Re: Continuum hypothesis

george <greeneg@xxxxxxxxxx> writes:

But people usually content themselves with saying "2nd-order"
because "higher- order" is in fact reducible to 2nd; 2nd is in some
sense high "enough", i.e., powerful enough that adding higher
"orders" doesn't actually confer new "powers".

People usually content themselves with saying "second-order" because
they're talking about second-order stuff. There is a sense in which
n-th order logic reduces to second-order logic of course, as observed
by Hintikka in the 50's: there's an effective transformation that
transforms a sentence A in n-th order logic into a sentence A* in
second-order logic such that A is valid just in case A* is. Also, for
pretty much any mathematical statement P there is a Sigma-1-1
statement P' with the property that P is true iff P' is valid.

However, in another sense going higher order does "confer new
powers". There are classes of structures definable in n+1-th order
logic not definable in n-th order logic -- being measurable is
third-order but not second-order definable --, and for example
third-order arithmetic is stronger than second-order arithmetic.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.