By Bruno Poizat

Translated from the French, this booklet is an creation to first-order version thought. ranging from scratch, it fast reaches the necessities, particularly, the back-and-forth approach and compactness, that are illustrated with examples taken from algebra. It additionally introduces common sense through the research of the types of mathematics, and it provides entire yet obtainable exposition of balance idea.

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**Sample text**

Proof. If s is a local isomorphism from R to R', and I add a to its domain, whether a is congruent modulo R to an element of dom( s) or a is in a new equivalence class, then since R' has infinitely many classes, and all of them are infinite, I can always extend s to a so as to have a local isomorphism. The conclusion follows. Conversely, we can easily see that if R has at least p classes and if R' is p-equivalent to it , then R' also has at least p classes; and that if every class of R has at least p elements and R' is p-equivalent to it, then R' has the same property.

Show that Rand R p are p-equivalent. 4. Show that Rand R' are elementarily equivalent iff f Rand f R' are equal, except possibly on 00 in the case where both take nonzero values infinitely often. 5. Let Rand R' be elementarily equivalent, and let R' be an extension of R; show that this extension is elementary iff every finite class of R' with one clement in the universe of R has all its elements in the universe of R and every infinite class of R' with at least one element in the domain of R has infinitely many such elements.

7. A binary relation 3-equivalent to a dens e chain without endpoin ts is also a dense chain without endpoints; conversely, any two such chains C and C' are elem entarily equivalen t, and every local isomorphism from C to C' is elem entary. Proof. Let C be a dense chain without endpoints. We know that any binary relation C' 3-equivalent to C is a chain. If C' is not dense, there exist consecutive elements a and b in C'. Let s be a I-isomorphism from C' to C defined at a and b, and let c be in C with s(a) < c < s(b); we cannot extend s -l to c, contradicting the hypothesis.