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# Downward Löwenheim-Skolem theorem

In 1915 Löwenheim gave a nearly complete proof of the fact that if a first-order formula
has a model, then it has a finite or countable model. A tidy, correct version of this was given by Skolem in 1922. We will give the more general version for arbitrary sets of formulas and arbitrary languages.

PROOF.
Let
, and
let
be a Skolemization of
.
Then expand
to a model
of
.
Let
be the subset of S generated by the constants and the range of
.
is closed under the operations of
; so let
be the
structure obtained by restricting the functions and relations of
to
. One sees that
, and that
. By reducing back to the original language we have the desired model of
.

A remarkable consequence of this result is the existence of small models of axiomatic first-order set theory (provided any model exists).

This seems to pose an obvious problem, because in such set theories one can prove
that the power set of a set has strictly greater cardinality than the set --
but if we have a countable model then all of our sets are countable. This is called
Skolem's paradox.

EXERCISES

## References

**1**-
L. Löwenheim,
*Über Möglichkeiten im Relativkalkül.*
Math. Ann. **68** (1915), 169-207.
[translated in *From Frege to Gödel*, van Heijenoort, Harvard Univ. Press, 1971.]

**2**-
Th. Skolem,
*Logisch-kombinatorische Untersuchungen über die
Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem
Theoreme über dichte Mengen.*
Videnskabsakademiet i Kristiania, Skrifter I, No. 4, 1919, 1-36.
Also in ``Selected Works in Logic by Th. Skolem'', ed. by Jens Erik Fenstak,
Scand. Univ. Books, Universitetsforlaget, Oslo, 1970, pp. 103-136.
[The first section is translated in: *From Frege to Gödel*, van Heijenoort, Harvard Univ. Press, 1971, 252-263.]

**3**-
Th. Skolem,
*Einige Bemerkung zur axiomatischen Begrundung der Mengenlehre.*
Proc. Scand. Math. Congr. Helsinki, 1922, 217-232.
[translation in *From Frege to Gödel*, van Heijenoort, Harvard Univ. Press, 1971.]

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Fri Jan 31 11:57:38 EST 1997

© Stanley Burris