FOM: Boolean algebra vs Boolean ring

Stephen G Simpson simpson at math.psu.edu
Thu Mar 12 10:42:52 EST 1998

```Jaap van Oosten writes:
> Algebra books define what an isomorphism of groups is, an isomorphism
> of rings, etc.; you will never find in an algebra book the statement that
> the group of integers is not isomorphic to the field of rationals.

OK, I take your point.  Your point is that classical algebra books
(Lang, etc.) don't define the concept of isomorphism in general; they
only define it piece-meal for specific structures such as groups,
rings, etc.  But what about universal algebra books, e.g. books by
P.M.Cohen, Graetzer, etc.?  In there I think they define isomorphism
in general, and part of the definition is that isomorphic structures
have the same signature.  Sorry, I don't have any of those books handy
right now to pull out a reference, but -- dare I ask -- do you agree?

> If two structures have different types you can't talk about
> isomorphism,

You can talk about things like this if you have a comprehensive
foundational scheme such as set-theoretic foundations, because
everything is a set or whatever.  Perhaps you are thinking of
something else, e.g. category-theoretic dys-foundations.  In catogory
theory, "isomorphism" is only defined within each category, not
generally.

> You created the confusion yourself with the off-base comment that
> Boolean algebras are not isomorphic to Boolean rings,

I still want to claim that Boolean algebras are not isomorphic to
Boolean rings.  Do you agree with this now?  Try to put yourself in
the appropriate frame of mind.

> that correct?") to your children.

I think questions such as "Do you agree?" are appropriate in the
context of my discussion with Pratt and other proponents of
categorical faux-foundations.  I am trying to establish a basis for
rational discussion, and I need to know how far back we have to go.

> kind of trivia?

Why are you so annoyed?

I'm annoyed too, but my reason for being annoyed is probably different
from yours.  The reason I'm annoyed is that proponents of categorical
dys-foundations seem to be willing to deny even the elements of basic
mathematics in order to make their dubious points.

-- Steve

```