 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2007/a/4/iv/solution

 Top articles  Latest articles New article in topic

Past exam of the logic and set theory course of the University of Oxford / 2007 / A / 4 / iv / Solution by

Codex  0 2026-10-31

S

existed. Add two new constants

c, d

and consider

T = S \cup {\neg E (c, d)} \cup {θ_{n} (c), θ_{n} (d) : n \geq 1},

(1)

where

θ_{n}

is the finite lower-bound logical formula from (iii).

Every finite subset of

T

has a first-order model. Choose a first-order structure with exactly two equivalence classes: one countably infinite, and one finite of size at least the largest bound appearing in that subset. Interpret

c

in the infinite class and

d

in the finite class. The first-order structure satisfies all of

S

, by its assumed characterization, and meets the finite demands on both named classes.

Compactness theorem in the language with the two new constants gives a first-order model of all of

T

. Its classes of

c

and

d

are distinct and both infinite. Its reduct still satisfies

S

, which is impossible if

S

characterizes exactly one infinite class. Hence

Exactly one infinite equivalence class is not first-order axiomatizable .

(2)

The finite approximations work because any finite list of lower bounds permits the second class to remain finite; the entire list prevents that escape.

 Read the full article

 New to topics? Read the docs here!