FANDOM


1. {}

2. [[]]

3. [(())]

4. [()()()]

5. ([()()][])

6. ([()()](()))

7. ([()()]()()())

8. ([()()]()())

9. ([()()]())

10. ([()()])

11. ([()][()][()][()][()])

12. ([()][()][()][()](([])))

13. ([()][()][()][()]([](()))) (I should have let this reduce to [][][], but I don't want to do whole analysis again)

14. ([()][()][()][()]([]()()()))

15. ([()][()][()][()]([]()())([]))

16. ([()][()][()][()]([]()())[][][])

17. ([()][()][()][()]([]()())[][](()))

18. ([()][()][()][()]([]()())[][]()()())

19. ([()][()][()][()]([]()())[][]()())

20. ([()][()][()][()]([]()())[][]())

21. ([()][()][()][()]([]()())[][])

22. ([()][()][()][()]([]()())[]tree(8))

tree(8). ([()][()][()][()]([]()())tree(tree(8)))

tree(tree(8)). ([()][()][()][()]([]())tree(tree(7)))

tree(tree(8))+1. ([()][()][()][()]([]())tree(tree(7))([])([]))

tree(tree(8))+2. ([()][()][()][()]([]())tree(tree(7))([])[][][])

tree(tree(8))*2. ([()][()][()][()]([]())tree(tree(7))([]))

tree(tree(8))*2. ([()][()][()][()]([]())tree(tree(7)))[]tree(tree(7)))

tree2(tree(8)). ([()][()][()][()]([]())tree(tree(7)))

tree2(tree(8)). ([()][()][()][()]([]())tree(tree(7))([])tree2(tree(8)))

tree2(tree2(tree(8))). ([()][()][()][()]([]())tree(tree(7)))

tree3(tree(tree(7))). ( [()][()][()](((...((([])))...))) )

...

Structures

tree function iteration

FGH ordinal

tree
tree(n) SVO []
tree2(n) SVO+1 ([])
tree3(n) SVO+2 ([]())
tree4(n) SVO+3 ([]()())
treeω(n) SVO+ω ([](()))
treeω+1(n) SVO+ω+1 ([](())())
treeω2(n) SVO+ω2 ([](())(()))

treeω2(n)

SVO+ω2 ([](()())
treeωω(n) SVO+ωω ([]((())))
treeε0(n) SVO+ε0 ([][])
treeε0*2(n) SVO+ε0*2 ([][][])
treeε0*ω(n) SVO+ε0*ω (([]))
treeε0*ω(n) SVO+ε0*ω2 (([]()))
treeε02(n) SVO+ε02 (([][]))
treeε0ω(n) SVO+ε0ω ((([])))
treeε1(n) SVO+ε1 [()]

However, I don't think this is optimal. I guess we can reach SVO*2 with ([][]), and then we reach ε(SVO+1) at [()]. But for now I don't see any way to do that, so here is my improved bound:

\(\text{TREE}(3) > f_{\vartheta(\Omega^\omega)+\varepsilon_1}(f_{\vartheta(\Omega^\omega)+\varepsilon_1}(f_{\vartheta(\Omega^\omega)+\varepsilon_1}(f_{\vartheta(\Omega^\omega)+\varepsilon_1}(\text{tree}_3(\text{tree}(\text{tree}(7)))))))\)

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