92 research outputs found

### Interval orders and reverse mathematics

We study the reverse mathematics of interval orders. We establish the logical
strength of the implications between various definitions of the notion of
interval order. We also consider the strength of different versions of the
characterization theorem for interval orders: a partial order is an interval
order if and only if it does not contain $2 \oplus 2$. We also study proper
interval orders and their characterization theorem: a partial order is a proper
interval order if and only if it contains neither $2 \oplus 2$ nor $3 \oplus
1$.Comment: 21 pages; to appear in Notre Dame Journal of Formal Logic; minor
changes from the previous versio

### The Veblen functions for computability theorists

We study the computability-theoretic complexity and proof-theoretic strength
of the following statements: (1) "If X is a well-ordering, then so is
epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where
alpha is a fixed computable ordinal and phi the two-placed Veblen function. For
the former statement, we show that omega iterations of the Turing jump are
necessary in the proof and that the statement is equivalent to ACA_0^+ over
RCA_0. To prove the latter statement we need to use omega^alpha iterations of
the Turing jump, and we show that the statement is equivalent to
Pi^0_{omega^alpha}-CA_0. Our proofs are purely computability-theoretic. We also
give a new proof of a result of Friedman: the statement "if X is a
well-ordering, then so is phi(X,0)" is equivalent to ATR_0 over RCA_0.Comment: 26 pages, 3 figures, to appear in Journal of Symbolic Logi

### Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension

We study some natural sets arising in the theory of ordinary differential
equations in one variable from the point of view of descriptive set theory and
in particular classify them within the Borel hierarchy. We prove that the set
of Cauchy problems for ordinary differential equations which have a unique
solution is $\Pi^0_2$-complete and that the set of Cauchy problems which
locally have a unique solution is $\Sigma^0_3$-complete. We prove that the set
of Cauchy problems which have a global solution is $\Sigma^0_4$-complete and
that the set of ordinary differential equation which have a global solution for
every initial condition is $\Pi^0_3$-complete. We prove that the set of Cauchy
problems for which both uniqueness and globality hold is $\Pi^0_2$-complete

### Epimorphisms between linear orders

We study the relation on linear orders induced by order preserving
surjections. In particular we show that its restriction to countable orders is
a bqo.Comment: 15 pages; in version 2 we corrected some typos and rewrote the
paragraphs introducing the results of subsection 3.3 (statements and proofs
are unchanged

### Borel quasi-orderings in subsystems of second-order arithmetic

AbstractWe study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques employed will be the reflection principles and Gandy forcing

### Invariantly universal analytic quasi-orders

We introduce the notion of an invariantly universal pair (S,E) where S is an
analytic quasi-order and E \subseteq S is an analytic equivalence relation.
This means that for any analytic quasi-order R there is a Borel set B invariant
under E such that R is Borel bireducible with the restriction of S to B. We
prove a general result giving a sufficient condition for invariant
universality, and we demonstrate several applications of this theorem by
showing that the phenomenon of invariant universality is widespread. In fact it
occurs for a great number of complete analytic quasi-orders, arising in
different areas of mathematics, when they are paired with natural equivalence
relations.Comment: 31 pages, 1 figure, to appear in Transactions of the American
Mathematical Societ

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