly contradicted by a statement of there being an edge, center, beginning, etcetera.
Edge
Infinity cannot have an edge by how we define it, normally for a universe with context an edge can refer to the observable universe since this is how we observe it. If a cosmic source however is stating they are at the edge of the universe then it is therefore not an infinite universe nor referring to the observable universe.
Center
Infinity cannot have a unique center point. While one may argue that it can have infinite centers, that in itself is the issue. In real life, because of the effects of the big bang everywhere can be the center of the universe because of any given objects perspective any location can be the center of the universe from their perspective it is expanding away from them and thus there is no real center to the universe. Thus, if there's one universal spatial center that's acknowledged by several characters the infinite centers argument does not work.
Beginning
Infinity cannot have a beginning as it is a borderless concept, one may argue that in some universes treated as infinite the big bang serves as a "beginning" but the big bang as a concept is something that expanded everywhere instantly and thus there cannot be an event before the big bang. Another argument would be the set of the infinite set of natural numbers (I.E. 1, 2, 3, 4, and so on), however even in these sets by all technicalities the "beginning" point is n, which isn't directly defined and can be any real number, thus it itself is not a true beginning border.
How we treat infinity on this wiki
Cosmology
This is how we treat infinite statements in regards to cosmology.
Universe
In a cosmology if a universe is stated to be endless, or goes on forever it is not considered to be "infinite" in size, it is considered to follow the model of our real life universe, where the universe is infinitely expanding thus giving the illusion that it's "infinite" to us. If the cosmology is stated to be infinite by a reliable source (check the Statements page for further info), then it can be used for a High 3-A+ universe. However, if there are contradictions such as the universe having an edge, center, beginning, etcetera, then it will be taken back to being an infinitely expanding universe. Along with this, if we get a visual representation of the universe with a clear end then we will not consider it infinite in size. Some cases where this won't be the case is if a far more cosmic source is stating something is infinite while another source with far less reliability states they're at an edge for the source very likely would not be aware of the size or if they're truly at the end. Along with this, the edge statement has to be noted by a cosmic source as the universe we observe is the observable universe which itself is not infinite even in an infinite-sized universe.
Multiverse
In a cosmology if it's stated there's countless or endless universes this is not enough to denote an infinite size, and it will simply be considered a 2-A multiverse. If the multiverse is stated to be infinite in size or hold infinite universes by a reliable source, then it can be used for a 2-A+ sized cosmology. However, if there are contradictions, such as there being a prime earth, original starting point for the universe, an object that holds all of it that is very clearly a finite sized place, etcetera, then it will be taken back to being a 2-A sized cosmology. The reason for this is that normally the definition of infinite that we're looking for to define 2-A+ is a state with no beginning or end, as infinity is boundless, thus it can't be bordered with a beginning nor end, for this reason it can't exist in the real world. Most authors will take infinite as just a big number, rather then a concept, which means they are going for the ad-infinitum interpretation where a finite number goes on forever rather then there being no beginning or end.
Tiering
Countable Infinity
A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time. For example, the set of integers {0,1,−1,2,−2,3,−3,…} is clearly infinite. However, as suggested by the above arrangement, we can count off all the integers. Counting off every integer will take forever. But, if you specify any integer, say −10,234,872,306, we will get to this integer in the counting process in a finite amount of time. Sometimes, we can just use the term “countable” to mean countably infinite. But to stress that we are excluding finite sets, we usually use the term countably infinite. Countably infinite is in contrast to uncountable, which describes a set that is so large, it cannot be counted even if we kept counting forever. This is essentially an ad infinitium line and is a technical infinitely finite increasing counter. A countable infinite increase his never considered anything beyond granting a user High 3-A+ in tiering.
Uncountable Infinity
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. It is impossible to visualized an uncountable set as there is no true starting point or end point for it. This is truly an infinitely sized set. One that is granted an uncountably increase in power is usually brought up in the tiering chain based off how many uncountably infinite boosts they've received.
Conclusion
Infinity is a very confusing concept that is hard to grasp as it's something that cannot exist in our real world. There are some takeaways that should be noted for a verse.
Firstly, even if a universe or multiverse is stated to be infinite or contain infinite universes, always pay attention to what it contains, if there's ever a starting point, the definition of infinity being used is normally an endless set or an ad infinitium set which is not 2-A+.
Secondly, no amount of stacks of countable infinities can reach higher infinites, as countable infinites can fit within countable infinites no matter how many countable infinites there are.