Re: Review of Mueckenheims book.

On Mar 16, 7:10 pm, "MoeBlee" <jazzm...@xxxxxxxxxxx> wrote:
On Mar 16, 4:25 pm, Virgil <vir...@xxxxxxxxxxx> wrote:

To my mind, if the consequences of differing definitions differ then the
definitions may fairly be said to disagree.

If the axioms are common, then different definitions give theories
that disagree syntactically but that can still be interpreted (if I am
not mistaken, that is the correct word in mathematical logic for this)
in each other, thus, in that sense, essentially the same theory.

P.S. Though, if I am stating this correctly, not only do we show each
interpretable in the other, but also that the interpretations of each
are a conservative extension of the other. Perhaps someone can help me
out with the precise statement of this. In any case, it's just pretty
much common sense that if two people start with the same axioms but
decide to use different definitions, then their theories "say" the
same thing except with different "choice of words", since the
mathematical semantics (what the theory "says" in terms of a domain of
discourse and relations therein) will "wash over" the specific
differences in choices of definitions. Thus, as for example, we (yes,
editorial 'we') try (never successfully) to get the crank to see that
whether we call it 'cardinality' or 'equinumerosity' or 'equipotence'
or 'schmardinality' or whatever is not what is of mathematical
importance in terms of what the theory "says".

I need to have a precise way of saying that in mathematical logic. I'm
interested two different kinds of situations:

(1) Where the axioms and thus the languages are identical, but the
theory then splits into two different conservative extensions by
definitions (i.e. at least one of the new defined symbols for one
theory has an incompatible definition in the other theory). What is
the way to state that these theories are nevertheless "saying" the
same thing?

(2) Where the axioms and even the signature of the languages are
different, but the theories "say" the same thing. For example, a group
theory T in which the signature is just <'+'> with axioms and the
identity element and the inverse operation are defined, as opposed to
a group theory T' where the signature is <'+' '-' '0'> and with
different axioms from T and with no need to define the identity
element or inverse operation. What is the way to describe that
situation in general and to state that the theories are "saying" the
same thing?

Perhaps Aatu or someone can help me out here.

MoeBlee


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