Chaotic Orderings of the Rationals and Reals
نویسندگان
چکیده
In this note we prove that there is a linear ordering of the set of real numbers for which there is no monotonic 3-term arithmetic progression. This answers the question (asked by Erdős and Graham) of whether or not every linear ordering of the reals must have a monotonic k-term arithmetic progression for every k.
منابع مشابه
Approximating Reals by Rationals of the Form A/b 2
In this note we formulate some questions in the study of approximations of reals by rationals of the form a/b arising in theory of Shrödinger equations. We hope to attract attention of specialists to this natural subject of number theory.
متن کاملComputable Approximations of Reals: An Information-Theoretic Analysis
How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the nite pre xes of the binary expansion of the real. Computable reals, whose binary expansions have a very low information content, can be approximated (very fast) with a computable convergence rate. Random reals, whose binary expan...
متن کاملA Constructive Proof of the Fundamental Theorem of Algebra without Using the Rationals
In the FTA project in Nijmegen we have formalized a constructive proof of the Fundamental Theorem of Algebra. In the formalization, we have first defined the (constructive) algebraic hierarchy of groups, rings, fields, etcetera. For the reals we have then defined the notion of real number structure, which is basically a Cauchy complete Archimedean ordered field. This boils down to axiomatizing ...
متن کاملAn involution of reals, discontinuous on rationals and whose derivative vanish almost everyhere
We study the involution of the real line, induced by Dyer’s outer automorphism of PGL(2,Z). It is continuous at irrationals with jump discontinuities at rationals. We prove that its derivative exists almost everywhere and vanishes almost everywhere.
متن کاملThe HoTT reals coincide with the Escardó-Simpson reals
Escardó and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy this universal property. We show that this is indeed the case in the category of sets of any universe. We also show that the type of HoTT reals is the least Cauch...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- The American Mathematical Monthly
دوره 118 شماره
صفحات -
تاریخ انتشار 2011