FOM: corrected conjecture
Harvey Friedman
friedman at math.ohio-state.edu
Wed Mar 18 01:57:07 EST 1998
In 8:13AM 3/16/98, in response to Feferman, I made the following
conjectures, reproduced here:
**********
Here are some conjectures that I have confidence in, but must fall short of
claiming:
PROPOSITION 1. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set are
vertices with the same number of ancestors.
PROPOSITION 2. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set with
the same first k-1 elements are vertices with the same entirely lower
ancestors.
PROPOSITION 3. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set with
the same min are vertices with the same entirely lower ancestors.
PROPOSITION 4. Let k,p >= 1. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set are
vertices with the same entirely lower ancestors.
1 and 4 are from the paper. In the paper, 4 was proved from ZFC + (for all
k)(there exists a k-subtle cardinal). And in the paper, 4 was proved to
imply the consistency of ZFC + {there exists a k-subtle cardinal}_k. Also 1
was proved in the paper from Z. I know how to prove 2 in ZFC + there exists
an inaccessible cardinal. I know how to prove 3 in ZFC + "for all k, there
exists a k-Mahlo cardinal."
CONJECTURES. 1 is provably equivalent to the 1-consistency of Z with
bounded separation (or type theory). 2 is provably trapped between the
1-consistency of ZFC and the 1-consistency of MKC. 3 is provably equivalent
to the 1-consistency of ZFC + {there exists a k-Mahlo cardinal}_k. 4 is
provably equivaletn to the 1-consistency of ZFC + {there exists a k-subtle
cardinal}_k.
*********
Proposition 2 should have read:
PROPOSITION 2. Let k,p >= 2. Every nonincreasing insertion domain in TR(k)
contains a k-tree in which all k element subsets of some p element set with
the same first k-2 elements are vertices with the same entirely lower
ancestors.
The weaker form of 2 that I wrote probably corresponds to, roughly,
omega_1, or second order arithmetic.
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