gameoflifemaniac wrote: November 5th, 2019, 1:56 pm
Can you give me some examples of values of Weiermann's ϑ? Maybe I'll understand it this way
Why that one? I've never even heard of that one! Maybe I can explain some simple OCF like Madore's ψ or some simpler one.
(that I made up)
Here's a
simple example:
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C_0(α)={0,1,Ω}
C_n+1(α)={γ+δ,ψ(η)|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ(α)=min{β∈Ω|β∉C(α)}
ψ(0) = w (the minimum value you can't get by adding finitely many finite numbers together)
ψ(1) = w^2 (The minimum value you cannot get by adding finitely many ws together)
ψ(n) = w^(n+1) for finite n
ψ(w) = w^w
etc.
ψ(e_0) = e_0
ψ(e_0 + 1) = e_0
... epsilon_0 for a long time
ψ(W) = e_0
ψ(W+1) = (e_0)w (I think)
The limit of ψ = e_w
Suppose we introduce multiplication:
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C_0(α)={0,1,Ω}
C_n+1(α)={γ+δ,γδ,ψ(η)|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ(α)=min{β∈Ω|β∉C(α)}
Now, it starts off at ψ(0):
Still w
ψ(1) = the smallest ordinal you can't get by multiplying 0,1, or w finitely many times:
w^w
ψ(n) for finite n = w^^(n+1)
ψ(w) = e_0
ψ(w2) = I think e_1
ψ(wn) for finite n = I think e_(n-1)
ψ(w^2) = e_w in that case
...
Adding exponentiation would make it, for all purposes,
madore's psi:
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That
C_0(α)={0,1,Ω}
C_n+1(α)={γ+δ,γδ,γ^δ,ψ(η)|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ(α)=min{β∈Ω|β∉C(α)}
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Madore's psi
C_0(α)={0,1,w,Ω}
C_n+1(α)={γ+δ,γδ,γ^δ,ψ(η)|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ(α)=min{β∈Ω|β∉C(α)}
It would start off more slowly but would be the same for all practical purposes.
If you want to see someone analyse it go to that link on the words Madore's psi.
Unfortunately you can't just extend it thru the hyper operators:
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C_0(α)={0,1,w,Ω}
C_n+1(α)={γ+δ,γδ,γ^δ,γ{ξ}δ,ψ(η)|γ,δ,ξ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ(α)=min{β∈Ω|β∉C(α)}
Unless you define a system for ordinal hyperoperators. If you do, awesome. (Technically since BEAF arrays of {a,b,c} = a{c}b you could use ordinal BEAF, mentioned
here, as a system for this. Don't ask me how ordinal beaf works because, I hate to tell you this, but I hardly understand normal BEAF.)
But you can define this extremely simple one:
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C_0(α)={0,Ω}
C_n+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ(η)|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ(α)=min{β∈Ω|β∉C(α)}
Which would get you to gamma_0 at ψ(0). Yes, ψ of
zero. Without so much as a 1 in the starter set.
With the finitely-many-entry Veblen function instead of the Veblen function, you could get to the SVO at 0, and with the any-amount-of-entries-in-the-function-including-infinitely-many you could get to the LVO.
Probably another good system would be
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C_0(α)={0,Ω}
C_n+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_0(η),w_γ|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ_0(α)=min{β∈Ω|β∉C(α)}
ψ_1(α)=sup(C(α))
In this case you could have two functions in one (and also the first one, a modification of the one in the code box above the one I'm talking about right now, would be more powerful than the other one in the other code box), and the uncountable-generating one would reach a|->w_a which I think is about inaccessible.
In LaTeX:
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\begin{eqnarray*}
C_0(\alpha) &=& \{0, \Omega\}\\
C_{n+1}(\alpha) &=& \{\gamma + \delta, \gamma\delta,\gamma^\delta,\phi_{\gamma}(\delta),\omega_\gamma,\psi_0(\eta) | \gamma, \delta, \eta \in C_n (\alpha); \eta < \alpha\} \\
C(\alpha) &=& \bigcup_{n < \omega} C_n (\alpha) \\
\psi_0(\alpha) &=& \min\{\beta \in \Omega|\beta \notin C(\alpha)\} \\
\psi_1(\alpha) &=& \sup(C(\alpha)) \\
\end{eqnarray*}