3,531 research outputs found
Incompleteness and jump hierarchies
This paper is an investigation of the relationship between G\"odel's second
incompleteness theorem and the well-foundedness of jump hierarchies. It follows
from a classic theorem of Spector's that the relation is well-founded. We provide an alternative proof of
this fact that uses G\"odel's second incompleteness theorem instead of the
theory of admissible ordinals. We then derive a semantic version of the second
incompleteness theorem, originally due to Mummert and Simpson, from this
result. Finally, we turn to the calculation of the ranks of reals in this
well-founded relation. We prove that, for any , if the rank of
is , then is the admissible
ordinal. It follows, assuming suitable large cardinal hypotheses, that, on a
cone, the rank of is .Comment: 11 pages. Corrects a mistake in the statements of two result
A uniform approach to fundamental sequences and hierarchies
In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited
We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w
w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x +
(4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no
recursion operator or master symmetry was known so far, and prove that the
system (*) admits infinitely many local generalized symmetries that are
constructed using a nonlocal {\em two-term} recursion relation rather than from
a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and
definitions adde
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