[FOM] Standard Language of Euclid

[Alasdair Urquhart]

In answer to Joe Shipman's question below,
the pure first-order theory of fields was proved
undecidable by Julia Robinson in her doctoral
thesis of 1948.  The main results of her thesis
were published as

"Definability and decision problems in arithmetic",
JSL Volume 14 (1949), 98-114.

This result of Robinson is obtained as a corollary
to her deep result that the first-order theory of the rationals
(in the language of + and x) is undecidable.

Alasdair Urquhart


> A related question: is it known whether the pure first-order theory of
> fields is undecidable? (By Tarski, if you add infinitely axioms to
> ensure characteristic 0 and infinitely many axioms to ensure
> real-closedness, the resulting theory is complete and decidable,but I
> am interested in the statements that must be true in all fields.)