I'm trying to describe Godel's Incompletemeness Theorem in 1 short sentence... We at the Unemployed Philosophers Guild are adding Kurt Godel to our line of illustrious finger puppets.  On the puppet tag we always have a short biography.  Below is what we have written.  Is our description of his Theorem acceptable to a mathematician/logician?

Austrian-born philosopher and logician Kurt Friedrich Gödel studied physics before publishing his famous (two) Incompleteness Theorem(s). According to Gödel, a mathematical system can’t prove or disprove every proposition within itself (it’s “incomplete”) and can’t prove itself both complete and consistent. Gödel fled Nazi Germany, renewed his friendship with fellow émigré Albert Einstein, and became a U.S. citizen. As “the most important logician since Aristotle,” Gödel influenced computer science, artificial intelligence, and philosophy of mathematics. He was devoted to operetta. 

Here is an alternate description of his work:  ...According to Gödel, if a mathematical system can prove every statement that can be constructed in the system, then there must be some contradictory statements in the system: and if there are no contradictory statements, then there are statements that cannot be proved...

Thank you!
It depends on the level of precision you want to reach. 

My first remark is that "According to Gödel" feels too much like "He thought so and he said so", but maybe I'm just misunderstanding this by adding a connotation that wasn't there.

More to the point, Gödel's theorem is about formal theories that can be computed. If you want to be precise this is a detail that cannot be missed because there are many theories that are complete and prove or refute every sentence. 

A quick summary would be 


  "Gödel proved that no effective [here the word "effective" allows you to vaguely state that it should be computable], expressive enough, consistent mathematical system can prove or refute every sentence: it will always be incomplete. In particular, he proved that such a system cannot prove its own consistency."


This allows you to be precise, while still being accessible (the words "effective", "expressive enough" are not detailed, so as to be understood by the non-logician), and is a good reflection of what Gödel actually did. My reformulation to "Gödel proved" (rather than "According to Gödel") is simply because I feel like this emphasizes more that this isn't just speculation or an idea he had but rather an actual theorem.
Austrian-born philosopher and logician Kurt Friedrich Gödel studied physics before publishing his famous (two) Incompleteness Theorem(s). 


Physics has nothing to do with the incompleteness theorems.  I doubt he spent as much time or effort in physics as logic, although Wikipedia suggests he did spend a lot more time on physics later in life.


  According to Gödel, a mathematical system can’t prove or disprove every proposition within itself (it’s “incomplete”)


No, there are logics, even rather complex ones, that are complete and consistent.  Propositional logic and "the first-order theory of Euclidean geometry is complete and decidable" (from the link below).


  and can’t prove itself both complete and consistent. 


No.  Sufficiently strong logics can not be both complete and consistent.  But any inconsistent logic can prove either of those.


  Gödel fled Nazi Germany, renewed his friendship with fellow émigré Albert Einstein, and became a U.S. citizen. 


Wikipedia agrees with this.  Why don't you just say "immigrant"?


  As “the most important logician since Aristotle,” 


I would say Hilbert, but some logicians would agree with you.


  Gödel influenced computer science, artificial intelligence, and philosophy of mathematics. He was devoted to operetta.


CS and philosophy are a given, the other two, no idea.


  Here is an alternate description of his work: ...According to Gödel, if a mathematical system can prove every statement that can be constructed in the system, then there must be some contradictory statements in the system: and if there are no contradictory statements, then there are statements that cannot be proved...


That doesn't even make sense to me.  Here is the first incompleteness theorem:

If a logic can be verified by a Turing Machine, is capable of proving all theorems of Robinson Arithmetic, and is 1-consistent; then there is a sentence $G$ such that neither $G$ is provable nor is $\lnot G$ provable in the logic.

(Small detail, but the strengthening of the statement from 1-consistent to consistent is actually due to Rosser.)

If I could say something a bit harsh, the Internet is rife with people attempting to summarize or explain Godel's Incompleteness Theorems who never bothered to really learn them.  Please don't be another.  Silence is better.  I'm not even confident about them, so I hesitated to write this.  If you wish to learn them, I find this link does a really thorough and detailed job without containing extraneous pontifications :

https://plato.stanford.edu/entries/goedel-incompleteness/


  On the puppet tag we always have a short biography. 


As far as logicians with interesting lives, I find Moses Schönfinkel really compelling.  Born a poor boy in Ukraine, talented enough to eventually study even under David Hilbert, invented one of the first completely formal logics that was in some sense the grandfather of all constructive logics.  Was committed to an asylum before he was 40 and spend his later life in poverty like how he was born.  His personal papers/work were burned by his neighbors trying to stay warm because of wartime conditions.