# FOM: Significance and significant people

Torkel Franzen torkel at sm.luth.se
Thu Mar 19 05:05:27 EST 1998

Neil Tennant says:

>Not so; the epochal advance intimated at this
>stage is that for any proposed large cardinal axiom, Friedman's method
>can be adjusted to provide a natural combinatorial principle that
>turns out to be provably equivalent to the consistency of that large
>cardinal.

OK, so what you regard as an epochal advance is a result "intimated"
but not stated. It would less misleading, I think, to speak of
an "intimated" or "hypothetical" epochal advance.

Setting aside the matter of "epochal advance", which surely is of
minor importance, you explain further how you look at the importance
of the stated result:

>Imagine writing down, in primitive notation,

>(B)	Con(ZFC + existence of appropriate large cardinals)

>so that its combinatorial character is explicit.  Would the result
>strike you as (i.e. look to you like) the kind of claim that ought,
>if true, to be true for simple reasons? I think not, since you know
>its provenance as, precisely, a consistency claim about an extremely
>powerful extension of currently accepted mathematics.

To this I can only reply that written in primitive notation, this
statement wouldn't strike me as anything at all, since I wouldn't be
able to make head or tail of it. As to whether (B) "ought, if true, be
true for simple reasons", I don't see any grounds for holding this.

You contrast (B) with (A):

>It would be several orders of syntactic magnitude easier to write
>down, in similarly primitive notation, Friedman's combinatorial
>principle (A). Of this statement I would say that it looks to me like
>the kind of claim that ought, if true, to be true for simple
>reasons. Again, this is because I know its provenance as a claim about
>inserting nodes in finite trees according to certain clearly stated
>rules.

You emphasize the "provenance" of (A), by which I assume you mean that
you know Friedman to have arrived at this statement, not in the course
of thinking about large cardinals, but in the course of thinking about
inserting nodes in finite trees. I suppose some such distinction also
explains why you are not equally impressed by an equivalent of B in
the form of a statement of the form "the Diophantine equation
p(k1,..kn)=0 has no solution". Would you say that such a statement
"ought, if true, to be true for simple reasons" if it has arisen in
the context of Diophantine studies, but not necessarily if it has been
first brought to our attention via recursion theory? I don't see
any justification for such a view.

You comment, finally:

>THEN one looks at Friedman's result, and learns that (A) implies
>(B). One realizes further that, if one were ever to prove the
>combinatorial principle (A), it could only be by dint of assuming the
>existence of a large cardinal even larger than those whose consistency
>is asserted by (B).

I don't realize anything of the kind. What is your justification for this
view?

---
Torkel Franzen