Question: As you know Fraïssé limit construction and its generalization, Hrushovski's construction, have many applications in model theory to build models with interesting property.

Now I would like to know the application of Fraïssé construction in set theory.

**Question**: What are the major applications of Fraïssé construction in set theory?

Any reference will be appreciated.

Answer: 
A Fraïssé construction lies at the heart of the proof of my embedding theorems.

**Theorem.** (J. D. Hamkins, [Every countable model of set theory
embeds into its own constructible universe](http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe/), JML 13(2), 2014).

1. For any two countable models of set theory $\langle
M,\in^M\rangle$ and $\langle N,\in^N\rangle$, one of them embeds
into the other.
2. Indeed, a countable model of set theory $\langle M,{\in^M}\rangle$ embeds into another model of set theory $\langle
N,{\in^N}\rangle$ if and only if the ordinals of $M$ order-embed
into the ordinals of $N$.
3. Consequently, every countable model $\langle M,\in^M\rangle$ of
set theory embeds into its own constructible universe $\langle
L^M,\in^M\rangle$.
4. Furthermore, every countable model of set theory embeds into
the hereditary finite sets $\langle \text{HF},{\in}\rangle^M$ of
any nonstandard model of arithmetic $M\models\text{PA}$. Indeed,
$\text{HF}^M$ is universal for all countable acyclic binary
relations.

The sense of embedding here is $j:M\to N$ for which $x\in y\iff j(x)\in j(y)$, which is the model-theoretic sense of embedding, and this is weaker than those usually considered in set theory because they need not be elementary nor even
$\Delta\_0$-elementary. An embedding $j:M\to N$ is simply an isomorphism
of $M$ with a substructure of $N$, not necessarily transitive.

The proof proceeds by finding sufficiently universal substructures
of any model of set theory, using essentially a Fraïssé limit
construction, as you can see in the paper. Basically, this construction shows that every
countable model of set theory $\langle M,\in^M\rangle$ has a
submodel that is universal for all countable $\text{Ord}^M$-graded
digraphs. A *grading* of a digraph is a linear pre-order on that
digraph, such that the linear order strictly increases when
following any edge of the digraph. Any model of set theory
$\langle M,\in^M\rangle$ admits an $\text{Ord}^M$ grading by means
of the von Neumann rank, since whenever $x\in y$, then the rank of
$y$ strictly exceeds that of $x$.

It happens that I am speaking today on this topic at the [CUNY set
theory seminar](http://jdh.hamkins.org/on-embeddings-of-the-universe-into-the-constructible-universe-set-theory-seminar-march-2015/), in about an hour.
I add a few more references and applications:

> 
> **(A)** Trevor Irwin's thesis ``Fraïssé limits and colimits with applications to continua'':
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**Abstract:** The classical Fraïssé construction is a method of taking a direct limit of a family of finite models of a language provided the family fulfills certain amalgamation conditions. The limit is a countable model of the same language which can be characterized by its (injective) homogeneity and universality with respect to the initial family of models. A standard example is the family of finite linear orders for which the Fraïssé limit is the rational numbers with the usual ordering.

We present this classical construction via category theory, and within this context we introduce the dual construction. This respectively constitutes the Fraïssé colimits and limits indicated in the title. We provide several examples.

We then present the projective Fraïssé limit as a special case of the dual construction, and as such it is the categorical dual to the classical (injective) Fraïssé limit. In this dualization we use a notion of model theoretic structure which has a topological ingredient. This results in the countable limit structures being replaced by structures which are zero-dimensional, compact, second countable spaces with the property that the relations are closed and the functions are continuous.

We apply the theory of projective Fraïssé limits to the pseudo-arc by first representing the pseudo-arc as a natural quotient of a projective Fraïssé limit. Using this representation we derive topological properties of the pseudo-arc as consequences of the properties of projective Fraïssé limits. We thereby obtain a new proof of Mioduszewski’s result that the pseudo-arc is surjectively universal among chainable continua, and also a homogeneity theorem for the pseudo-arc which is a strengthening of a result due to Lewis and Smith. We also find a new characterization of the pseudo-arc via the homogeneity property.

We continue with further applications of these methods to a class of continua known as pseudo-solenoids, and achieve analogous results for the universal pseudo-solenoid.

> 
> **(B)** Wieslaw Kubiś, ''Fraisse sequences - a category-theoretic approach to universal homogeneous structures'':
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**Abstract:** We present a category-theoretic approach to universal homogeneous objects, with applications in the theory of Banach spaces and in set-theoretic topology,