A note on the expressive power of linear orders

This article shows that there exist two particular linear orders such that first-order logic with these two linear orders has the same expressive power as first-order logic with the Bit-predicate FO(Bit). As a corollary we obtain that there also exists a built-in permutation such that first-order logic with a linear order and this permutation is as expressive as FO(Bit)

Elementary Properties Of The Finite Ranks

. This note investigates the class of finite initial segments of the cumulative hierarchy of pure sets. We show that this class is first-order definable over the class of finite directed graphs and that this class admits a first-order definable global linear order. We apply this last result to show that FO(!; BIT) = FO(BIT): This note establishes some elementary properties of the finite initial segments of the cumulative hierarchy of pure sets. We define the sets Vn ; n 2 ! by induction as follows: V 0 = ;; Vn+1 = P(Vn ): (Here, P(X) is the power set of X; that is, the set of all subsets of X:) The collection of nonempty finite ranks, FR, is fVn : n ? 0g: We will often use Vn to denote the directed graph whose set of nodes is Vn and whose edge relation is 2, the set membership relation; similarly, we will use FR to denote the collection of such graphs Vn for n ? 0: More generally, if M is a collection of sets, we will also use M to denote the directed graph whose set of nodes is M and..