Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories.(In 80s)

Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:

a) Trivial (No structure)

b) Linear (Locally modular): Vector Spaces.

c) Non-Linear (Non Locally modular): Algebraically Closed Fields(ACF).

In other words Zilber conjectured that any non-locally modular strongly minimal set interprets an ACF.

Hrushovski construct a new class of $\aleph_{1}$- categorical structures, disproving Zilber's conjecture.(A New Strongly Minimal Set - Annals of Pure and Applied Logic 1993 ).

Hrushovski generalized the Frasse's method to construct his counter example. Hrushovski construct an $\omega$- saturated strongly minimal set whose geometry is flat.

Definition: A combinatorial geometry J is flat if whenever $E_{i}(i\in I)$ are finite number of finite-dimensional closed subset of J, s ranges over the subsets of I, then $\sum_{s}(-1)^{card(s)}\leq0$.

Hrushovski proved the following theorem:

Theorem: Let D be a saturated strongly minimal set whose geometry is flat. Then D does not interpret an infinite group. So D does not interpret an ACF.

Question: Can we use Hrushovski's method to find a saturated strongly minimal set whose geometry is not flat ?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.