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W0, W1, and W2.

W0, W1, and W2 are "transfinite ordinals" defined by Joseph Holland and Arnold Beckonson in the order of increasing ordinality. These ordinals are considered to be large. The existence of a ordinal W1 implies the existence of such a ordinal W0, therefore, W1 must be greater than W0. This existence axiom holds true with W2.

I'll also call W0, W1, and W2 as the rank-into-rank ordinals, because of their numbers at the end that asserts as a sort of rank onto it, and that a larger rank is larger in size. Not to be confused with the "rank-into-rank" (See [1])

https://en.wikipedia.org/wiki/Transfinite_number


Definition Of W0.

The first ordinal, W0, is a very large ordinal as previously stated.

Define a set X, containing a boundless (not w) amount of arguments, in which every argument X[n] is a subset. These subsets contain the list of every ordinal, such that all ordinals within X[n] is uncomputable towards all X[i] where i<n.

The ordinal I is the limit of set X.

A trivial result says that omega(1)CK is equivalent to the last argument in X[1], and w is equivalent to the last argument in X[0].

Definition of W1

The second ordinal, W1, is also extremely large. W1 is an extension of W0.

Consider a hierarchy H_n(x,...,x), with any amount of arguments definable. Define I as H_0(1), and the second I as H_0(2). The second W0 is reached with the original definition of W0, modified to be much larger.

"Define a set X, containing a boundless (not w) amount of arguments, in which every argument X[n] is a subset. These subsets contain the list of every ordinal, such that all ordinals within X[n] is uncomputable towards all X[i] where i<n. X[0] is equivalent to W0."

H_0(3) is defined similarly.

Now define H_0(1,0) as the limit of H_0(x), or X = H_0(X). The rest of the definition of H_0 is similar to Veblen function. See: https://en.wikipedia.org/wiki/Veblen_function for more details.

A limit H_1(0) is defined as the limit of H_0. H_1(1) is reached with the sentences: With the use of the previous function H_0, enumerate all steps from it to get from H_0 to H_1(0). Instead of H_0 in the previous sentence, H_1(0) is used, and instead of H_1(0), H_1(1) is used. After following the steps, the result is equivalent to H_1(1).

This works similarly for the rest.

In general, all of H_n is defined as the modification of H_1:

"A limit H_n(0) is defined as the limit of H_n-1. H_n(x) is reached with the sentences: With the use of the previous function H_n-1, enumerate all steps from it to get from H_n-1 to H_n(0). Instead of H_n-1 in the previous sentence, H_n(x-1) is used, and instead of H_n(0), H_n(x) is used. After following the steps, the result is equivalent to H_n(x)."

Now let's define H_(1,0)(0), H_(1,0)(0) is defined as the limit for H_x, that is x=H_x. Continue and define H_(1,1)(0), by enumerating all steps from H_0(0) to H_x. Continue with H_(1,2)(0), and define X=H_(2,0)(0) as X=H_(1,X)(0), continue with the same pattern as the Veblen function.

H_(1,0,0,0,0,0,0,0,...X...,0,0,0)(M)=H_(1)(0)(X)

H_(1)(1)(X) is achieved by enumerating all steps from H_0 to H_(1)(0)(X). Continue with the same pattern as the Veblen function.

H_(1)(0)(0)(X)=H_(1,0,0,0,0,0,0,0,...X...,0,0,0)(0)(X).

Continuing with H(1)(0)(0)(0)(X), and beyond, we get W1. W1 is defined as H(1)(0)...W1...(0)(0)(W1,W1)

Although it seems that W1 is around the value of W0 from a certain relative perspective, W1 is obviously much, much greater than W0.

Definition of W2

W2 has a relatively simple definition. It is defined as the first ordinal such that it is equivalent to the limit ordinal in H+"There Exists Recursive Subsets".

H+"There Exists Recursive Subsets" is equivalent to H with the extra properties:

Subsets and Sub-Subsets, or even Sub-Sub-Sub-...with n Subs...-Subsets:

(((1,0),(1,0)),((1,0),(1,0))) or ((1,0),(1,0)).

Sub-semisets and Sub-Sub-semisets, or even Sub-Sub-Sub-...with n Subs...-Sub-semisets

((1)(0)(0)(0)(0))

The existence of nesting as indices.

H_((2)_(n))=H_((2),...H_((2),(n-1)),...,H_(2))

H_((2)_(2)_(n))=H_((2)_(1)_(H_((2),...H_((2),(2),(n-1)),...,H_(2)))

W2 >> W1 >> W0.