Original motivations of Fraïssé's amalgamation construction

Roland Fraïssé introduced in the 50's his famous construction of Fraïssé limits, and then Ehud Hrushowski modified it in the early 90's to construct new structures.

The motivations for the latter was clear, but what were the original motivations of Fraïssé to reconstruct a countable structure from its finite pieces ?

• Have you tried to read the original paper?
– YCor
Jan 25 at 8:52
• I wanted to, but couldn't find the reference of that original paper, whence my question here. Can you give me a link to that paper ? Jan 25 at 8:59
• I found a reference for his 1953 thesis but I don't know if it's available online (it would be certainly worthwhile).
– YCor
Jan 25 at 11:54

1 Answer

There are two source papers by Fraïssé, a brief note and a longer text:

The abstract of the 1954 paper summarises Fraïssé's motivation, to generalize the ordering of the rationals:

This article is an elementary study of certain classes of relations, $$\Gamma$$ and $$\gamma$$, which generalize the class of orders and that of finite orders. We generalize the fact that any set can be ordered, by generalizing the following two propositions: given two total orders A, B, there exists an extended order (e.g. A + B) common to A and B or their isomorphs; given a set of orders A, there exists an extended order R common to A (or their isomorphs). In this way we generalize two propositions on the order $$\eta$$ of the rational numbers: any countable order is isomorphic to a restriction of $$\eta$$; Order $$\eta$$ is the only countable dense order with no first or last element.

For an English text, see volume 2, chapter 11 of Fraïssé's Course of Mathematical Logic.

• Shouldn't he have said at the end instead that $\eta$ is the only countable dense order with no first or last element? Jan 25 at 14:14
• @JoelDavidHamkins --- indeed, the omission of the word dense was my mistake (now corrected), not Fraïssé's Jan 25 at 18:19
• Ok so as far as I understand, the motivations for this original construction was to present a general method allowing to construct omega-categorical structures. Am I right ? Feb 5 at 9:01