Let M be a nonstandard model of PA. (a) For all a, b = M, define a~ b ⇒ |a−b| € NM, where z = : |a − b] is the unique element in M such that x + z = y or y + z = x. Show that is an equivalence relation on M and that it is NOT definable in M.
Let M be a nonstandard model of PA. (a) For all a, b = M, define a~ b ⇒ |a−b| € NM, where z = : |a − b] is the unique element in M such that x + z = y or y + z = x. Show that is an equivalence relation on M and that it is NOT definable in M.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...
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![Let M be a nonstandard model of PA.
(a) For all a, b = M, define
a~b ↔ |a − b| € NM,
where z =
|a − b] is the unique element in M such that x + z = y or y + z = x. Show
that ~ is an equivalence relation on M and that it is NOT definable in M.
(b) Let Q = M/~denote the quotient by this equivalence relation, i.e. Q := {[a] : a ≤ M},
where [a] denotes the equivalence class of a. Define the relation <q on Q as follows:
for all [a], [b] € Q,
E
[a] <q [b] ⇒ there is c E M \ NM such that a + c = b.
Show that <Q is well-defined (does not depend on the representatives a, b) and is a
strict linear order on Q.
(c) Show that the order (Q, <q) has a least element but no greatest element, and it is a
dense (in itself), i.e. u <q v ⇒ ]w(u <q w <q v) for all u, v ¤ Q.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b945f66-390e-417d-9fc3-363b543734f1%2Fb3c324b8-221c-4254-804f-e49057f61437%2F7rj0k86_processed.png&w=3840&q=75)
Transcribed Image Text:Let M be a nonstandard model of PA.
(a) For all a, b = M, define
a~b ↔ |a − b| € NM,
where z =
|a − b] is the unique element in M such that x + z = y or y + z = x. Show
that ~ is an equivalence relation on M and that it is NOT definable in M.
(b) Let Q = M/~denote the quotient by this equivalence relation, i.e. Q := {[a] : a ≤ M},
where [a] denotes the equivalence class of a. Define the relation <q on Q as follows:
for all [a], [b] € Q,
E
[a] <q [b] ⇒ there is c E M \ NM such that a + c = b.
Show that <Q is well-defined (does not depend on the representatives a, b) and is a
strict linear order on Q.
(c) Show that the order (Q, <q) has a least element but no greatest element, and it is a
dense (in itself), i.e. u <q v ⇒ ]w(u <q w <q v) for all u, v ¤ Q.
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