FOM: Re: Neo-Fregean reverse mathematics

[Roger Bishop Jones]

In response to Alasdair Urquhart March 23, 2001 11:03 PM


> ... But a more
> serious problem in the "Neo-Fregean" literature is the persistence
> with which people refer to "Hume's Principle" as a
> "contextual definition."
..
> But how can it be a "contextual definition" when it has
> the axiom of infinity as a direct logical consequence?
> How can it be a definition of any kind at all?

Clearly it can be considered a definition in contexts in which
it is conservative. e.g. (HOL or Zermelo set theory) with infinity.

There is also precedent for counting non-conservative axioms as
"defining" the terms they introduce.
Isn't this idea attributable to Hilbert?
(it also appears in Computer Science under the heading "axiomatic
semantics").

> This brings me to my logical question.  Does anyone know
> what is the exact proof-theoretical strength of the
> cardinality principle?  That is to say, starting from
> the conventional first-order language of set theory,
> can we find a system that is in some sense equivalent to (some form of)
> higher order logic + the cardinality principle?  A result
> along this line would clarify the status of the cardinality
> principle.

Of course, by itself it has no proof theoretic strength.
The choice of context is significant, and Neil Tennant has observed
that in the absence of powersets it delivers more value.
Surely HOL + the cardinality principle has just the same
strength as HOL + infinity, which is <= Z (with inf)?

Roger Jones