You have a partial embedding f: A → N (where A ⊂ M) and want to extend it to include a new element m ∈ M \ A. What does the extension lemma guarantee?
AThat m itself embeds into N directly, without any modification to N
BThat there exists an extension N' ⊇ N containing an element that plays the role of m
CThat f can always be extended to a full automorphism of M
DThat the extension is possible only if A is an algebraically closed substructure
The extension lemma does not promise that N already contains a copy of m — N may need to be extended. The conclusion is that some N' ⊇ N exists in which the extended embedding lands. This is the key: you get a larger target structure, not a free embedding into the original one. The compactness argument constructs N' by showing the type of m over A is finitely satisfiable in extensions of N, then invoking compactness to guarantee a model.
Question 2 Multiple Choice
What role does compactness play in the proof of the extension lemma?
AIt guarantees that the diagram of any infinite structure is equivalent to a finite one
BIt allows inferring satisfiability of the full type of m over A from satisfiability of each finite subset
CIt ensures that every embedding extends to an automorphism of a sufficiently large structure
DIt eliminates the need for constants in the diagram expansion
The type of m over A is an infinite set of atomic and negated-atomic conditions. Compactness says: if every finite subset is satisfiable, the whole set is satisfiable. Since each finite fragment of the type can be satisfied in some extension of N (using that f already embeds A correctly), compactness delivers a model satisfying all conditions at once — that model is N'. Without compactness you could only extend finitely many conditions at a time.
Question 3 True / False
The extension lemma provides a 'local-to-global' principle: small, one-element extensions guaranteed by compactness can be iterated to build a global isomorphism between countable structures.
TTrue
FFalse
Answer: True
This is exactly the back-and-forth method. Each stage extends a partial isomorphism by one element from one side, then one element from the other. The extension lemma guarantees each single step is possible. Iterating countably many times — alternating between the two structures — builds a total isomorphism. The 'local' is each one-element extension; the 'global' is the finished isomorphism.
Question 4 True / False
The extension lemma guarantees that a partial embedding f: A → N can typically be extended within the same structure N, without passing to a larger structure.
TTrue
FFalse
Answer: False
This is the key misreading to avoid. The lemma guarantees an extension into some N' that extends N — it does not promise the extension lands inside the original N. In general, N may not contain the necessary element. The construction produces a new structure N' ⊇ N. Requiring the extension to stay in N would be a much stronger (and often false) claim.
Question 5 Short Answer
Why does building a global isomorphism via back-and-forth require the extension lemma to apply at every stage, not just once?
Think about your answer, then reveal below.
Model answer: Each stage of back-and-forth produces a new partial isomorphism that is larger than the previous one, with new elements on both sides. The extension lemma is invoked fresh at each stage to extend the current partial map by one more element. If the lemma failed at any stage — if some element could not be matched — the construction would halt and no total isomorphism would result. The entire argument is a countable iteration, and its validity rests on the lemma holding unconditionally at each step.
This is why the extension lemma is 'foundational' for amalgamation and homogeneity results: those constructions are essentially back-and-forth arguments, and the lemma is the engine that makes every local extension work. A single failure would break the transfinite induction.