# [FOM] 427: Finite Games and Incompleteness 1

Harvey Friedman friedman at math.ohio-state.edu
Thu Jun 10 16:08:10 EDT 2010

```THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION.

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The game theoretic approach was mentioned in

425: Well Behaved Reduction Functions 4 http://www.cs.nyu.edu/pipermail/fom/2010-May/014790.html
426: Well Behaved Reduction Functions 5 http://www.cs.nyu.edu/pipermail/fom/2010-June/014801.html

Here we refine this approach.

It appears that we have reached a new level of simplicity, clarity,
and relationship with mathematical subjects.

Which mathematical subjects?

i. Discrete and Finite Game Theory. We make direct use of rudimentary
forms of games (relation games) and winning strategies.
ii. Ramsey theory. Ramsey's original 1930 theorems can be viewed, very
naturally, as well behavedness theorems. This is also true of many
subsequent developments in Ramsey theory.

These two areas are combined in an entirely transparent way in order
to create striking incompleteness from ZFC - in fact, necessary uses
of large cardinals for proving explicitly Pi01 statements.

TEMPLATE. Every downward relation game on N^k has a length k tower of
restricted uniform winning strategies which are well behaved over a
common infinite set.

TEMPLATE. Every downward relation game on N^k has a length k tower of
finite restricted uniform winning strategies which are well behaved
over a common k element set.

PROPOSITION. Every downward relation game on N^k has a length k tower
of restricted uniform wining strategies which are lower shift
invariant over a common infinite set.

PROPOSITION. Every downward relation game on N^k has a length k tower
of finite restricted uniform winning strategies which are lower shift
invariant over a common k element set.

And also various forms that are explicitly Pi01.

THIS POSTING IS SELF CONTAINED.

FINITE GAMES AND INCOMPLETENESS 1
by
Harvey M. Friedman
June 10, 2010

1. GAMES AND WINNING STRATEGIES.
2. TOWERS AND LOWER SHIFT INVARIANCE.
3. INFINITE GAME THEOREM.
4. FINITE GAME THEOREMS.
5. ORDER INVARIANT GAME THEOREMS.

1. GAMES AND WINNING STRATEGIES.

Let A be a set. We say that G is a relation game on A if and only if G
is a subset of A x A, and we use the following elemental rules of play.

Alice and Bob alternately choose elements of A, with Alice making the
first move. This opening move of G is an arbitrary element of A. If a
player makes the move x, and the opponent responds with y such that
not x G y, then the game stops, and the player wins and the opponent
loses.

In general, the play of G on A may last for infinitely many moves,

To avoid this situation, we say that G is a win/loss relation game on
A if and only if all plays of the game are of finite length.

The moves of Alice (Bob) are called winning or losing according to the
fundamentals of game theory. Since the game is entirely symmetric, x
in A is winning (losing) for Alice, if and only if it is winning
(losing) for Bob.

We say that f is a uniform winning strategy for the win/loss relation
game G on the set A if and only if f:A into A, where for all x in A,
if x is losing then f(x) is winning.

The restrictions of G are the games G intersect B x B on B, where B is
a subset of A.

The restricted uniform winning strategies of a win/loss relation game
G on A are the uniform winning strategies of the restrictions of G.

We say that G is a downward relation game on N^k if and only if G is a
game on N^k, where x G y implies max(x) > max(y). This obviously
implies that G, and all of its restrictions, are win/loss games.

A cube is a set of the form E^k, E contained in N. The length of E^k
is the cardinality of E.

A tower of restricted uniform winning strategies for a downward
relation game G on N^k consists of restricted uniform winning strategies

f_1 contained in f_2 contained in ... contained in f_p

where each dom(f_i) is contained in a subcube of dom(f_i+1).

We can now state the general form of some of our Propositions.

TEMPLATE. Every downward relation game on N^k has a length k tower of
restricted uniform winning strategies that are well behaved over some
common infinite set.

TEMPLATE. Every downward relation game on N^k has a length k tower of
finite restricted uniform winning strategies that are well behaved
over some common k element set.

2. LOWER SHIFT INVARIANCE.

For E contained in N, we define the E shift as the map that sends x in
E to the next largest element of E. Obviously, the E shift is
undefined at the largest element of E. If E is infinite then the E
shift maps E into E.

The E shift lifts naturally to E^k, by acting on coordinates.

We say that partial f:N^k into N^k is shift invariant over E if and
only if E is contained in N, and for all x,y in E^k, if y is the E
shift of x, then f(x) = f(y).

This notion is too strong for our purposes. We are interested in a
weakening of this notion that works well with many familiar
mathematical functions.

We say that partial f:N^k into N^k is lower shift invariant over E if
and only if E is contained in N, and for all x,y in E^k and 1 <= i <=
k, if the E shift of x is y and f_i(x) < min(x), then f_i(x) = f_i(y).
Here f_i is the i-th coordinate function of f.

There are some standard mathematical contexts where lower shift
invariance arises.

THEOREM 2.1. Every affine transformation T:N^k into N^k is lower shift
invariant over N. Every polynomial P;N^k into N^k with nonnegative
integer coefficients is lower shift invariant over N. Every piecewise
affine transformation T:N^k into N^k is lower shift invariant over
some infinite geometric progression. Every piecewise polynomial P:N^k
into N^k with nonnegative integer coefficients is lower shift
invariant over a tail of the double factorials.

In the theory of large cardinals, we have the following.

THEOREM 2.2. Let k >= 1. SRP proves that there exists lambda such that
every f:lambda^k into lambda^k is lower shift invariant over an
unbounded subset of lambda. SRP+ proves that for every k, there exists
lambda such that every f:lambda^k into lambda^k is lower shift
invariant over an unbounded subset of lambda. SRP and SRP+ are best
possible here over ZFC.

3. INFINITE GAME THEOREM.

PROPOSITION 3.1. Every downward relation game on N^k has a length k
tower of restricted uniform winning strategies that are lower shift
invariant over some common infinite set.

THEOREM 3.2. Proposition 3.1 is provable in SRP+ but not from any
consequence of SRP that is consistent with RCA_0. Propositions 3.1 is
provably equivalent, over WKL_0, to Con(SRP).

Here SRP+ = ZFC + "for all k there exists a limit ordinal with the k-
SRP. SRP = ZFC + {there exists a limit ordinal with the k-SRP}_k. The
k-SRP asserts that every partition of the unordered k-tuples from
lambda into two pieces has a homogeneous set that is a stationary
subset of lambda.

4. FINITE GAME THEOREMS.

We write [t] for {0,1,...,t}.

PROPOSITION 4.1. Every downward relation game on N^k has a length k
tower of restricted uniform winning strategies that are lower shift
invariant over some common k element set.

PROPOSITION 4.2. Let t > 2^[8k]. Every downward relation game on [t]^k
has a tower of restricted uniform winning strategies that are lower
shift invariant over some common k element set.

Note that Proposition 4.2 is explicitly Pi01.

THEOREM 4.3. Proposition 4.1, 4.2 are provable in SRP+ but not from
any consequence of SRP that is consistent with RCA_0. Propositions
4.1, 4.2 are provably equivalent, over RCA_0, to Con(SRP). In the case
of Proposition 4.2, SEFA suffices.

SEFA is super exponential function arithmetic.

5. ORDER INVARIANT GAME THEOREM.

We say that x,y in N^k are order equivalent if and only if for all 1
<= i,j <= k, x_i < x_j iff y_i < y_j.

We say that V contained in N^k is order invariant if and only if for
all order equivalent x,y in N^k, x in V iff y in V.

We say that a game G on N^k is order invariant if and only if G is
order invariant as a subset of N^2k.

We say that a partial f:N^k into N^k is <= p if and only if every
coordinate of every coordinate of its domain and range is <= p.

PROPOSITION 5.1. Every downward order invariant relation game on N^k
has a tower of restricted uniform winning strategies <= k(8k)!! that
are lower shift invariant over {(8k)!!,2(8k)!!,...,k(8k)!!}.

Note that Proposition 5.1 is explicitly Pi01.

THEOREM 5.2. Proposition 5.1 is provable in SRP+ but not from any
consequence of SRP that is consistent with EFA. Proposition 5.1 is
provably equivalent, over EFA, to Con(SRP).

EFA is exponential function arithmetic.

**********************

manuscripts. This is the 427th 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 #1-349 can be found athttp://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.

350: one dimensional set series  7/23/09  12:11AM
351: Mapping Theorems/Mahlo/Subtle  8/6/09  10:59PM
352: Mapping Theorems/simpler  8/7/09  10:06PM
353: Function Generation 1  8/9/09  12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1  8/9/09  6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2  8/10/09  6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem  8/14/09  9:31AM
357: HIGH SCHOOL Games/Update  8/20/09  10:42AM
358: clearer statements of HIGH SCHOOL Games  8/23/09  2:42AM
359: finite two person HIGH SCHOOL games  8/24/09  1:28PM
360: Finite Linear/Limited Memory Games  8/31/09  5:43PM
361: Finite Promise Games  9/2/09  7:04AM
362: Simplest Order Invariant Game  9/7/09  11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1  9/24/09  1:06PM
366: Free Reductions and Large Cardinals/polished  9/28/09 2:19PM
367: Upper Shift Fixed Points and Large Cardinals  10/4/09 2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction 10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement  10/29/09 2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09  12:14PM
371: Vector Reduction and Large Cardinals  11/21/09  1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09  5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals  12/7/09  9:17AM
374: Upper Shift Greedy Chain Games  12/12/09  5:56AM
375: Upper Shift Clique Games and Large Cardinals 1graham
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09  2:23PM
377: The Polynomial Shift Theorem  12/25/09  2:39PM
378: Upper Shift Clique Sequences and Large Cardinals  12/25/09 2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems  12/28/09  7:06AM
381: Trigonometric Shift Theorem  12/29/09  11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals  12/30/09 2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09  3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION 12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences  1/1/10  7:35PM
386: Terrifically and Extremely Long Finite Sequences  1/1/10 7:35PM
387: Better Polynomial Shift Translation/typos  1/6/10  10:41PM
388: Goedel's Second Again/definitive?  1/7/10  11:06AM
389: Finite Games, Vector Reduction, and Large Cardinals 1 2/9/10
3:32PM
390: Finite Games, Vector Reduction, and Large Cardinals 2 2/14/09
10:27PM
391: Finite Games, Vector Reduction, and Large Cardinals 3 2/21/10
5:54AM
392: Finite Games, Vector Reduction, and Large Cardinals 4 2/22/10
9:15AM
393: Finite Games, Vector Reduction, and Large Cardinals 5 2/22/10
3:50AM
394: Free Reduction Theory 1  3/2/10  7:30PM
395: Free Reduction Theory 2  3/7/10  5:41PM
396: Free Reduction Theory 3  3/7/10  11:30PM
397: Free Reduction Theory 4  3/8/10  9:05AM
398: New Free Reduction Theory 1  3/10/10  5:26AM
399: New Free Reduction Theory 2  3/12/10  9:36AM
400: New Free Reduction Theory 3  3/14/10  11:55AM
401: New Free Reduction Theory 4  3/15/10  4:12PM
402: New Free Reduction Theory 5  3/19/10  12:59PM
403: Set Equation Tower Theory 1  3/22/10  2:45PM
404: Set Equation Tower Theory 2  3/24/10  11:18PM
405: Some Countable Model Theory 1  3/24/10  11:20PM
406: Set Equation Tower Theory 3  3/25/10  6:24PM
407: Kernel Tower Theory 1  3/31/10  12:02PM
408: Kernel tower Theory 2  4/1/10  6:46PM
409: Kernel Tower Theory 3  4/5/10  4:04PM
410: Kernel Function Theory 1  4/8/10  7:39PM
411: Free Generation Theory 1  4/13/10  2:55PM
412: Local Basis Construction Theory 1  4/17/10  11:23PM
413: Local Basis Construction Theory 2  4/20/10  1:51PM
414: Integer Decomposition Theory  4/23/10  12:45PM
415: Integer Decomposition Theory 2  4/24/10  3:49PM
416: Integer Decomposition Theory 3  4/26/10  7:04PM
417: Integer Decomposition Theory 4  4/28/10  6:25PM
418: Integer Decomposition Theory 5  4/29/10  4:08PM
419: Integer Decomposition Theory 6  5/4/10   10:39PM
420: Reduction Function Theory 1  5/17/10   2:53AM
421: Reduction Function Theory 2  5/19/10   12:00PM
422: Well Behaved Reduction Functions 1  5/23/10  4:12PM
423: Well Behaved Reduction Functions 2  5/27/10  3:01PM
424: Well Behaved Reduction Functions 3  5/29/10  8:06PM
425: Well Behaved Reduction Functions 4  5/31/10  5:05PM
426: Well Behaved Reduction Functions 5  6/2/10  12:43PM

Harvey Friedman

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