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*To*: fom@math.psu.edu*Subject*: FOM: 75:Finite Reverse Mathematics*From*: Harvey Friedman <friedman@math.ohio-state.edu>*Date*: Tue, 28 Dec 1999 13:21:46 -0500*Sender*: owner-fom@math.psu.edu

We have renamed Reverse Arithmetic by Finite Reverse Mathematics. We could have used the name Reverse Finite Mathematics, but that is grammatical awkward, and names of potential subjects are just too critical for that. 1. AXIOMS FOR FINITE SETS OF INTEGERS/REVIEW. The approach in #74 was to use basic axioms for integers, and basic axioms for finite sets of integers. Everything was unimpeachable except possibly the two axioms constructing A+B and A*B, for finite sets of integers A,B. Now A+B is around. For example, there is the celebrated alphabeta theorem of Mann (originally conjectured in 1931) discussed in H.B. Mann, "A proof of the fundamental theorem on the density of sums of sets of positive integers," Annals of Math. (2) 43, 523-527 (1942). E. Artin and P. Scherk, "On the sums of two sets of integers," Annals of Math. (2) 44, 138-142 (1943). F.J. Dyson, "A theorem on the densities of sets of integers," Journal of the London Math. Soc., 20, 8-14 (1945). (References from Niven and Zuckerman, An Introduction to the Theory of Numbers, Wiley, 1962.) But what about A*B? It's probably around too, but the closest I have seen in my personal library is the discussion in Graham, Rothschild, Spencer, Ramsey Theory, Wiley, 1980, page 68, where for sets S of positive integers, the set of all nontrivial products of elements of S is discussed in connection with extensions of Folkman's theorem. Some results are mentioned of the author's and Hindman involving this construction. So I said that one can get away with A+B and {1,4,9,16,...,n^2}. However, that was not quite right: I left out that one needs scalar multiplication n*A; i.e., {n*m: m in A}. But that is unimpeachable. So in review, we can use the following from #74: 1. Additive identity. x+0 = x. 2. Additive commutativity. x+y = y+x. 3. Additive associativity. x+(y+z) = (x+y)+z. 4. Additive inverse. x+(-x) = 0. 5. Multiplicative identity. x*1 = x. 6. Multiplicative commutativity. x*y = y*x. 7. Multiplicative associativity. x*(y*z) = (x*y)*z. 8. Distributivity. x*(y+z) = (x*y)+(x*z). 9. Linearity. x = 0 or 0 < x or 0 < -x. 10. Additive positivity. (0 < x and 0 < y) implies 0 < x+y. 11. Multiplicative positivity. (0 < x and 0 < y) implies 0 < x*y. 12. Unit positivity. 0 < 1. 13. Irreflexivity. not(0 < 0). 14. Finite intervals. (therexists A)(forall x)(x epsilon A iff (y < x and x < z)). 15. Boolean difference. (therexists C)(forall x)(x epsilon C iff (x epsilon A and not(x epsilon B)). 16. Set addition. (therexists C)(forall x)(x epsilon C iff (therexists y)(therexists z)(y epsilon A and z epsilon B and x = y+z)). 17. Least elements. (therexists x)(x epsilon A) implies (therexists x)(x epsilon A and ((forall y)(y epsilon A implies not(y < x)). 18. Common multiples. (therexists x)(0 < x and (forall y)((y epsilon A and 0 < y) implies (therexists z)(x = y*z))). 19. Set multiplication. (therexists C)(forall x)(x epsilon C iff (therexists y)(therexists z)(y epsilon A and z epsilon B and x = y*z)). Or we can use the following modification of #74: 1. Additive identity. x+0 = x. 2. Additive commutativity. x+y = y+x. 3. Additive associativity. x+(y+z) = (x+y)+z. 4. Additive inverse. x+(-x) = 0. 5. Multiplicative identity. x*1 = x. 6. Multiplicative commutativity. x*y = y*x. 7. Multiplicative associativity. x*(y*z) = (x*y)*z. 8. Distributivity. x*(y+z) = (x*y)+(x*z). 9. Linearity. x = 0 or 0 < x or 0 < -x. 10. Additive positivity. (0 < x and 0 < y) implies 0 < x+y. 11. Multiplicative positivity. (0 < x and 0 < y) implies 0 < x*y. 12. Unit positivity. 0 < 1. 13. Irreflexivity. not(0 < 0). 14. Finite intervals. (therexists A)(forall x)(x epsilon A iff (y < x and x < z)). 15. Boolean difference. (therexists C)(forall x)(x epsilon C iff (x epsilon A and not(x epsilon B)). 16. Set addition. (therexists C)(forall x)(x epsilon C iff (therexists y)(therexists z)(y epsilon A and z epsilon B and x = y+z)). 17. Least elements. (therexists x)(x epsilon A) implies (therexists x)(x epsilon A and ((forall y)(y epsilon A implies not(y < x)). 18. Common multiples. (therexists x)(0 < x and (forall y)((y epsilon A and 0 < y) implies (therexists z)(x = y*z))). 19. Scalar set multiplication. (therexists B)(forall y)(x epsilon B iff (therexists z)(z epsilon A and y = x*z)). 20. Squares. (therexists A)(forall y)(y in A iff (therexists z)(0 < z and z < x and y = z*z)). The two groups have the same first 18 axioms, and differ after that. In the first group, Axioms 1-19 are equivalent to ISigma_0(exp) = EFA and ISigma_0 is obtained by dropping axiom 18. In the second group, Axioms 1-20 are equivalent to ISigma_0(exp) = EFA and ISimga_0 is obtained by dropping axiom 18. Now jsut how unimpeachable is {1,4,9,...,n^2}? I said in posting #74 that I was worried about it. But fortunately already in the same Niven and Zuckerman book that I found the H.B. Mann stuff with A+B, I also found the following on page 232 right after the definition of A+B: "As an example let us take S to be the set of squares 0,1,4,9,... and I the set of all nonnegative integers. Then by Theorem 5.6 we see that S + S + S + S = I." Of course, strictly speaking, we cannot use the set of squares 0,1,4,9,... for our purposes since this is infinite. However, this seems like a minor objection. Another minor objection is that Niven and Zuckerman only define A+B if A,B are sets of nonnegative integers and 0 lies in both A and B. The natural definition is for sets A,B of integers without restriction, and the restrictions they place are merely an artifact of the fact that they are developing a proof of Mann's theorem. Nevertheless, with a little care, one can derive the existence of A+B for finite A,B containedin Z from the existence of A+B for finite A,B containedin N with 0 in A,B, anyways - using the existence of integer translates. And the existence of translates is explicitly in a lot of places in the literature. So in conclusion, both of these axiom systems are more or less unimpeachable, with an edge perhaps given to the second system. 2. PRACTICAL PROPOSAL OF A BASE THEORY FOR FINITE REVERSE MATHEMATICS. By practical, I mean a) a program which is clearly doable to some extent, with a number of initial results. b) a program which has a number of open problems with various common themes. c) the general flavor and structure of the program is the same as reverse mathematics as presented in Simpson's book, with statements in finite mathematics expected to be equivalent to a handful of systems. d) just as in existing reverse mathematics, the formal systems are finitely axiomatizable, though presented in terms of schemes. Thus in this section, we do not concentrate on the major foundational challenge of making the base theory consist of the most unimpeachably essential universally used mathematical statements - which was addressed in postings #73, #74, and in section 1 above. In fact, we follow the tradition represented in Simpson's book by presenting formal systems in the most efficient way for logicians, freely using schemes, obvious codings, etcetera. The (at present most practical) base theory of existing reverse mathematics is RCA_0. The analogous practical base theory for finite reverse mathematics is proposed to be FCA (finite comprehension axiom with restricted induction). Recall that RCA_0 is a two sorted system based on variables over nonnegative integers, variables over sets of nonnegative integers, 0,S.+,*,<,= among nonnegative integers, and epsilon between nonnegative integers and sets of nonnegative integers. Here FCA is a two sorted system based on variables over nonnegative integers, variables over finite sets of nonnegative integers, 0,S.+,*,<,= among nonnegative integers, and epsilon between nonnegative integers and finite sets of nonnegative integers. The axioms of FCA are as follows. 1. x+0 = x. 2. x+Sy = S(x+y). 3. x*0 = 0. 4. x*Sy = x*y + x. 5. Sx not= 0. 6. Sx = Sy implies x = y. 7. not(x < 0). 8. x < Sy iff (x < y or x = y). 9. Finiteness. (therexists x)(forall y)(y in A implies y < x). 10. Every nonempty finite set has a least element. 11. Bounded comprehension. (therexists A)(forall y)(x in A iff (y < x and phi)), where phi is strictly bounded and A is not free in phi. In 11, by strictly bounded we mean that all quantifiers are over integers, and are bounded to terms of either sort, and of course free variables are allowed of either sort. As an obvious consequence of FCA, we have 12. Strictly bounded induction. I.e., induction with respect to all strictly bounded formulas. We could make FCA a weak subsystem of RCA_0 by simply dropping axiom 9. However, for the moment, we prefer to think of finite reverse mathematics as a self contained autonomous subject, and leave 9 in. NOTE: There is work of Buss, Krajicek, Takeuti, etc. in this direction. Was this system considered by one or more of them? Note that we are presenting this system in the context of a proposal for a new incarnation of reverse mathematics, something that may not have been concerned with. END. THEOREM 1. FCA is a conservative extension of ISigma_0. FCA is finitely axiomatizable. FCA is equivalent to the two systems presented in section 1, without their axiom of multiples, but with this axiom of finiteness added (where we are converting from integers to nonnegative integers). We believe that FCA is a practical base theory for finite reverse mathematics. Just as in reverse mathematics, we can use the quadratic pairing function on the natural numbers to code finite sequences of nonnegative integers and finite functions of several variables. Assuming we use FCA as the base theory, what should some of the principal stronger theories be? We postpone discussion of this, and instead now disucss the issues involved in using a weaker base theory than FCA. 3. THE CHALLENGE OF WEAKENING FCA_0 FOR FINITE REVERSE MATHEMATICS. RCA_0 can be challenged as too strong a base theory for reverse mathematics, because too much exciting mathematics with intricate logical structure is already provable, and thus lost to reverse mathematics with RCA_0 as the base theory. Work is under way to deal with this. In fact, the whole enterprise of finite reverse mathematics can be viewed as addressing a small corner of this issue. I say small corner because RCA_0 can be challenged to be too strong not only in the realm of finite mathematics, but also in the realm of infinite mathematics, and finite reverse mathematics does not address the infinite context at all. But the same criticism of RCA_0 can be made of FCA. So in this section, we explore the possibility of weakening the base theory FCA. Some obvious fragments of FCA are obtained in a couple of ways. Firstly, we can get rid of sets entirely and work in arithmetic only. So we are looking at fragments of ISigma_0. I.e., fragments of 1. x+0 = x. 2. x+Sy = S(x+y). 3. x*0 = 0. 4. x*Sy = x*y + x. 5. Sx not= 0. 6. Sx = Sy implies x = y. 7. not(x < 0). 8. x < Sy iff (x < y or x = y). 9. Induction for all bounded formulas in the language 0,S,+,*,<. The most obvious fragments are what is called lE_i, i >= 0. This is induction with respect to all bounded formulas in prenex form which are in Sigma_k form in the obvious sense. Thus IE_0 is just what is called open induction. IE_0 has been seriously studied, but I do not know of any reasonable set of mathematical theorems that are provably equivalent to IE_0. So we might take IE_0 as a base theory. Then the question is whether we can find any reasonable set of mathematical theorems that are provably equivalent to IE_1 over IE_0? Actually, it is not even clear (to me) whether these fragments IE_i are finitely axiomatizable. If not, then one certainly is not going to get any single or small group of theorems to bootstrap from one to the next. So it seems to be quite difficult to preserve the reverse mathematics spirit in this setup. It still could be the case that large groups of theorems provable in ISigma_0 but not in IE_0 might themselves be provably equivalent over IE_0 to each other, and there just might be a manageable number of equivalence classes. And just what these appropriate fragements of ISigma_0 are that might emerge are not clear to me. It is perhaps somewhat easier to imagine the previous paragraph with IE_0 replaced by IE_1. Thus one would be looking at theorems of ISigma_0 that are not theorems of IE_1, and then classify their status over IE_1 as the base theory. CAUTION: It is known that IE_1 proves more than IE_0, but it is not known if IE_1 is equivalent to ISigma_0, although it is known that if the polynomial time hierarchy does not collapse then IE_1 is weaker than ISigma_0, and in fact the IE_i hierarchy does not collapse. I think this might lead to some very interesting questions. Already IE_1 is a pretty powerful fragment of ISigma_0. It proves the existence of gcd's and lcm's, has no recursive nonstandard models, etc. Just what number theory can be proved in IE_1 is an interesting question. More generally, finding a good n for which various fundamental number theory can be proved in IE_n is an interesting project. One could hope to find the least n under some assumption that these systems don't collapse, which I gather is an open question. In complexity theory, one dodges the fundamental open question of whether things collapse, and still get computational completeness at levels. The question is to what extent this can be done here for finite reverse mathematics. However, the fact that the IE_i don't appear to be finitely axiomatizable seems to be a problem. Perhaps one has to subdivide each IE_i into subsclasses by counting the length of quantifier blocks and the degree of the single polynomial used in the matrix. There is another way of getting around the perhaps not finitely axiomatizable issue in the context of arithmetic. And that is to introduce variables over tuples of integers and talk directly about polynomials with integer coefficients. There are a lot of possibilities here. Of course, it would be more striking not to have to do this. Now suppose that sets are used. We can start with the base theory 1. x+0 = x. 2. x+Sy = S(x+y). 3. x*0 = 0. 4. x*Sy = x*y + x. 5. Sx not= 0. 6. Sx = Sy implies x = y. 7. not(x < 0). 8. x < Sy iff (x < y or x = y). 9. Finiteness. (therexists x)(forall y)(y in A implies y < x). 10. Every nonempty finite set has a least element. 11. Open comprehension. (therexists A)(forall x)(x in A iff phi), where phi is quantifier free and A is not in phi. This is conservative over IE_0. If we allow phi to be bounded existential, even with at most two quantifiers, then we get all of FCA because we get closure under addition and subtraction and multiplication of sets. This is strong enough to derive FCA even in the context of nonnegative integers, not just in the context of integers. So here we have a limited window of opportunity for a kind of finite reverse mathematics. We could look at closure conditions on the finite sets of natural numbers, and determine what kind of bounded comprehension those statements are equivalent to over the above base theory. Still, there are many unexplored ways where a finite reverse mathematics may emerge if done just right that uses a base theory below FCA and where lots of striking phenomena appear at or below FCA. I.e., where a large number of theorems coalesce into a relatively small set of equivalence classes under provable equivalence. It will need considerable experimentation and perhaps some serious technical ideas. 4. DIGRESSION: ANTI FINITE REVERSE MATHEMATICS. Anti (finite) reverse mathematics is simply the subject of showing that various proposed base theories for (finite) reverse mathematics are unsuitable. This is done by constructing various finite sets of mathematical statements from the literature which form independent sets of sentences over the proposed base theories, in the sense that no one in the list is derivable from all of the others (or even that no nontrivial Boolean combination of them can be derived). This can be particularly convincing if there is an important mathematical theorem of the form (forall n)(phi(n)) such that there is an infinite set of sentences of the form phi(n-bar) which is an independent set of sentences over the proposed base theory. For example, suppose we took the axioms of discrete ordered rings as the base theory. Look at the theorem (for all integers n)(if n is the square of a rational then n is the square of an integer). For primes n, this forms an infinite independent set over discrete ordered rings. I think that we need a lot more anti reverse mathematics and anti finite reverse mathematics, in order to clarify the boundary between workable base theories and nonworkable base theories. CAUTION: I am convinced that there are some good base theories for reverse mathematics that are fragments of RCA_0. I am not as sure that there are any good base theories for finite reverse mathematics that are fragments of FCA. 5. FINITE POWER SET. A proposed principal theory beyond the proposed base theory FCA is FPS, given as follows. 1. FCA. 2. Finite power set. For every n there exists a finite binary relation whose cross sections include every subset of [0,n]. A convenient consequence of this theory is bounded induction. Recall that FCA uses strictly bounded induction; i.e., where all quantifiers are over integers, and are bounded to terms of either sort, and of course free variables are allowed of either sort. In bounded induction, set quantifiers are allowed, but they must be restricted to the subsets of some set. Or equivalently, they can be written in the form (forall x containedin [0,n])... This system is equivalent to the two systems in section 1 in the appropriate sense, and so is equivalent to ISigma_0(exp) = EFA in the appropriate sense. There are plenty of nice mathematical statements that are equivalent to FPS over FCA; this follows the pattern of reverse mathematics. E.g., a. Every finite set of positive natural numbers has a positive common multiple (or least positive common multiple). b. Finite Ramsey theorem for 3-tuples with 2 colors. I.e., if n >> k then any coloring of the unordered 3-tuples from [0,n] with 3 colors has a monochromatic set of cardinality k. c. Finite Ramsey theorem for 2-tuples with k colors. I.e., if n >> k then any coloring of the unordered 2-tuples from [0,n] with k colors has a monochromatic set of cardinality 3. I don't know the status of the finite Ramsey theorem for 2-tuples with 2 colors (asking for a homogenous set of cardinality k) over FCA. 6. FINITE ITERATED POWER SET. Another proposed principal theory beyond FPS is the system FIPS given as follows. 1. FCA. 2. Finite iterated power set. For every k,n there exists a sequence x_0,...,x_n of nonnegative integers of length n, where x_0 = k and for each 0 < i < n, there is a binary relation on [0,x_i+1] where every subset of [0,x_i] is a cross section of that relation. There are nice mathematical statements that are equivalent to FIPS over FCA. E.g., the full finite Ramsey theorem. 7. STRONGER SYSTEMS. Standard recursive defining equations for various natural functions, including exponentiation and superexponentiation, can be interpreted as Turing machine instructions for computing a function. The corresponding axiom asserts that the computation is well defined - i.e., has a numerical code - and halts at all inputs. In this sense, we can define the theories FCA + exp, FCA + superexp, and we can prove that these systems are equivallent to FPS and FIPS. We also take a standard version of the Ackerman hierarchy, say unary functions A_1,A_2,..., starting with A_1 = doubling, and define each FCA + A_i in the obvious way. And we take a version of the binary Ackerman function A(n,m), and form the system FCA + A. We can carry out finite reverse mathematics for genuine strictly mathematical theorems of finite mathematics in the same way that it is done in reverse mathematics (see Simpson's book), by employing coding. The coding of finite mathematical statements is much more straightforward and much less problematic than it is for reverse mathematics, where, e.g., various kinds of subsets of complete separable metric spaces have to be considered. There are even issues there about real numbers and sequences of real numbers. They are appropriately coded in Simpson's book, but there are still delicate issues as to just why, and what alternatives look like. But here, it is comparatively unproblematic. I.e., sequences are coded as finite functions, finite functions of several variables are coded as finite sets of finite sequences, ordered pairs of integers are coded using the quadratic pairing function, finite sets of finite sets are coded as finite sets, etcetera. THEOREM. The following statement is provably equivalent to FCA + A over FCA. Let k >= 1 and x in Z+k. In every sufficiently long walk w in Z+k starting at x there is a term w_i which points outward to a later term w_j where w_j is at least twice as far as w_i from the origin. See posting #34. WIth some of the strong Pi-0-2 sentences, it is awkward and technical to write down a defining equation. But with a little bit of care one can show the following kind of result, which we just state for any of my finite forms of Kruskal's theorem just to give an idea. Call them generically FKT. THEOREM. FKT is equivalent to the 1-consistency of Pi-1-2-BI_0 over FCA. Of course, if we are embarked on some sort of computationally sensitive project - not finite reverse mathematics in the sense meant here - then we may view these codings as problematic, or even outright inappropriate. But that is another matter. 8. ELIMINATION OF CODING: SECTION 1 TREATMENT. Even though for the purposes of the development of finite reverse mathematics along the general lines of reverse mathematics in Simpson's book, there are no problematic coding issues to worry about, we still would like to also have a coding free development. This is particularly needed if we want to carry out what we did in section 1 in order to show that high proof theoretic strength exists among the unimpeachably mathematical. For this purpose, we need a base theory that has more primitives than the Section 1 systems, and is still a conservative extension of them, and above all, is still made up of unimpeachable mathematical statements. We will carry this out in a minimalistic manner for three examples of theorems of relatively high strength. a. The block subsequence theorem of postings #19 and #21. b. A version of the walk theorem cited in section 7 above. c. The 1998 finite form of Kruskal's theorem in postings #22 and #27. d. Monotone shift theorem. e. Paris/Harrington theorem. Monotone shift theorem says the following. Let n >> k,r and F:[0,n]^k into [0,n]^r. There exists 0 <= x_1 < ... < x_k+1 <= n such that F(x_1,...,x_k) <= F(x_2,...,x_k+1). Here <= means <= in each of the r coordinates. This has strength PA = Peano Arithmetic like e. But not in this posting. This is long enough. In a later posting. 9. INTERMEDIATE SYSTEMS. Actually, there is a need for intermediate systems, especially between FCA and FPS. This makes sense in light of the intense study in the formal arithmetic community of systems between ISigma_0 and ISigma_0(exp). In particular, FCA does not prove the pigeonhole principle "any one-one function on an interval into itself is onto." This follows from work of Ajtai, The Complexity of the Pigeonhole Principle, 29th Annual Symposium on Foundations of Computer Science, 1988, 346-358. Woods proved that there are infinitely many primes from the pigeonhole principle. In Paris, Wilkie, Woods JSL 1988, there is a proof that there are infinitely many primes using the existence of n^log(n), representing yet another extension of FCA properly contained in FPS. In particular, the status of "there are infinitely many primes" and "there is a prime between every n and 2n" are open. So there is a real question of whether there is an untameable mess between FCA and FPS (for strictly mathematical theorems) from the finite reverse mathematics point of view. We don't know yet. ********** This is the 75th in a series of self contained postings to FOM covering a wide range of topics in f.o.m. Previous ones are: 1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM. 2:Axioms 11/6/97. 3:Simplicity 11/14/97 10:10AM. 4:Simplicity 11/14/97 4:25PM 5:Constructions 11/15/97 5:24PM 6:Undefinability/Nonstandard Models 11/16/97 12:04AM 7.Undefinability/Nonstandard Models 11/17/97 12:31AM 8.Schemes 11/17/97 12:30AM 9:Nonstandard Arithmetic 11/18/97 11:53AM 10:Pathology 12/8/97 12:37AM 11:F.O.M. & Math Logic 12/14/97 5:47AM 12:Finite trees/large cardinals 3/11/98 11:36AM 13:Min recursion/Provably recursive functions 3/20/98 4:45AM 14:New characterizations of the provable ordinals 4/8/98 2:09AM 14':Errata 4/8/98 9:48AM 15:Structural Independence results and provable ordinals 4/16/98 10:53PM 16:Logical Equations, etc. 4/17/98 1:25PM 16':Errata 4/28/98 10:28AM 17:Very Strong Borel statements 4/26/98 8:06PM 18:Binary Functions and Large Cardinals 4/30/98 12:03PM 19:Long Sequences 7/31/98 9:42AM 20:Proof Theoretic Degrees 8/2/98 9:37PM 21:Long Sequences/Update 10/13/98 3:18AM 22:Finite Trees/Impredicativity 10/20/98 10:13AM 23:Q-Systems and Proof Theoretic Ordinals 11/6/98 3:01AM 24:Predicatively Unfeasible Integers 11/10/98 10:44PM 25:Long Walks 11/16/98 7:05AM 26:Optimized functions/Large Cardinals 1/13/99 12:53PM 27:Finite Trees/Impredicativity:Sketches 1/13/99 12:54PM 28:Optimized Functions/Large Cardinals:more 1/27/99 4:37AM 28':Restatement 1/28/99 5:49AM 29:Large Cardinals/where are we? I 2/22/99 6:11AM 30:Large Cardinals/where are we? II 2/23/99 6:15AM 31:First Free Sets/Large Cardinals 2/27/99 1:43AM 32:Greedy Constructions/Large Cardinals 3/2/99 11:21PM 33:A Variant 3/4/99 1:52PM 34:Walks in N^k 3/7/99 1:43PM 35:Special AE Sentences 3/18/99 4:56AM 35':Restatement 3/21/99 2:20PM 36:Adjacent Ramsey Theory 3/23/99 1:00AM 37:Adjacent Ramsey Theory/more 5:45AM 3/25/99 38:Existential Properties of Numerical Functions 3/26/99 2:21PM 39:Large Cardinals/synthesis 4/7/99 11:43AM 40:Enormous Integers in Algebraic Geometry 5/17/99 11:07AM 41:Strong Philosophical Indiscernibles 42:Mythical Trees 5/25/99 5:11PM 43:More Enormous Integers/AlgGeom 5/25/99 6:00PM 44:Indiscernible Primes 5/27/99 12:53 PM 45:Result #1/Program A 7/14/99 11:07AM 46:Tamism 7/14/99 11:25AM 47:Subalgebras/Reverse Math 7/14/99 11:36AM 48:Continuous Embeddings/Reverse Mathematics 7/15/99 12:24PM 49:Ulm Theory/Reverse Mathematics 7/17/99 3:21PM 50:Enormous Integers/Number Theory 7/17/99 11:39PN 51:Enormous Integers/Plane Geometry 7/18/99 3:16PM 52:Cardinals and Cones 7/18/99 3:33PM 53:Free Sets/Reverse Math 7/19/99 2:11PM 54:Recursion Theory/Dynamics 7/22/99 9:28PM 55:Term Rewriting/Proof Theory 8/27/99 3:00PM 56:Consistency of Algebra/Geometry 8/27/99 3:01PM 57:Fixpoints/Summation/Large Cardinals 9/10/99 3:47AM 57':Restatement 9/11/99 7:06AM 58:Program A/Conjectures 9/12/99 1:03AM 59:Restricted summation:Pi-0-1 sentences 9/17/99 10:41AM 60:Program A/Results 9/17/99 1:32PM 61:Finitist proofs of conservation 9/29/99 11:52AM 62:Approximate fixed points revisited 10/11/99 1:35AM 63:Disjoint Covers/Large Cardinals 10/11/99 1:36AM 64:Finite Posets/Large Cardinals 10/11/99 1:37AM 65:Simplicity of Axioms/Conjectures 10/19/99 9:54AM 66:PA/an approach 10/21/99 8:02PM 67:Nested Min Recursion/Large Cardinals 10/25/99 8:00AM 68:Bad to Worse/Conjectures 10/28/99 10:00PM 69:Baby Real Analysis 11/1/99 6:59AM 70:Efficient Formulas and Schemes 11/1/99 1:46PM 71:Ackerman/Algebraic Geometry/1 12/10/99 1:52PM 72:New finite forms/large cardinals 12/12/99 6:11AM 73:Hilbert's program wide open? 12/20/99 8:28PM 74:Reverse arithmetic beginnings 8:33AM 12/22/99

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