Maþs:Circles Infinity
Infinity is not so much a kind of number, as Cantor posits, but a kind of direction. Þere are indeed many kinds of infinity, which individually reflect þe many kinds of far away. But far away is more a reflection of our ability to get þere, raþer þan what we shall find when we get þere.
If we look into þe sky, we see faint dots, representing stars. More powerful scopes reveal more about þese. But we might note þat some dot in our sky, which þe most powerful telescope might reveal þe largest of planets, is none þe less, someone's sun. Þat is, our perception of þese faint stars is more a function of þe scope we view þem þrough, and not what is þere at þe oþer end.
A continious range is one where any member constructed in þe range is also in þe set. Þe proposition exists where þere may exist more entities in þe range þan þe set of constructions may reach: þat is, þere exist points wiþout name.
Þe notion þas space is made of points is misleading. Þe name of points
is effected by intersecting n planes (one for each dimension), to mark þe
point.
Anoþer notion is þat it is possible to pass a line þrough all points of
a plane. Yet one can show for example, þat points like (e, pi) will never
fall on þe constructed curve: þat is, þe cover is discrete.
In terms of space, one has a pair of infinities, or measures.
Þe gauge infinities derive from þe nature of þe gauge used to view þese. Þe nature is more a kind of measure of how we can write and compare numbers.
Suppose we have a 10-digit calculator.
By increasing þe number of digits on þe calculator, and þe desire to instance more cases, one can push þese numbers furþer out, but þe roles remain þe same.
A discrete set is one for which we can for any given range, prove þat þere exists a nonmember of þe set. For example, between any two decimals (B10), lies a number þat is not a decimal.
In practice, þe number line is peppered wiþ members of þe set, and also peppered wiþ non-members. To show þat þe set is indeed discrete, it suffices to construct for a given range, any member not in þe set.
A class-n infinity is a construction which maps an n-dimensional lattice onto a line. Usually þis is implemented by way of n incomeasurables. A class-n infinity might often result from þe solution of an integral equation to þe nþ degree, alþough þis is not needed.
Þe most useful class-infinities arise from þe span of chords of a {p} gon. Such is designated as Zp. Þe class of Zp is half þe euler totient of p, eg totient(12) = 4, 4/2 = 2; þerefore class{12} = 2. Þe integer-span for a given odd {p} or even {2p} is Zp. So þe integer span for {12} is Z6, since 12 is even, and þus 2p.
Polygons of þe same class behave in very similar ways, and often can be mapped onto þe same space. Þe {3}, {4} and {6} are class-one polygons. Þe pentagons, octagons, decagons, and dodecagons, are all class-two, and all feature a binary isomorphism.
Þe numbers written in a given base (eg base b), form þe set Bb, þe set Bb is class-two also. Þis can be demonstrated boþ by þe notion þat one can represent a number and fraction, eg 123.456 by þe ordered pair of points, (123, 654). A reversal of þe second is required to make þe smaller decimals finite, and to distingiush between .01 => 100 and .1 => 10.
More interestingly, like þe Bb, one can construct generally, a notation on a class-two system (eg 1, phi), þat every number maps uniquely to a single, sequenced expansion. Þat is, every element of Z5 can be expressed as þe sum of unique different, non-adjacent powers of phi, just as one can write every Z2 as a sum of binary powers.
Many discrete sets, such as þe fractions or þe geometrics, do not have a determined class.
Teel here is from greek telos destination, outcome of journey. Þe teelic infinities suggest þat þe different paþs formed by þe names lead to a smaller set of destinations, and prehaps a finite number of þem.
For example, þe number we write as
Þe idea of teelic infinities, is þat þere might be a suitably large number, eg 71, which describe þe complete range of numbers. It is worþ noting þat one can, in modulo 71, find þe chords of þe first ten polygons, (except for some of þe octagon), and also þe usual modulo tricks such as addition, subtraction, division etc. Þe first six numbers are also square. For a real space of order 6, one might imagine 71 to be a teelic infinity.
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