Why do Lua arrays(tables) start at 1 instead of 0?

Lua is descended from Sol, a language designed for petroleum engineers with no formal training in computer programming. People not trained in computing think it is damned weird to start counting at zero. By adopting 1-based array and string indexing, the Lua designers avoided confounding the expectations of their first clients and sponsors.

Although I too found them weird at the beginning, I have learned to love 0-based arrays. But I get by OK with Lua's 1-based arrays, especially by using Lua's generic for loop and the ipairs operator—I can usually avoid worrying about just how arrays are indexed.

In Programming in Lua's first discussion of tables, they mention:

Since you can index a table with any value, you can start the indices of an array with any number that pleases you. However, it is customary in Lua to start arrays with 1 (and not with 0, as in C) and several facilities stick to this convention.

Later on, in the chapter on data structures, they say almost the same thing again: that Lua's built-in facilities assume 1-based indexing.

Anyway, there are a couple conveniences to using 1-based indexing. Namely, the # (length) operator: t[#t] access the last (numeric) index of the table, and t[#t+1] accesses 1 past the last index. To someone who hasn't already been exposed to 0-based indexing, #t+1 would be more intuitive to move past the end of a list. There's also Lua's for i = 1,#t construct, which I believe falls under the same category as the previous point that "1 to the length" can be more sensible than indexing "0 to the length minus 1".

But, if you can't break the mindset of 0-based indexing, then Lua's 1-based indexing can certainly be more of a hindrance. Ultimately, the authors wanted something that worked for them; and I'll admit I don't know what their original goal was, but it's probably changed since then.

My understanding is that it's that way just because the authors thought it would be a good way to do it, and after they rolled the language out to the public that decision calcified considerably. (I suspect there would be hell to pay were they to change it today!) I've never seen a particular justification beyond that.