# Why does one need to specify that the language is "well-orderable" for first order logic to be complete?

While reading Wikipedia I noticed the phrase "well-ordered language" in the following related to Gödel's completeness theorem:

The completeness theorem then says that for any first-order theory T with a well-orderable language, and any sentence s in the language of T

if s is a semantic consequence of T then s is a syntactic consequence of T.

Why did the authors need to specify a "well-ordered language" and what are the characteristics of a first order language that show that it is well-ordered?

Wikipedia contributors. (2019, June 10). Gödel's completeness theorem. In Wikipedia, The Free Encyclopedia. Retrieved 20:07, June 19, 2019, from https://en.wikipedia.org/w/index.php?title=G%C3%B6del%27s_completeness_theorem&oldid=901289398

• They are just being overtechnical. For a non-well-orderable language one needs an uncountable alphabet, and has to work in a set theory without the axiom of choice. Not a language most ever met, or care about. Jun 19, 2019 at 20:29
• @Conifold The link does explain what a well-ordered language is but not why it is needed for the completeness proof. Jun 20, 2019 at 12:10
• To run (transfinite) induction on the set of formulas when constructing its model, see an example on Quora. Jun 20, 2019 at 15:46