If you post some examples of ordinals without a definite article it will be easier for us to help, but here are some examples for you. (Note that these are all examples of something that looks like an ordinal but isn't.)
1. He got a first from his university.
(A "first" means "a first-class degree" and is not an ordinal.)
2. This is a first-class restaurant.
(In this case "first-class" is a compound adjective, not an ordinal.)
3. We tried to arrive first but actually came second.
(In this case, "first" and "second" are adverbs not ordinals.)
4. John is our first, and Mary is our second.
(In this case, "first" and "second" are adjectives that modify the implied noun "child" not ordinals.)
5. She ate a third of the cake.
(In this case "a third" refers to a part of something (33.3%) and it's a noun not an ordinal.)
Like I said, if you can post some other examples I'm sure that I or somebody could help you understand.