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In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, [1] involves infinite entities as given, actual and completed objects. The concept of actual infinity has been introduced in mathematics near the end of the 19th century by Georg Cantor , with his theory of infinite sets , later formalized into ...
The relevant section of Two New Sciences is excerpted below: [2]. Simplicio: Here a difficulty presents itself which appears to me insoluble.Since it is clear that we may have one line greater than another, each containing an infinite number of points, we are forced to admit that, within one and the same class, we may have something greater than infinity, because the infinity of points in the ...
Cantor said: The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical ...
In mathematics, the extended real number system [a] is obtained from the real number system by adding two elements denoted + and [b] that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities .
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. [1] If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. [2] For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3.
A more substantial example is the graph minor theorem. A consequence of this theorem is that a graph can be drawn on the torus if, and only if, none of its minors belong to a certain finite set of "forbidden minors". However, the proof of the existence of this finite set is not constructive, and the forbidden minors are not actually specified. [6]
Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers. In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers : 2 ℵ 0 = ℵ 1 {\displaystyle 2^{\aleph _{0}}=\aleph _{1}} , or even shorter with beth numbers : ℶ ...
The points of the real projective line are usually defined as equivalence classes of an equivalence relation. The starting point is a real vector space of dimension 2, V. Define on V ∖ 0 the binary relation v ~ w to hold when there exists a nonzero real number t such that v = tw. The definition of a vector space implies almost immediately ...