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Class

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In googology, the classes are a way of grouping positive real numbers by magnitude.[1] Invented by Robert Munafo, the system is inspired by the way humans perceive sizes of groups of objects. Concisely speaking, the class-0 numbers are those greater than or equal to 0 and less than 6, and the class-n numbers are the numbers whose base-10 logarithms are in class n-1. Superclasses use the same concept but with a different execution.

In set theory, which is closely related to some googology, a class refers to an arbitrary collection of sets, even if it's not a set itself, for example the class of all sets or the class of all ordinals.


Class-0 numbers Edit

Class-0 numbers are those that are so simple that they can easily be recognized in a very small amount of time. For most humans, these numbers range from 1 to 6.

Class-1 numbers Edit

Class-1 numbers are small enough to be possible to perceive as a group of objects, but are larger than class-0 numbers. In other words, if x is a class-1 number, it is possible to see x objects in a single scene. Class-1 numbers range from 6 to 106 (one million), as it is difficult, but not impossible, to see a million objects in a single scene.

Class-2 numbers Edit

Class-2 numbers are small enough to be able to be exactly represented in decimal form, but are larger than class-1 numbers. Class-2 numbers begin at \(10^6\) to \(10^{10^6}\) (known as a maximusmillion or millionplex). This is simply a continuation of the pattern that can be seen in the relationship between class-0 and -1 numbers: the logarithm of a class-x number can be represented as a class-(x - 1) number. Googol, therefore, is a number in this class, as 101 digits can be represented in decimal form.

Class-3 numbers Edit

Class-3 numbers can be approximately represented in scientific notation. They range from \(10^{10^6}\) to \(10^{10^{10^6}}\) (known as a millionduplex), following the patterns of classes 0, 1, 2, and 3. Googolplex is a class-3 number.

When represented as a power tower in a computer, a class-3 number x is practically indistinguishable from x + 1.

Class-4 numbers Edit

Class-4 numbers have class-3 base-10 logarithms. They range from \(10^{10^{10^6}}\) to \(10^{10^{10^{10^6}}}\).

When represented as a power tower in a computer, a class-4 number x is practically indistinguishable from 2x.

Higher classes Edit

Class-5 numbers have class-4 base-10 logarithms. They range from \(10^{10^{10^{10^6}}}\) to \(10^{10^{10^{10^{10^6}}}}\).

When represented as a power tower in a computer, a class-5 number x is practically indistinguishable from x2.

In general, class-n numbers are those numbers that are larger than class-n-1 numbers, and whose have class-n-1 base-10 logarithms.

Sources Edit

  1. [1]

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