# [FOM] 358: clearer statements of HIGH SCHOOL Games

Harvey Friedman friedman at math.ohio-state.edu
Sun Aug 23 02:42:48 EDT 2009

```This is a continuation of http://www.cs.nyu.edu/pipermail/fom/2009-August/013960.html
on solitaire games of a deceptively elementary nature.

We make better use of the "invert" terminology in the statement of the
solitaire games. This leads to clearer and more memorable statements.

###############################

As usual, SMAH+ = ZFC + "for all k there is a strongly k-Mahlo
cardinal". SMAH = ZFC + {there is a strongly k-Mahlo cardinal}_k. That
various solitaire games can be won, is provable in SMAH+ but not in ZFC.

NOTE: The current plan is to have a manuscript with proofs later in
the calendar year. This will iron out all of the bugs. The development
is closely related to the book on Boolean Relation Theory, an advanced
draft of which exists on my website http://www.math.ohio-state.edu/%7Efriedman/

I apologize in advance for any remaining bugs.

1. Copy/Invert game with a binary relation, addition, and
multiplication.
2. Copy/Invert game with a binary function, addition, and
multiplication.
3. Copy/Invert game with a multivariate function and addition.
4. CopyInvert game with a polynomial.
5. Copy/Invert games with multivariate relations.
6. Finite Copy/Invert games.
7. Mapping Theorem - Explicitly Pi01.

1. COPY/INVERT GAME WITH A BINARY RELATION, ADDITION, AND
MULTIPLICATION.

Let R be a binary relation on Z+. An inversion of R at x is a y < x
such that R(y,x).

A nontrivial product is a product of two positive integers, none of
which is 1.

We define a solitaire game with n rounds, n >= 1. The rounds result in
subsets X[1],...,X[n] of Z+. It will be arranged that X[1]
containedin ... containedin X[n].

At the first round, you can play any subset of Z+ that includes 1.

Let 1 <= i < n, and suppose X[i] has been played. X[i+1] includes X[i]
together with integers obtained by the following specific
nondeterministic process.

You first enumerate the sums and nontrivial products of pairs from
X[i] in numerical order (although this is natural but not really
important). We allow both coordinates in pairs to be the same. These
are called the CANDIDATES (candidates for insertion into X[i+1]).

For each successive candidate x, you throw x or an R inversion of x
into X[i+1]. The first option is called "copy" and the second option
is called "invert". It may be the case that integers thrown into X[i
+1] under either option are already in X[i+1] or even already in X[i].

Note that the aggregate of candidates during the play of the game are
just the sums and nontrivial products of pairs of elements of X[r-1].
In case n = 1, there are no candidates.

There are obviously no obstructions to play thus far.

You are considered to have won if and only if

NO CANDIDATE IN X[n] HAS AN R INVERSION IN X[n].

This winning condition implicitly imposes profound obstructions during
play which are very difficult to control.

REMARK. We are demanding that 1 be included in the first play. We seem
to need this for some of the independence proofs. We can get away
without this for many of the results, but have decided to include it
throughout the abstract, as it is quite natural.

THEOREM 1.1. For all R contained in Z+ x Z+ and n >= 1, the copy/
invert game with R,+,x,n can be won where X[1] is infinite.

PROPOSITION 1.2. For all R contained in Z+ x Z+, n >= 1, and
sufficiently large integers t, the copy/invert game with R,+,x,n can
be won where X[1] is an infinite set of integers including t.

PROPOSITION 1.3. For all R contained in Z+ x Z+ and n >= 1, the copy/
invert game with R,+,x,n can be won where X[1] is an infinite set of
odd integers.

PROPOSITION 1.4. For all R contained in Z+ x Z+, n >= 1, and infinite
E contained in Z+, the copy/invert game with R,+,x,n can be won where
X[1] is an infinite subset of E together with 1.

THEOREM 1.5. Theorem 1.1 is provable in RCA_0. Propositions 1.2-1.4
are provable in SMAH+ but not in any consistent fragment of SMAH. They
are provably equivalent, over ACA, to 1-Con(SMAH).

We use both addition and multiplication. It is a stretch to be able to
use addition only (or multiplication only), and we are not expecting
that.

There is the question of how nice the relations R can be so that we
have this strong unprovability. We give an answer to this using
bounded Diophantine relations. We can use much more restricted classes
in sections 3-5.

Recall that a Diophantine relation is a relation on Z of the form

R(n_1,...,n_k) iff (there exists m_1,...,m_t in Z)
(P(n_1,...,n_k,m_1,...,m_t) = 0)

where P is a polynomial with integer coefficients.

A bounded Diophantine relation is a relation on Z of the form

R(n_1,...,n_k) iff (there exists m_1,...,m_t in [-
Q(n_1,...,n_k),Q(n_1,...,n_k)])(P(n_1,...,n_k,m_1,...,m_t) = 0)

where P,Q are polynomials with integer coefficients.

THEOREM 1.6. Theorem 1.5 holds even if we restrict to bounded
Diophantine relations. In Propositions 1.2 - 1.4, the winning sets can
be taken to be arithmetic (in the sense of recursion theory).

There is a way to get good finite independent statements without
restricting the class of functions, and looking only on initial
segments of integers. This involves the placement of integers on
finite initial segments of the integer number line. We will take this
matter up later in its own section - FINITE GAMES. See section 6 below.

2. COPY/INVERT GAME WITH A BINARY FUNCTION AND ADDITION, MULTIPLICATION.

Let f:Z+^2 into Z+. An f inversion at x is some pair y,z < x such that
f(y,z) = x.

We define a solitaire game with n rounds, n >= 1. The rounds result in
subsets X[1],...,X[n] of Z+. It will be arranged that X[1]
containedin ... containedin X[n].

At the first round, you can play any subset of Z+ that includes 1.

Let 1 <= i < n, and suppose X[i] has been played. X[i+1] includes X[i]
together with integers obtained by the following specific
nondeterministic process.

You first enumerate the sums and products of pairs from X[i] in
numerical order (although this is natural but not really important).
We allow both coordinates in pairs to be the same. These are called
the CANDIDATES (candidates for insertion into X[i+1]).

For each successive candidate x, you throw x or the two components of
some f inversion of x into X[i+1]. The first option is called "copy"
and the second option is called "invert". It may be the case that
integers thrown into X[i+1] under either option are already in X[i+1]

There are obviously no obstructions to play thus far.

You are considered to have won if and only if

NO ELEMENT OF X[n] HAS AN f INVERSION WHOSE TWO COMPONENTS ARE FROM
X[n].

This winning condition implicitly imposes profound obstructions during
play which are very difficult to control.

THEOREM 2.1. For all f:Z+^2 into Z+ and n >= 1, the copy/invert game
with f,+,x,n can be won where X[1] is infinite.

PROPOSITION 2.2. For all f:Z+^2 into Z+, n >= 1, and sufficiently
large integers t, the copy/invert game with f,+,x,n can be won where
X[1] is an infinite set of integers containing t.

PROPOSITION 2.3. For all f:Z+^2 into Z+ and n >= 1, the copy/invert
game with f,+,x,n can be won where X[1] is an infinite set of odd
integers.

PROPOSITION 2.4. For all f:Z+^2 into Z+, n >= 1, and infinite E
contained in Z+, the copy/invert game with f,+,x,n can be won where
X[1] is an infinite subset of E together with 1.

THEOREM 2.5. Theorem 2.1 is provable in RCA_0. Propositions 2.2-2.4
are provable in SMAH+ but not in any consistent fragment of SMAH. They
are provably equivalent, over ACA, to 1-Con(SMAH).

THEOREM 2.6. Theorem 2.5 holds even if we restrict to f whose graph is
a bounded Diophantine relation. In Propositions 2.2 - 2.4, the winning
sets can be taken to be arithmetic (in the sense of recursion theory).

NOTE: It is a reasonable challenge here to eliminate the use of
multiplication. This should be clarified when I write the manuscript
later this year.

3. COPY/INVERT GAME WITH A MULTIVARIATE FUNCTION AND ADDITION.

Let f:Z+^k into Z+. An f inversion at x is a k-tuple y_1,...,y_k < x
such that f(y_1,...,y_k) = x.

We define a solitaire game with n rounds, n >= 1. The rounds result in
subsets X[1],...,X[n] of Z+. It will be arranged that X[1]
containedin ... containedin X[n].

At the first round, you can play any subset of Z+ that includes 1.

Let 1 <= i < n, and suppose X[i] has been played. X[i+1] includes X[i]
together with integers obtained by the following specific
nondeterministic process.

You first enumerate the sums of pairs from X[i] in numerical order
(although this is natural but not really important). We allow both
coordinates in pairs to be the same. These are called the CANDIDATES
(candidates for insertion into X[i+1]).

For each successive candidate x, you throw x or the components of some
f inversion of x into X[i+1]. The first option is called "copy" and
the second option is called "invert". It may be the case that integers
thrown into X[i+1] under either option are already in X[i+1] or even

There are obviously no obstructions to play thus far.

You are considered to have won if and only if

NO ELEMENT OF X[n] HAS AN f INVERSION WHOSE k COMPONENTS ARE FROM X[n].

This winning condition implicitly imposes profound obstructions during
play which are very difficult to control.

THEOREM 3.1. For all f:Z+^k into Z+ and n >= 1, the copy/invert game
with f,+,n can be won where X[1] is infinite.

PROPOSITION 3.2. For all f:Z+^k into Z+, n >= 1, and sufficiently
large integers t, the copy/invert game with f,+,n can be won where
X[1] is an infinite set of integers containing t.

PROPOSITION 3.3. For all f:Z+^k into Z+ and n >= 1, the copy/invert
game with f,+,n can be won where X[1] is an infinite set of odd
integers.

PROPOSITION 3.4. For all f:Z+^k into Z+, n >= 1, and infinite E
contained in Z+, the copy/invert game with f,+,n can be won where X[1]
is an infinite subset of E together with 1.

THEOREM 3.5. Theorem 3.1 is provable in RCA_0. Propositions 3.2-3.4
are provable in SMAH+ but not in any consistent fragment of SMAH. They
are provably equivalent, over ACA, to 1-Con(SMAH). This is true even
if we restrict to piecewise linear f. In fact, if we restrict to
piecewise linear f, then we can win the game by playing rather
concrete sets; i.e., piecewise <,+,exp sets (base 2 exp). This results
in arithmetic independence results, which become Pi02 if we invoke the
well known decision procedure for base 2 exponential Presburger
arithmetic.

PROPOSITION 3.6. Let f:Z+^k into Z+ be piecewise linear, n >= 1, and
p,t be sufficiently large. The copy/invert game with f,+,n can be won
where X[1] is {1,p,p^2,...,p^t}.

Note that if you play finite X[1], then that puts a bound on all of
your subsequent plays - both on the number of subsequent integers
played, and the actual integers themselves. So the game becomes truly
finite after you have played a finite X[1]. Proposition 3.6 is
therefore explicitly Pi03. These remarks also apply to sections 4,5,6.

THEOREM 3.7. Proposition 3.6 is provable in SMAH+ but not in any
consistent fragment of SMAH. It is provably equivalent, over ACA, to 1-
Con(SMAH).

4. COPY/INVERT GAME WITH A POLYNOMIAL.

Let P:Z^k into Z be a polynomial. A special P inversion at x is a k-
tuple y_1,...,y_k from [1,x/2] such that P(y_1,...,y_k) = x.

We define a solitaire game with n rounds, n >= 1. The rounds result in
subsets X[1],...,X[n] of Z. It will be arranged that X[1]
containedin ... containedin X[n].

At the first round, you can play any subset of Z that includes 0,1.

Let 1 <= i < n, and suppose X[i] has been played. X[i+1] includes X[i]
together with integers obtained by the following specific
nondeterministic process.

You first enumerate the elements of P[X[i]^k] in increasing magnitude,
with negatives before positives (although this is natural but not
really important). These are the CANDIDATES for insertion into X[i+1].

For each successive candidate x, you throw x or some special P
inversion of x into X[i+1], The first option is called "copy" and the
second option is called "invert". It may be the case that integers
thrown into X[i+1] under either option are already in X[i+1] or even

There are obviously no obstructions to play thus far.

You are considered to have won if and only if

NO ELEMENT OF X[n] HAS A SPECIAL P INVERSION WHOSE COMPONENTS LIE IN
X[n].

THEOREM 4.1. For all polynomials P:Z^k into Z and n >= 1, the copy/
invert game with P,n can be won where X[1] is infinite.

PROPOSITION 4.2. For all polynomials P:Z^k into Z and n >= 1, the copy/
invert game with P,n can be won where X[1] is an infinite set of
nonnegative integers.

THEOREM 4.3. Theorem 4.1 is provable in RCA_0. Proposition 4.2 is
provable in SMAH+ but not in any consistent fragment of SMAH.
Proposition 4.2 is provably equivalent, over ACA, to 1-Con(SMAH). In
fact, we can win the game by playing rather concrete sets; i.e.,
piecewise <,+,exp sets (base 2 exp). This results in arithmetic
independence results, which become Pi02 if we invoke the well known
decision procedure for base 2 exponential Presburger arithmetic.

PROPOSITION 4.4. Let P:Z+^k into Z+ be a polynomial, n >= 1, and p,t
be sufficiently large. The copy/invert game with P,n can be won where
X[1] is {1,p,p^2,...,p^t}.

THEOREM 4.5. Proposition 4.4 is provable in SMAH+ but not in any
consistent fragment of SMAH. It is provably equivalent, over ACA, to 1-
Con(SMAH).

5. COPY/INVERT GAMES WITH MULTIVARIATE RELATIONS.

Let R be a binary relation on Z+^k. An R inversion at x a y in Z+^k
such that max(y) < max(x) and R(y,x).

We define a solitaire game with n rounds, n >= 1. The rounds result in
subsets X[1],...,X[n] of Z+^k. It will be arranged that X[1]
containedin ... containedin X[n].

At the first round, you can play E^k, for any E contained in Z+.

Let 1 <= i < n, and suppose X[i] has been played. X[i+1] includes X[i]
together with integers obtained by the following specific
nondeterministic process.

You first enumerate the k-tuples (x_1,...,x_k) such that

i. for all 1 <= i <= k, x_i or x_i + 1 is a coordinate of an element
of X[i].
ii. there exists 1 <= i <= k such that x_i + 1 is a coordinate of an
element of X[i].

It is suggested that you enumerate these k-tuples first according to
their maximum coordinate, and second lexicographically (although this
is not really important). These k-tuples are called the CANDIDATES
(candidates for insertion into X[i+1]).

For each successive candidate x, you throw x or an R inversion of x
into X[i+1]. The first option is called "copy" and the second option
is called "invert". It may be the case that vectors thrown into X[i+1]
under either option are already in X[i+1] or even already in X[i].

There are obviously no obstructions to play thus far.

You are considered to have won if and only if

NO CANDIDATE IN X[n] HAS AN R INVERSION IN X[n].

PROPOSITION 5.1. For all R contained in Z+^k x Z+^k and n >= 1, the
copy/invert game with R,n can be won where X[1] is infinite.

PROPOSITION 5.2. For all order invariant R contained in Z+^k x Z+^k
(as a subset of Z+^2k) and n >= 1, the copy/invert game with R,n can
be won where X[1] is infinite.

PROPOSITION 5.3. For all order invariant R contained in [1,t!]^k x
[1,t!]^k (as a subset of [1,t!]^k) and n >= 1, the copy/invert game
with R,n can be won where X[1] = [8kn,t]!^k.

Note that Proposition 5.3 is explicitly Pi01.

THEOREM 5.4. Propositions 5.1-5.3 are provable in SMAH+ but not in any
consistent subsystem of SMAH. They are provably equivalent, over ACA,
to Con(SMAH).

6. FINITE COPY/INVERT GAMES.

We have not given finite games associated with sections 1-2.

Instead of just playing infinite subsets of Z+ in the solitaire games
in sections 1-2 above, we can be required to place the elements on the
positive integer number line. As each new positive integer is played,
it is placed at a unique unused position on the positive integer
number line.

As the positive integers in numerical order during the first round,
they must be placed at the positions called POSTS:

1 (8n)!! (8n+1)!! (8n+2)!! ...

In subsequent rounds, every new positive integer x thrown in must be
placed to the right of any POST whose content is less than x, and to
the left of any POST whose content is greater than x.

For Propositions 1.3 and 2.3, the addition of this PLACEMENT
REQUIREMENT does not affect the metamathematical status.

PROPOSITION 6.1. Let R be a binary relation on [1,t] and n,p >= 1. The
copy/invert game for R,+,x,n, with PLACEMENT, can be won where X[1]
consists of p odd integers including 1.

PROPOSITION 6.2. Let f:Z^2 into Z and n,p >= 1. The copy/invert game
for f,+,x,n, with PLACEMENT, can be won where X[1] consists of p odd
integers including 1.

THEOREM 6.3. Propositions 6.1-6.2 are provable in SMAH+ but not in any
consistent subsystem of SMAH. They are provably equivalent, over ACA,
to 1-Con(SMAH).

7. MAPPING THEOREM - EXPLICIT Pi01

Unchanged from http://www.cs.nyu.edu/pipermail/fom/2009-August/013960.html

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

manuscripts. This is the 356th 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 #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM. NOTE: The title of #269 has been corrected
from the original.

250. Extreme Cardinals/Pi01  7/31/05  8:34PM
251. Embedding Axioms  8/1/05  10:40AM
252. Pi01 Revisited  10/25/05  10:35PM
253. Pi01 Progress  10/26/05  6:32AM
254. Pi01 Progress/more  11/10/05  4:37AM
255. Controlling Pi01  11/12  5:10PM
256. NAME:finite inclusion theory  11/21/05  2:34AM
257. FIT/more  11/22/05  5:34AM
258. Pi01/Simplification/Restatement  11/27/05  2:12AM
259. Pi01 pointer  11/30/05  10:36AM
260. Pi01/simplification  12/3/05  3:11PM
261. Pi01/nicer  12/5/05  2:26AM
262. Correction/Restatement  12/9/05  10:13AM
263. Pi01/digraphs 1  1/13/06  1:11AM
264. Pi01/digraphs 2  1/27/06  11:34AM
265. Pi01/digraphs 2/more  1/28/06  2:46PM
266. Pi01/digraphs/unifying 2/4/06  5:27AM
267. Pi01/digraphs/progress  2/8/06  2:44AM
268. Finite to Infinite 1  2/22/06  9:01AM
269. Pi01,Pi00/digraphs  2/25/06  3:09AM
270. Finite to Infinite/Restatement  2/25/06  8:25PM
271. Clarification of Smith Article  3/22/06  5:58PM
272. Sigma01/optimal  3/24/06  1:45PM
273: Sigma01/optimal/size  3/28/06  12:57PM
274: Subcubic Graph Numbers  4/1/06  11:23AM
275: Kruskal Theorem/Impredicativity  4/2/06  12:16PM
276: Higman/Kruskal/impredicativity  4/4/06  6:31AM
277: Strict Predicativity  4/5/06  1:58PM
278: Ultra/Strict/Predicativity/Higman  4/8/06  1:33AM
279: Subcubic graph numbers/restated  4/8/06  3:14AN
280: Generating large caridnals/self embedding axioms  5/2/06 4:55AM
281: Linear Self Embedding Axioms  5/5/06  2:32AM
282: Adventures in Pi01 Independence  5/7/06
283: A theory of indiscernibles  5/7/06  6:42PM
284: Godel's Second  5/9/06  10:02AM
285: Godel's Second/more  5/10/06  5:55PM
286: Godel's Second/still more  5/11/06  2:05PM
287: More Pi01 adventures  5/18/06  9:19AM
288: Discrete ordered rings and large cardinals  6/1/06  11:28AM
289: Integer Thresholds in FFF  6/6/06  10:23PM
290: Independently Free Minds/Collectively Random Agents 6/12/06
11:01AM
291: Independently Free Minds/Collectively Random Agents (more) 6/13/06
5:01PM
292: Concept Calculus 1  6/17/06  5:26PM
293: Concept Calculus 2  6/20/06  6:27PM
294: Concept Calculus 3  6/25/06  5:15PM
295: Concept Calculus 4  7/3/06  2:34AM
296: Order Calculus  7/7/06  12:13PM
297: Order Calculus/restatement  7/11/06  12:16PM
298: Concept Calculus 5  7/14/06  5:40AM
299: Order Calculus/simplification  7/23/06  7:38PM
300: Exotic Prefix Theory   9/14/06   7:11AM
301: Exotic Prefix Theory (correction)  9/14/06  6:09PM
302: PA Completeness  10/29/06  2:38AM
303: PA Completeness (restatement)  10/30/06  11:53AM
304: PA Completeness/strategy 11/4/06  10:57AM
305: Proofs of Godel's Second  12/21/06  11:31AM
306: Godel's Second/more  12/23/06  7:39PM
307: Formalized Consistency Problem Solved  1/14/07  6:24PM
308: Large Large Cardinals  7/05/07  5:01AM
309: Thematic PA Incompleteness  10/22/07  10:56AM
310: Thematic PA Incompleteness 2  11/6/07  5:31AM
311: Thematic PA Incompleteness 3  11/8/07  8:35AM
312: Pi01 Incompleteness  11/13/07  3:11PM
313: Pi01 Incompleteness  12/19/07  8:00AM
314: Pi01 Incompleteness/Digraphs  12/22/07  4:12AM
315: Pi01 Incompleteness/Digraphs/#2  1/16/08  7:32AM
316: Shift Theorems  1/24/08  12:36PM
317: Polynomials and PA  1/29/08  10:29PM
318: Polynomials and PA #2  2/4/08  12:07AM
319: Pi01 Incompleteness/Digraphs/#3  2/12/08  9:21PM
320: Pi01 Incompleteness/#4  2/13/08  5:32PM
321: Pi01 Incompleteness/forward imaging  2/19/08  5:09PM
322: Pi01 Incompleteness/forward imaging 2  3/10/08  11:09PM
323: Pi01 Incompleteness/point deletion  3/17/08  2:18PM
324: Existential Comprehension  4/10/08  10:16PM
325: Single Quantifier Comprehension  4/14/08  11:07AM
326: Progress in Pi01 Incompleteness 1  10/22/08  11:58PM
327: Finite Independence/update  1/16/09  7:39PM
328: Polynomial Independence 1   1/16/09  7:39PM
329: Finite Decidability/Templating  1/16/09  7:01PM
330: Templating Pi01/Polynomial  1/17/09  7:25PM
331: Corrected Pi01/Templating  1/20/09  8:50PM
332: Preferred Model  1/22/09  7:28PM
333: Single Quantifier Comprehension/more  1/26/09  4:32PM
334: Progress in Pi01 Incompleteness 2   4/3/09  11:26PM
335: Undecidability/Euclidean geometry  4/27/09  1:12PM
336: Undecidability/Euclidean geometry/2  4/29/09  1:43PM
337: Undecidability/Euclidean geometry/3  5/3/09   6:54PM
338: Undecidability/Euclidean geometry/4  5/5/09   6:38PM
339: Undecidability/Euclidean geometry/5  5/7/09   2:25PM
340: Thematic Pi01 Incompleteness 1  5/13/09  5:56PM
341: Thematic Pi01 Incompleteness 2  5/21/09  7:25PM
342: Thematic Pi01 Incompleteness 3  5/23/09  7:48PM
343: Goedel's Second Revisited 1  5/27/09  6:07AM
344: Goedel's Second Revisited 2  6/1/09  9:21PM
345: Thematic Pi01 Incompleteness 4 6/15/09  1:15PM
appears misnumbered as 344.
346: Goedel's Second Revisited 3  6/16/09  11:04PM
347: Goedel's Second Revisited 4  6/20/09  1:25AM
348: Goedel's Second Revisited 5  6/22/09  11:00AM
349: Pi01 Incompleteness/set series  7/20/09  11:21PM
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

Harvey Friedman

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