[FOM] 216:New Pi01 Statements

Harvey Friedman friedman at math.ohio-state.edu
Sun Jun 6 18:33:35 EDT 2004

We present a new explicitly Pi01 statement provable from Mahlo cardinals of
finite order but not from ZFC.

Let R containedin [1,n]^k x [1,n]^k = [1,n]^2k. We say that R is strictly
dominating if and only if for all x,y in [1,n]^k, R(x,y) implies max(x) <

For A containedin [1,n]^k, we write R[A] = {y: (there exists x in

THEOREM 1. Let k,n >= 1 and R containedin [1,n]^k x [1,n]^k = [1,n]^2k.
There exists nonempty A containedin [1,n]^k such that A = [1,n]^k\R[A].
Furthermore, A is unique.

We say that R is order invariant if and only if for all x,y in [1,r]^2k, if
x,y have the same order type then x in R iff y in R.

Let A,B containedin [1,n]^t, t >= 1. We say that A,B are order equivalent if
and only if every element of A has the same order type as an element of B,
and every element of B has the same order type as an element of A.

PROPOSITION 2. Let n,r >= k^16k >= 2 and R containedin [1,n]^k x [1,n]^k =
[1,n]^2k be strictly dominating and order invariant. There exists nonempty
A containedin {1,...,r-2,r,...,n}^k such that

{(r,r^2,...,r^n)} x A x A and
{(r,r^2,...,r^n)} x A x [1,n]^k\R[A]

are order equivalent.

Note that if we write [1,n] instead of {1,...,r-2,r,...,n}, then we can
obviously write "equal" instead of "order equivalent". This is an obvious
consequence of Theorem 1.

THEOREM 3. Proposition 2 is provably equivalent, over ACA to the consistency
of MAH = ZFC + {there exists a k-Mahlo cardinal}_k. In particular,
Proposition 2 is provable in MAH+ = ZFC + "for all k there exists a k-Mahlo
cardinal", and cannot be proved in ZFC (provided ZFC is consistent).

We now introduce an additional parameter, p, in Proposition 2 as follows.

PROPOSITION 4. Let n,r >= k^16kp >= 2 and R containedin [1,n]^k x [1,n]^k =
[1,n]^2k be strictly dominating and order invariant. There exists A
containedin {1,...,r-2,r,...,n}^k such that

{(r,r^2,...,r^n)} x A^p x A and
{(r,r^2,...,r^n)} x A^p x [1,n]^k\R[A]

are order equivalent.

THEOREM 5. Proposition 4 for any fixed k is provable in MAH. This is false
for ZFC together with any "there exists a k-Mahlo cardinal", k fixed.
Proposition 4 for k = 3 is not provable in ZFC (provided ZFC is consistent).

These results are related to BRT = Boolean relation theory, but serve a
somewhat different purpose. BRT has a particularly strong thematic
character, with potential points of contact with perhaps all areas of
mathematics. These Propositions are simply the most mathematically natural
explicitly Pi01 statements independent of ZFC that we have been able to find
- yet.


I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.
This is the 186th in a series of self contained numbered postings to
FOM covering a wide range of topics in f.o.m. The list of previous
numbered postings #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html  in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:

150:Finite obstruction/statistics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
152:sin  10:35PM  6/8/02
153:Large cardinals as general algebra  1:21PM  6/17/02
154:Orderings on theories  5:28AM  6/25/02
155:A way out  8/13/02  6:56PM
156:Societies  8/13/02  6:56PM
157:Finite Societies  8/13/02  6:56PM
158:Sentential Reflection  3/31/03  12:17AM
159.Elemental Sentential Reflection  3/31/03  12:17AM
160.Similar Subclasses  3/31/03  12:17AM
161:Restrictions and Extensions  3/31/03  12:18AM
162:Two Quantifier Blocks  3/31/03  12:28PM
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169:New PA Independence  5:11PM  8:35PM
170:New Borel Independence  5/18/03  11:53PM
171:Coordinate Free Borel Statements  5/22/03  2:27PM
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Harvey Friedman

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