User blog:Rgetar/Higher weakly inaccessible and weakly Mahlo cardinals

I tried to understand what is Mahlo cardinal using Wikipedia, Cantor's Attic, Googology Wiki, but I failed.

Definition of Mahlo cardinal
On these sites Mahlo cardinal is explained similarly: a cardinal κ is called weakly Mahlo if κ is weakly inaccessible and the set of weakly inaccessible cardinals less than κ is stationary in κ.

And what is "stationaty set"?

Stationary set in cardinal k of uncountable cofinality is set S such as S ⊆ k and S intersects every club set in k.

And what is "club set"?

Club set in k is closed set in k and unbounded set in k.

And what is "closed set" and "unbounded set"?

Closed set in limit ordinal α is set containing all its limits less than α.

Unbounded set in limit ordinal α is set containing element larger than any ordinal less than α.

Research of inaccessible cardinals
Since I did not understand the sence of this definition, I did my own research of inaccessible cardinals. Let we have a function
 * f(x) = Ωx

where Ωx is x-th infinite cardinal. Then weakly inaccessible cardinal is Ωx such as x is limit ordinal and Ωx is regular, that is
 * cof(f(x)) = f(x)
 * cof(Ωx) = Ωx

If cof(f(x)) = f(x) then
 * x = f(x)

Higher inaccessibles
Then I thought that we can get higher inaccessibles similarly, but instead of Ωx we should take another strictly increasing function f(x), containing its limits:
 * if y > x then f(y) > f(x)
 * f(sup{x[n]}) = sup{f(x[n])}

(In this comment Denis Maxudov wrote that such functions are called normal functions).

Then we get some cardinals f(x) such as x is limit ordinal and f(x) is regular, that is cof(f(x)) = f(x).

For example, let I*(2, x) is function enumerating weakly inaccessible cardinals and their limits. Then weakly 2-inaccessible cardinal is regular I*(2, x) such as x is limit ordinal.

Let I*(3, x) is function enumerating weakly 2-inaccessible cardinals and their limits. Then weakly 3-inaccessible cardinal is regular I*(3, x) such as x is limit ordinal.

And so on.

Ωx can be designated as I*(1, x).

Let I*(ω, 0) is limit of I*(1, 0), I*(2, 0), I*(3, 0), ...; I*(ω, x + 1) is limit of least I*(1, y1) > I*(ω, x), least I*(2, y2) > I*(ω, x), least I*(3, y3) > I*(ω, x), ...; for limit x I*(ω, x) is sup{I*(ω, y) | y < x}.

Generally,


 * I*(0, x) = x
 * for non-limit x I*(y + 1, x) is least regular I*(y, z) > I*(y + 1, w) for any w < x, where z is limit ordinal
 * for non-limit x and limit y I*(y, x) is supremum of least I*(v, z) > I*(y, w) for any w < x, where v < y
 * for limit x and y > 0 I*(y, x) is supremum of I*(y, w), where w < x

So, if y and x are not limit ordinals, then I*(y, x) is regular.

Fixed points
Let x[n] are fixed points of a normal function f(x):
 * f(x[n]) = x[n]

and x is their limit:
 * sup{x[n]} = x

Since
 * f(sup{x[n]}) = sup{f(x[n])}

then
 * f(x) = sup{x[n]} = x

That is limit of fixed points of a noraml function is again fixed point of this function.

Inaccessibles are fixed points of Ωx, and every I*(2, x) is inaccessible or limit of inaccessibles. So, every I*(2, x) is omega number.

Similarly, every I*(y, x) is I*(z, ...) number for any z < y. (For limit y and non-limit x I*(y, x) is limit of I*(z, ...) for any z < y).

For non-limit x I*(y + 1, x) is y-inaccessible (and regular; "0-inaccessibles" are regular omega numbers), so it is regular I*(z + 1, ...) for z < y.

So, any y-inaccessible is z-inaccessible for any z < y, and any regular limit of y-inaccessibles is y-inaccessible.

R
Let R is functional operator, which input and output are normal function:
 * for non-limit y R(f(x); y) is least regular f(x) > R(f(x); z) such as x is limit ordinal for any z < y
 * for limit y R(f(x); y) = sup{R(f(x); z)|z < y}

Then we can add functional power parameter n:
 * R(f(x); n; y) = Rn(f(x); y)

that is
 * R(f(x); 0; y) = f(y)
 * R(f(x); 1; y) = R(f(x); y)
 * R(f(y); n + 1; z) = R(R(f(x); n; y); z)
 * for limit n R(f(x); n; y) is least w such as w = R(f(x); m; z) for all m R(f(x); n; v) for any v < y

For f(x) = x
 * R(f(x) = x; n; y) = I*(n, y)

Beyond

 * R(f(x) = I*(x, 0); y) are hyperinaccessibles and their limits.

So we can define classes of regular cardinals using some normal function f(x). We can obtain inaccessibles, higher inaccessibles, hyperinaccessibles and so on.

Then I thought: what if there exist such regular cardinal that they cannot be obtained this way, and there is no corresponding normal function? What if such cardinals are Mahlo cardinals?

For regular cardinal R(f(x); y), where y is non-limit, normal function f(x) can be used as fundamental sequence:
 * R(f(x); 0)[n] = f(n)
 * R(f(x); y + 1)[n] = f(R(f(x); y) + 1 + n)

Such fundamental sequences are closed, that is they contain their limits. And these fundamental sequences does not contain regular cardinals for limit n. If we add 1 to all elements of a fundamental sequences for non-limit n or remove all its element with non-limit n, then we get closed fundamental sequence without regular cardinals.

Usually regular cardinals have closed fundamental sequences without regular elements. But if corresponding normal function does not exist, then such fundamental sequences do not exist.

I made up a definition: regular cardinal without closed fundamental sequence without regular elements.

Club sets
Then I tried to find out: is it definition of Mahlo cardinal?

I thought that club sets can be considered as closed fundamental sequences. Because unbounded set in α "reaches" α, so it is just a fundamental sequence. And closed set in α includes all its limits less than α. For example, fundamental sequence of Ω
 * Ω[n] = n

that is
 * 0, 1, 2, ... ω, ω + 1, ω + 2, ...

is closed, because it contains all its limits less than Ω. But if we remove ω, that is
 * 0, 1, 2, ... ω + 1, ω + 2, ω + 3, ...

then we get non-closed fundamental sequence, because it does not contain limit ω of its part 0, 1, 2, ...

Stationary sets
If there is no closed fundamental sequence without regular elements, then every closed fundamental sequence (club set) has regular element, that is every club set intersects set of lesser regular cardinals, that is set of lesser regular cardinals is stationary.

In Wikipedia it is said that weakly Mahlo cardinal is weakly inaccessible and the set of weakly inaccessible cardinals less than it is stationary in it. But in Googology Wiki it is said: "If we weaken "inaccessible" to merely "regular," we get the weakly Mahlo cardinals". I think that these two definitions are equivalent.

So, my definition is definition of Mahlo cardinal.