[FOM] hierarchy of methods

Harvey Friedman friedman at math.ohio-state.edu
Thu Feb 2 18:46:28 EST 2006

On 2/2/06 3:59 PM, "hendrik at topoi.pooq.com" <hendrik at topoi.pooq.com> wrote:

> The trouble is, the path of *ever* stronger methods is not unique.

It is an absolutely crucial feature of these ever stronger methods that the
path upwards appears to be unique for

arithmetical consequences.

Thus, even though there are many natural examples of pairs of systems in the
logical hierarchy ranging from EFA through j:V into V that are incompatible,
in the sense that neither proves the other, we always have (in any case that
is not artificially created to be otherwise) that

T1 and T2 prove the same arithmetic sentences or

T1 proves properly more arithmetic sentences than T2 or

T2 proves properly more arithmetic sentences than T1.

(Of course, in the microstructure of weak fragments of arithmetic, this
isn't quite true, but it is true even there for Pi01 sentences.)

I often state this in terms of the trichotomy principle for
*interpretability* in the very strong sense:

T1,T2 are mutually interpretable (and stronger) or

T1 proves the consistency of T2 or

T2 proves the consistency of T1.

This is probably the deepest, most mysterious, and as yet totally
unexplained, phenomenon in the whole of f.o.m. Probably one of the top few
such in the whole range of human intellectual activity.

At some point, we will have an explanation, or at least a theorem that says
that all systems obeying certain conditions are comparable in these senses.

Then it will become obvious that an informed discussion of f.o.m. matters
will require full awareness of the explanation.

This is a perfect example of the kind of fundamental f.o.m. information that
we simply do not have at this point, that will affect f.o.m. issues in
unexpected ways. 

Harvey Friedman

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