[FOM] Eradicating Evil?

[martdowd at aol.com]

 Here are some comments on Dr. Friedman's posting on eradicating evil and flat foundations.

> So let me start by trying to state the major issue in clear, but
> probably oversimplified terms.

One issue which should be considered is that of extending ZFC with
"strongly justified" axioms.  I1-I4 of "Some New Axioms for Set Theory"
is an obvious place to start.  I don't feel a great urge to learn
enough homotopy type theory to translate these into it (although I
admit to being curious).

> So immediately at least one simple form of the challenge arises - how
> do we do convenient f.o.m. without being committed to specific
> representations of fundamental mathematical structures?

There is no need to do this, but a successful such theory would only confirm
that the standard foundations provide (bu some fundamental fact of
nature) a method for elaborating all of mathematics.

> Actually, for a different purpose entirely (however, one can argue
> that the purpose is closely related), I have already written
> extensively about what I (almost have previously) called "flat
> foundations".
> Flat Mental Pictures, September 5, 2014, 5 pages.

Formalizing I1-I4 (in my notation) in some version of this theory might be
worth the trouble.  There's a notation conflict with the rank into rank
axioms, which do not qualify as "strongly justified" in my view.

> Before I give the formalization of MK in this setup, I would like to
> hear from radicals and moderates whether this is moving toward the
> "eradication of evil" that is so ardently desired.

It wouldn't be surprising if this could be done.  Working mathematicians
who use the epsilon relation routinely would of course regard it as a fact
of secondary interest.

- Martin Dowd
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