Infinity Topics

• johann benedict
• infinity symbol
• riemann sphere
• arithmetic operations
• projective geometry
• plane geometry
• leading to the one
• projective line
• meromorphic functions
• infinite sets
• algebraic properties
• tangible example
• zfc
• infinite time
• rock carvings
• higher dimensions
• topological space
• real numbers
• ouroboros
• fraenkel

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Infinity, Infinity

Infinity is related to limit, aleph numbers, class in set theory, Dedekind-infinite sets, large cardinal, Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC).

Again, one can imagine walking along its surface forever. However, this explanation is not plausible, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Mbius and Johann Benedict Listing discovered the Mbius strip in 1858.

For instance, it has been found in Tibetan rock carvings, and the ouroboros, or infinity snake, is often depicted in this shape.

Obviously, this action would cause the hourglass to take infinite time to empty thus presenting a tangible example of infinity. The invention of the hourglass predates the existence of the infinity symbol allowing this theory to be plausible.

Points labeled \infty and -\infty can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat \infty and -\infty as the same, leading to the one-point compactification of the real numbers, which is the real projective line. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions.

Arithmetic operations similar to those given below for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely z/0 = \infty for any complex number z . In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of \infty at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Mbius transformations.

An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.

Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite.

Source: Wikipedia > Infinity