Carnap has begun with the new logical methods presented first by Frege and then more extensively in Principia Mathematica.
The next most conspicuous source appears to be Hilbert's programme of metamathematics, both in its conception and in more detailed techniques which Carnap learned through Gödel and Tarski, who could be said to be logicians working in the context of Hilbert's programme.
The impact of Hilbert's programme seems pervasive.
Hilbert had proposed that problems in the foundations of mathematics should be resolved by a program of metamathematical work made possibly by considering mathematics as conducted in formal languages which could themselves be the subject of mathematical studies.
Carnap generalises this metalinguistic view of foundational problems from its application by Hilbert in mathematics to broader philosophical purposes.
The domain of the meaningful is split into analytic and synthetic judgements.
The former are logical truths, the latter belong to science (in a very general sense) rather than philosophy, but the metatheory remains a part of philosophy as the logical study of the syntax of language.
The two features of Hilbert's programme which are most prominent in Carnap's work on Logical Syntax are firstly the metalinguistic aspect, but also the confidence in the sufficiency of syntax.
It is this latter feature which was most precarious in the period when Carnap was working on Logical Syntax.
Göel had perhaps first lent it greater credibility by his proof of the completeness of first order logic, but quickly delivered a terminal blow with his proof of the incompleteness of arithmetic.
Of these it is Gödel's contribution which is easier to spot in Carnap's work on logical syntax.
The methods used by Gödel in his