User blog comment:Rgetar/Idea for FGH for larger transfinite ordinals/@comment-35470197-20190702232339/@comment-32213734-20190708053541

I think that there may be recursion, which does not terminates.

I redefined Si sets: Si + 1 is set of FGH expressions fα k βk(...(fα 2 β2(fα 1 β1(x)))...), where
 * k is natural number
 * x is ω or Ω
 * αi, βi ∈ Si
 * α1 ≥ α2 ≥ ... ≥ αk
 * if αi = αi + 1 then βi > βi + 1

And S0 is set of all natural numbers, ω and Ω. Comparison of elements of S0: in a pair of natural numbers larger expression is larger number; Ω > ω > n for any natural n.

Sj < i are subsets of Si.

S are union of all Si for natural i.

Si is well-founded, and S is not.

Let mis ("minimal S") of an FGH expression α ∈ S is minimal i such as α ∈ Si.

Let
 * mis(fαβ(x)) = n

Then
 * mis(α) < n
 * mis(β) < n
 * mis(x) ≤ n

There are 9 cases for cofinality function cof(y) and fundamental sequence function fs(y, n) = y[n].

cof(y)


 * 1. if y = 0 then cof(y) = 0


 * 2. else if y is natural number then cof(y) = 1


 * 3. else if y is ω or Ω then cof(y) = y


 * else y = fαβ(x). Then


 * 4. if cof(β) > 1 then cof(y) = cof(β)


 * 5. else if cof(β) = 1 then cof(y) = cof(fα0(fαβ[0](x)))


 * else cof(β) = 0. Then


 * 6. if cof(α) = 0 then cof(y) = 1


 * 7. else if cof(α) = 1 then cof(y) = cof(fα[0]x(x))


 * 8. else if cof(α) < x then cof(y) = cof(α)


 * 9. else cof(α) ≥ x. Then cof(y) = cof(fα[x]0(x))

fs(y, n) = y[n]

n < cof(y)


 * 1. if y = 0 then there are no n.


 * 2. else if y is natural number then y[0] = y - 1


 * 3. else if y is ω or Ω then y[n] = n


 * else y = fαβ(x). Then


 * 4. if cof(β) > 1 then y[n] = fαβ[n](x)


 * 5. else if cof(β) = 1 then y[n] = fα0(fαβ[0](x))[n]


 * else cof(β) = 0. Then


 * 6. if cof(α) = 0 then y[0] = x


 * 7. else if cof(α) = 1 then y[n] = fα[0]x(x)[n]


 * 8. else if cof(α) < x then y[n] = fα[n]0(x)


 * 9. else cof(α) ≥ x. Then y[n] = fα[x]0(x)[n]

Both functions cof and fs have recursion: they invoke themselves. And they have mutual recursion: cof invokes fs, and fs invokes cof.

But recursion should terminate. That is at each level of recursion fs and cof should be invoked finite number of times, and at least one parameter of each function should form terminating sequence with increasing recursion level.

For first parameters of cof and fs in such sequences either mis of the parameter decreases, or mis does not increase, but the parameter decreases. Since mis is finite and set of expressions with fixed mis is well-founded, such sequence terminates.

Except Case 9: it can increase mis. So I am not sure does this recursion terminate. But I did not prove that it does not terminate.

In program Ordinal Explorer v4.0, which uses these expressions and functions, lists are created fast for not very large expressions, where Case 9 is not used. But when Case 9 is used, lists are created slowly, since expressions become very long.

I tried to do triple expansion of an expression with fundamental sequence
 * fΩ0(ω)
 * ff Ω0(Ω) 0(ω)
 * ff f Ω0(Ω) 0(Ω) 0(ω)

After 300 steps I interrupted the process and got list of 301 expressions (incomplete triple expansion). Used format:
 * [α.β]x = fαβ(x)
 * c = ω
 * u = Ω

Here is expression which was added to the list last:

[u.[[[u.0]u.0]c.[[[u.0]u.0]c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[[[u.0]u.0]c.[c.[c.[u.0][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[c.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.[c.[c.[c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.[[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c][[u.0]u.0]c.[c.c][[u.0]u.0]c][u.c][[u.0]u.0]c][[c.c]u.c][[u.0]u.0]c

Length of expressions during the process grew (sometimes insignificantly decreased). There is repeating pattern ("c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c.[c."), and number of "c.["'s slowly, but constantly grows. But I am not sure, will it grow infinitely, because there were other places with similar patterns, where number of "c.["'s slowly grew up to 6, but then quickly decreased down to 0. So I do not know if this expansion infinite or finite.

Case 7 also causes problems: expressions are not such long, but some single expansions may be very long (thousands of expressions).

So I think it make sense to modify and simplify FGH for transfinite ordinals.

Old FGH for countable x:
 * f00(x) = x + 1
 * fα + 10(x) = fαx(x)
 * fα0(x) = sup(fγ < α0(x)) for cof(α) = ω
 * fα0(x) = fα[x]0(x) for cof(α) = Ω

New "FGH" for countable x:
 * f00(x) = x + 1
 * fα + 10(x) = fαω(x)
 * fα0(x) = sup(fγ < α0(x)) for cof(α) = ω
 * fα0(x) = sup(yi ∈ ℕ), where y0 = ω, yi + 1 = fα[y i] 0(x), for cof(α) = Ω

(Actually, this new "FGH" is not different from my "booster-base" system).

Now I think that "old" FGH for transfinite ordinals can be used for given system of fundamental sequences, but it is not very useful for creating a system of fundamental sequences. So, a system of fundamental sequences can be created some other way, for example, using "new" FGH, and then be used in it.