What is this flaw

No, I was still inactive today
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Fast growing hierarchy:
f[a](b) = f[a-1]^b(b)
f[α](n) = [left blank]
f[0](b) = b+1
f[w](b) = f[b](b)
f[w+1](b) = f[w]^b(b)
Where is this flaw? Did I make a mistake, or it replied me a comment?