math into logic How does one translate Godel sentence about the integers into "This sentence is not provable" and Rosser's sentence into "If this sentence is provable, there is a shorter proof of its negation".
If I write down a sentence in logic, how can one translate it into a statement about the natural integers?
Which words am I allowed to use such that it can be translated into a mathemtical statement abouth the integers?
What is a precise way to translate logical statements into mathemtical statements about integers? For instance the AND operator and "This sentence" and "X is provable".
 A: It's rather a long story, and the best way to understand it is probably to work your way through a proof in a logic textbook or in a popularization such as Nagel and Newman. 
A: An important tool used in these types of constructions is Gödel's β function.  Also these Wikipedia articles cover the main ideas: 
Gödel numbering
Proof sketch for Gödel's first incompleteness theorem
A: The main points are these:


*

*you need to have a complete formalization of the language and the deductive system (i.e. the deduction rules)

*once the language is completely formalized you can define a codification which attach a number to any formal statement (aka formula)

*once the deductive system is formalized and the language is coded with numbers you can express all the deductive rules as complex arithmetical operations on the code-numbers

*the statement "from $A$ follows $B$ throught the deduction rules" now can be translated into an arithmetical statement of the form "from the number $Code(A)$ you can derive the number $Code(B)$ by the application of these arithmetical operations"

*so at this stage any statement about provability of formulas has been converted into a statement about numbers and arithmetics.


To carry on each of these steps several technical problems need to be solved, see the references above for the details.
