Kripke - Naming and Necessity Lecture II [Hayley Thompson]

"...when asked what the theory of relativity is, they will say: 'Einstein's theory' and thus be led into the most straightforward sort of vicious cycle."

Kripke uses this example to argue names may break either the unique individual thesis, in which a set properties are believed to pick out one unique individual, or the non-circularity thesis and retain meaning. The violation of the non-circularity thesis, described in Kripke's quote above, that happens when certain entities are forced to comply with the unique individual thesis, may best be described with the following diagram:

In this case, Einstein refers to a man who had a theory, namely that of relativity.  But due to its complexity, for many people the 'the theory of relativity' has little meaning. When asked what this theory refers to, they would likely say 'a theory by Einstein'.

I agree with Kripke that the non-circularity thesis is not required for an name to have reference. In the case described, at least for some people, 'the theory of relativity' and 'Einstein'  are two names that might only be uniquely referenced as properties of each other, leading to the cycle. Still, this cycle establishes a semantic connection that allows both the names 'Einstein' and 'the theory of relativity' to refer to unique entities.

"We have to be referring to Godel if we say 'Godel proved the incompleteness of arithmetic.'"

In this quote, Kripke is explaining that the speaker intends only one specific man, who either was given the name Godel at birth or came to be known as such through a game of transliteration telephone, to fulfill the predicate of the statement. Would this necessitate that the man who proved the incompleteness of arithmetic had to be Godel? No. The statement may have a false truth value. While agree with this to some extent, in common usage, when not used as an equivocal or identity statement, 'Godel' may be used as a generic term for 'whoever it was who proved the incompleteness of arithmetic' or vise-verse. For example, if one were to say "Godel must have been a coffee-drinker, since he got coffee stains all over the papers containing his incompleteness proof", they are making this assertion about the actual man who wrote the incompleteness proof for arithmetic and not just some random person named Godel who was credited with proving said theory.