# [FOM] 357: HIGH SHCOOL Games/Update

Harvey Friedman friedman at math.ohio-state.edu
Thu Aug 20 10:42:59 EDT 2009

```This is a continuation of http://www.cs.nyu.edu/pipermail/fom/2009-August/013932.html
on solitaire games of a deceptively elementary nature.

I have discussed solitaire games based on unary functions,
multivariate functions, piecewise linear functions, polynomials,
multivariate relations, piecewise linear multivariate relations, and
order invariant multivariate relations. I also discussed the Mapping
Theorem.

I am still convinced of the main claims - EXCEPT for the claims
concerning games based on unary functions.

HOWEVER, it appears that the difficulties with unary functions are
overcome using binary relations - which are arguably as good or
better, anyway.

There are also some improvements. One is a new punch line - see "any
sufficiently large integer" below. Another is an improved FINITE
version of the solitaire games.

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

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. Finite Copy/Invert games.
6. Copy/Invert games with relations.
7. Mapping Theorem - Explicitly Pi01.

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

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

At the first round, any subset of Z+ can be played that includes 1.

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

The player first enumerates 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 n, the player either throws n into X[i
+1], or throws some m < n such that R(m,n), 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

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

A *successful candidate* is a candidate that was thrown in.

There are obviously no obstructions to play thus far.

The player is considered to have won if and only if NO TWO DISTINCT
SUCCESSFUL CANDIDATES ARE RELATED BY R.

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+, the copy/invert game with
R,+,x,r can be won where X is infinite.

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

PROPOSITION 1.3. For all R contained in Z+ x Z+, the copy/invert game
with R,+,x,r can be won where X is an infinite set of odd integers.

PROPOSITION 1.4. For all R contained in Z+ x Z+, and infinite E
contained in Z+, the copy/invert game with R,+,x,r can be won where
X 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 only use the candidates for the criterion for winning. This is
because the relation may, for example, be all of Z+ x Z+, in which
case obviously no infinite set (or set with at least two elements) is
free of R.

In addition, 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.

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 R.

There is a (new) 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 5 below.

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

Let f:Z+^2 into Z+. We define a solitaire game with r rounds, r >= 1.
The rounds result in subsets X,...,X[r] of Z+. It will be arranged
that X containedin ... containedin X[r].

At the first round, any subset of Z+ can be played that includes 1.

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

The player first enumerates 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 n, the player either throws n into X[i
+1], or throws some a,b < n such that f(a,b) = n, 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

There are obviously no obstructions to play thus far.

The player is considered to have won if and only if

f(a,b) = n and a,b < n

fails for all a,b,n in X[r].

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+, the copy/invert game with f,+,x,r
can be won where X is infinite.

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

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

PROPOSITION 2.4. For all f:Z+^2 into Z+, and infinite E contained in Z
+, the copy/invert game with f,+,x,r can be won where X 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.

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

At the first round, any subset of Z+ can be played that includes 1.

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

The player first enumerates 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 n, the player either throws n into X[i
+1], or throws some m_1,...,m_k < n such that f(m_1,...,m_k) = n, 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].

There are obviously no obstructions to play thus far.

The player is considered to have won if and only if

f(m_1,...,m_k) = n and m_1,...,m_k < n

fails for all m_1,...,m_k,n in X[r].

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+, the copy/invert game with f,+,r
can be won where X is infinite.

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

PROPOSITION 3.3. For all f:Z+^k into Z+, the copy/invert game with f,
+,r can be won where X is an infinite set of odd integers.

PROPOSITION 3.4. For all f:Z+^k into Z+, and infinite E contained in Z
+, the copy/invert game with f,+,r can be won where X 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).

THEOREM 3.6. Theorem 3.5 holds even if we restrict to semilinear f. We
can win the game by playing rather concrete sets; e.g., piecewise <,
+,exp sets (base 2 exp). This results in arithmetic independence
results. In fact, we can play any interval (finite or infinite) of
double powers of 2, together with 1, in the first round, provided the
least double power of 2 in the interval is sufficiently large relative
to the given f. This results in a Pi03 sentence provably equivalent,
over ACA, to 1-Con(SMAH). It can be reduced to Pi02 using the well
known decision procedure for base 2 exponential Presburger arithmetic.

4. COPY/INVERT GAME WITH A POLYNOMIAL.

Let P:Z^k into Z be a polynomial. We define a solitaire game with r
rounds, r >= 1. The rounds result in subsets X,...,X[r] of Z. It
will be arranged that X containedin ... containedin X[r].

At the first round, any subset of Z can be played that includes 0,1.

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

The player first enumerates 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 n, the player either throws n into X[i
+1], or throws some 0 < m_1,...,m_k < n/2 such that P(m_1,...,m_k) =
n, 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].

There are obviously no obstructions to play thus far.

The player is considered to have won if and only if

P(m_1,...,m_k) = n and 0 < m_1,...,m_k < n/2

fails for all m_1,...,m_k,n in X[r].

THEOREM 4.1. For all polynomials P:Z^k into Z, the copy/invert game
with P,r can be won where X is infinite.

PROPOSITION 4.2. For all polynomials P:Z^k into Z, the copy/invert
game with P,r can be won where X is an infinite set of nonnegative
integers.

PROPOSITION 4.3. For all polynomials P:Z^k into Z, and sufficiently
large integers n, the copy/invert game with P,r can be won where X
is an infinite set of nonnegative integers containing n.

PROPOSITION 4.4. For all polynomials P:Z^k into Z, the copy/invert
game with P,r can be won where X is any interval (finite or
infinite) of triple powers of 2, together with 1, provided the first
triple power of 2 in the interval is sufficiently large relative to
the given f. This results in a Pi03 sentence provably equivalent, over
ACA, to 1-Con(SMAH).

THEOREM 4.5. Theorem 4.1 is provable in RCA_0. Propositions 4.2-4.4
are provable in SMAH+ but not in any consistent subsystem of SMAH.
They are provably equivalent, over ACA, to 1-Con(SMAH).

5. FINITE COPY/INVERT GAMES.

We describe this finite game in detail just for the case of the Copy/
Invert Game for R,+,x, where R is a binary relation on Z+, with r
rounds. There is a second parameter p for this finite game.

There are totally analogous finite forms of the other games, as will
be evident below.

The game is played on the integer number line from 1 through (8r+p
+1)!. As usual, the rounds are written X,...,X[r], where X
containedin ... containedin X[r].

Play proceeds for r rounds just as in the original game. However, at
the end of each round 1,...,r, X[i] is entered onto the number line
from 1 through (8r+p+1)! in the following way.

X is required to consist of p+1 positive integers, including 1.
These numbers are entered in strictly increasing order at the positions

1, (8r)!, (8r+1)!, ..., (8r+p)!

on the integer number line from 1 through (8r+p+1)!. Obviously the
integer 1 must be placed at position 1. The remaining (8r+p+1)!-p-1
positions are left blank.

These positions above are called the POSTS.

As the integers in the plays X,...,X[r] are generated, they are
placed on the number line in such a way that

i. There can be at most one integer in any position.
ii. An integer n must be placed to the right of any post whose
contents is smaller than n.
iii. An integer n must be placed to the left of any post whose
contents is greater than n.

Integers may be generated that have already been placed, in which case
no action is taken for that integer.

We use the same winning condition on X[r] that we used for the
original game.

THEOREM 5.1. For all R contained in Z+ x Z+, the finite copy/invert
game with R,+,x,r,p can be won.

PROPOSITION 5.2. For all R contained in Z+ x Z+, and sufficiently
large integers n, the finite copy/invert game with R,+,x,r,p can be
won where n is included in X.

PROPOSITION 5.3. For all R contained in Z+ x Z+, the finite copy/
invert game with R,+,x,r,p can be won where X consists of odd
integers.

PROPOSITION 5.4. For all R contained in Z+ x Z+, and infinite E
contained in Z+, the finite copy/invert game with R,+,x,r,p can be won
where X is a subset of E together with 1.

THEOREM 5.5. Theorem 5.1 is provable in RCA_0. Propositions 5.2-5.4
are provable in SMAH+ but not in any consistent fragment of SMAH. They
are provably equivalent, over ACA, to 1-Con(SMAH).

We can even arrange for play to proceed on a finite initial segment of
the positive integers.

PROPOSITION 5.6. For all r,p there exists t so large that for all R
contained in [1,t] x [1,t], the finite copy/invert game with R,
+,x,r,p, played on [1,t], can be won where X consists of odd
integers.

THEOREM 5.7. Propositions 5.2-5.4 are provable in SMAH+ but not in any
consistent fragment of SMAH. They are provably equivalent, over ACA,
to 1-Con(SMAH). The growth rate of the least t as a function of r,p,
is greater than that of the provably recursive functions of SMAH.

6. COPY/INVERT GAMES WITH RELATIONS.

We can also apply these ideas to the relation games. Recall that these
do not use addition or multiplication, but do use +1. Recall that
these games involve R contained in Z+^k x Z+^k, and lead to Pi01
independent statements, some of them being explicitly Pi01.

We can use the idea present in section 1 above, which is to look for R
freedom among candidates only, as the best idea for these relation
games.

We will revisit this matter at a later date.

7. MAPPING THEOREM - EXPLICIT Pi01

We present this for the sake of completeness of the discussion. There
has been no improvement.

Recall Propositions 2.1-2.3 and Theorem 2.4 in http://www.cs.nyu.edu/pipermail/fom/2009-August/013907.html
Also recall 3.1-3.3 there.

PROPOSITION 7.1. For all R contained in N^2k x N^k, some R intersect
A^3 inverts some infinite subcube of A to some subcube of (A-1 delta
R<[A^2])^2.

PROPOSITION 7.2. For all (unit coefficient) piecewise linear R
contained in N^2k x N^k, some R intersect A^3 inverts some infinite
subcube of A some subcube of (A-1 delta R<[A^2])^2. Furthermore, A and
its
aforementioned infinite subcube can be taken to be piecewise (base 2)
exponential.

PROPOSITION 7.3. For all unit coefficient piecewise linear R contained
in [0,t!]^2k x [0,t!]^k, some R intersect A^3 inverts some [16k,t]!^k
containedin A to some subcube of (A-1 delta R<[A^2])^2.

Note that Proposition 7.3 is explicitly Pi01.

THEOREM 7.4. Propositions 7.2-,.3 are provable in SMAH+ but not in any
consistent fragment of SMAH. Propositions 7.2,7.3 are provably
equivalent, over ACA, to Con(SMAH).

We can also use regressive values as in http://www.cs.nyu.edu/pipermail/fom/2009-August/013907.html

PROPOSITION 7.5. For all R contained in N^2k x N^k, some R intersect
A^3 has finitely many regressive inverse values on some infinite
cube in N^k that it inverts to some subcube of (A delta R<[A^2])^2.

This supports another kind of explicitly Pi01 independent statement:

PROPOSITION 7.6. For all R contained in [0,!(8rk)]^2k, some R
intersect A^3 has at most (8k)! regressive inverse values on some
length r cube in [0,!(8rk)]^3 that it inverts to some subcube of (A
delta R<[A^2])^2.

THEOREM 7.7. Propositions 7.5,7.6 are provable in SUB+ but not in any
consistent fragment of SUB. Propositions 7.5,7.6 are provably
equivalent, over ACA, to Con(SUB).

Here there are also natural ways to adjust parameters to get
explicitly Pi01 statements corresponding to the consistency of
fragments of ZF (and higher).

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

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

Harvey Friedman

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