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Once I have used a non-numerical table index, is it possible to delete the index itself so that it is not returned from a call to next()?

For example:

table = {}
table["index"] = true    -- arbitrary value
table["index"] = nil -- delete the value, so shouldn't the index itself be non-existent?
i = next(table)    -- programmer expects resulting i to be nil
if(table[i] ~= nil) then
   print("Why am I here?")