[FOM] re: synonymity

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Mon Feb 10 23:21:44 EST 2003

  (Apologies if this just repeats stuff posted long ago.)
  The best source I know for some EXAMPLES of synonymous theories is the
article "String Theory" by Corcoran, Frank and Shapiro, "Journal of
Symbolic Logic" 1974: proving that second order arithmetic is synonymous
with some second order theories of strings of symbols.  John Corcoran
discussed the general concept in a talk that generated a JSL abstract about
the same time, but I don't know if there is a longer publication.  (I
suspect that some of the famous old examples of mutually interpretable
theories-- e.g. Robinson Arithmetic and Tarski's "baby" set theory
(described in Tarski, Mostowski and Robinson)-- can be shown to be examples
of synonymous theories, but this is just a suspicion.)
   Examples on the other side-- pairs of theories which are mutually
interpretable but not synonymous-- have been mentioned in the literature,
but I don't know of any published example with details.  I **think** A and
B, below, may provide such an example: when I asked David Kaplan (who is
reported in the literature as finding examples) about it a couple of years
ago, he said he couldn't remember the details, but that A and B sounded as
if they were on the  right track.  I don't know if there are any well-known
non-artificial examples.
    A: First Order theory (with nodes of tree as domain and predicate for
the parent-of relation) of a tree in which the origin has three (3)
children, each of those children has two (2), with subsequent "generations"
alternating between generations in which every node has three children and
generations in which every node has two.
    B: Same as A, except that this time the origin (and nodes in subsequent
even-numbered generations) has (have) two (2) children, and nodes in
odd-numbered generations have three.
    But  I haven't ever worked out the details of a proof.
Allen Hazen
Philosophy Department
University of Melbourne

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