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1,800 |
A note on canonical functions
|
math.LO
|
We construct a generic extension in which the aleph_2 nd canonical function
on aleph_1 exists.
|
math
|
1,801 |
Categoricity over P for first order T or categoricity for phi in L_{omega_1 omega} can stop at aleph_k while holding for aleph_0, ..., aleph_{k-1}
|
math.LO
|
Suppose L is a relational language and P in L is a unary predicate. If M is
an L-structure then P(M) is the L-structure formed as the substructure of M
with domain {a: M models P(a)}. Now suppose T is a complete first order theory
in L with infinite models. Following Hodges, we say that T is relatively
lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then
there is an isomorphism i:M-> N which is the identity on P(M). T is relatively
categorical if it is relatively lambda-categorical for every lambda. The
question arises whether the relative lambda-categoricity of T for some lambda
>|T| implies that T is relatively categorical.
In this paper, we provide an example, for every k>0, of a theory T_k and an
L_{omega_1 omega} sentence varphi_k so that T_k is relatively
aleph_n-categorical for n < k and varphi_k is aleph_n-categorical for n<k but
T_k is not relatively beth_k-categorical and varphi_k is not
beth_k-categorical.
|
math
|
1,802 |
The primal framework. I
|
math.LO
|
This the first of a series of articles dealing with abstract classification
theory. The apparatus to assign systems of cardinal invariants to models of a
first order theory (or determine its impossibility) is developed in [Sh:a]. It
is natural to try to extend this theory to classes of models which are
described in other ways. Work on the classification theory for nonelementary
classes [Sh:88] and for universal classes [Sh:300] led to the conclusion that
an axiomatic approach provided the best setting for developing a theory of
wider application. In the first chapter we describe the axioms on which the
remainder of the article depends and give some examples and context to justify
this level of generality. The study of universal classes takes as a primitive
the notion of closing a subset under functions to obtain a model. We replace
that concept by the notion of a prime model. We begin the detailed discussion
of this idea in Chapter II. One of the important contributions of
classification theory is the recognition that large models can often be
analyzed by means of a family of small models indexed by a tree of height at
most omega. More precisely, the analyzed model is prime over such a tree.
Chapter III provides sufficient conditions for prime models over such trees to
exist.
|
math
|
1,803 |
Full reflection of stationary sets below aleph_omega
|
math.LO
|
It is consistent that for every n >= 2, every stationary subset of omega_n
consisting of ordinals of cofinality omega_k where k = 0 or k <= n-3 reflects
fully in the set of ordinals of cofinality omega_{n-1}. We also show that this
result is best possible.
|
math
|
1,804 |
The Hanf numbers of stationary logic. II. Comparison with other logics
|
math.LO
|
We show that the ordering of the Hanf number of L_{omega, omega}(wo) (well
ordering), L^c_{omega, omega} (quantification on countable sets), L_{omega,
omega}(aa) (stationary logic) and second order logic, have no more restraints
provable in ZFC than previously known (those independence proofs assume
CON(ZFC) only). We also get results on corresponding logics for L_{lambda, mu} .
|
math
|
1,805 |
Strong partition relations below the power set: consistency, was Sierpinski right, II?
|
math.LO
|
We continue here [Sh276] but we do not relay on it. The motivation was a
conjecture of Galvin stating that 2^{omega} >= omega_2 + omega_2->
[omega_1]^{n}_{h(n)} is consistent for a suitable h: omega-> omega. In section
5 we disprove this and give similar negative results. In section 3 we prove the
consistency of the conjecture replacing omega_2 by 2^omega, which is quite
large, starting with an Erd\H{o}s cardinal. In section 1 we present iteration
lemmas which are needed when we replace omega by a larger lambda and in section
4 we generalize a theorem of Halpern and Lauchli replacing omega by a larger
lambda .
|
math
|
1,806 |
Viva la difference I: Nonisomorphism of ultrapowers of countable models
|
math.LO
|
We show that it is not provable in ZFC that any two countable elementarily
equivalent structures have isomorphic ultrapowers relative to some ultrafilter
on omega .
|
math
|
1,807 |
The primal framework. II. Smoothness
|
math.LO
|
This is the second in a series of articles developing abstract classification
theory for classes that have a notion of prime models over independent pairs
and over chains. It deals with the problem of smoothness and establishing the
existence and uniqueness of a `monster model'. We work here with a predicate
for a canonically prime model.
|
math
|
1,808 |
On a conjecture of Tarski on products of cardinals
|
math.LO
|
We look at an old conjecture of A. Tarski on cardinal arithmetic and show
that if a counterexample exists, then there exists one of length omega_1 +
omega .
|
math
|
1,809 |
A partition theorem for pairs of finite sets
|
math.LO
|
Every partition of [[omega_1]^{< omega}]^2 into finitely many pieces has a
cofinal homogeneous set. Furthermore, it is consistent that every directed
partially ordered set satisfies the partition property if and only if it has
finite character.
|
math
|
1,810 |
Coding and reshaping when there are no sharps
|
math.LO
|
Assuming 0^sharp does not exist, kappa is an uncountable cardinal and for all
cardinals lambda with kappa <= lambda < kappa^{+ omega}, 2^lambda = lambda^+,
we present a ``mini-coding'' between kappa and kappa^{+ omega}. This allows us
to prove that any subset of kappa^{+ omega} can be coded into a subset, W of
kappa^+ which, further, ``reshapes'' the interval [kappa, kappa^+), i.e., for
all kappa < delta < kappa^+, kappa = (card delta)^{L[W cap delta]}. We sketch
two applications of this result, assuming 0^sharp does not exist. First, we
point out that this shows that any set can be coded by a real, via a set
forcing. The second application involves a notion of abstract condensation, due
to Woodin. Our methods can be used to show that for any cardinal mu,
condensation for mu holds in a generic extension by a set forcing.
|
math
|
1,811 |
Cardinal arithmetic for skeptics
|
math.LO
|
We present a survey of some results of the pcf-theory and their applications
to cardinal arithmetic. We review basics notions (in section 1), briefly look
at history in section 2 (and some personal history in section 3). We present
main results on pcf in section 5 and describe applications to cardinal
arithmetic in section 6. The limitations on independence proofs are discussed
in section 7, and in section 8 we discuss the status of two axioms that arise
in the new setting. Applications to other areas are found in section 9.
|
math
|
1,812 |
The universality spectrum of stable unsuperstable theories
|
math.LO
|
It is shown that if T is stable unsuperstable, and aleph_1< lambda
=cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)<
mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g.
aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These
results are generalized to kappa =cf(kappa) < kappa (T) in the place of
aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa
< kappa (T) then there is a universal tree of height kappa +1 in cardinality
lambda .
|
math
|
1,813 |
Constructing strongly equivalent nonisomorphic models for unsuperstable theories. Part B
|
math.LO
|
We study how equivalent nonisomorphic models of unsuperstable theories can
be. We measure the equivalence by Ehrenfeucht-Fraisse games. This paper
continues [HySh:474].
|
math
|
1,814 |
On the Singular Cardinal Hypothesis
|
math.LO
|
We use the core model for sequences of measures to prove a new lower bound
for the consistency strength of the failure of the SCH:
THEOREM
(i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa
> kappa^+$ then there is an inner model with a cardinal $\kappa$ such that for
all ordinals $\alpha<\kappa$ there is an ordinal $\nu < \kappa$ with $o(\nu) >
\alpha$.
(ii) If there is a singular strong limit cardinal $\kappa$ of uncountable
cofinality such that $2^\kappa > \kappa^+$ then there is an inner model with
$o(\kappa) = \kappa^{++}$.
Since this paper was originally submitted, Gitik has improved this result to
give exact lower bounds.
|
math
|
1,815 |
A division Algorithm for the Free Left Distributive Algebra
|
math.LO
|
The normal form theorem, proved in R. Laver, On the left distributive law and
the freeness of an algebra of elementary embeddings, Advances in Mathematics 91
(1992), 209-231, for the free algebra $\Cal A$ on one generator $x$ satisfying
the left distributive law $a(bc) = (ab)(ac)$ is extended by showing that
members of $\Cal A$ can be put into a "division form."
|
math
|
1,816 |
On the Algebra of Elementary Embeddings of a Rank into Inself
|
math.LO
|
Let $j:V_\lambda---> V_\lambda$ be an elementary embedding, with critical
point $\kappa$, and let $f(n)$ be the number of critical points of embeddings
in the algebra generated by $j$ which lie between $j^n(\kappa)$ and
$j^{n+1}(\kappa)$. It is shown that $f(n)$ is finite for all $n$.
|
math
|
1,817 |
Maximal Chains in {}^ωωand Ultrapowers of the Integers
|
math.LO
|
Various questions posed by P. Nyikos concerning ultrafilters on $\omega$ and
chains in the partial order $(\omega,<^*)$ are answered. The main tool is the
oracle chain condition and variations of it.
|
math
|
1,818 |
A short proof of the irreflexivity conjecture
|
math.LO
|
Gives a short proof of Dehornoy's latest result. The same simple argument
(and more) was discovered by Laver's student Larue.
|
math
|
1,819 |
On Gödel's second incompleteness theorem
|
math.LO
|
A very short proof of G\"odel's second incompleteness theorem (for set
theory, second order arithmetic etc.)
|
math
|
1,820 |
On Braid Words and Irreflexivity
|
math.LO
|
The purpose of this note is to prove irreflexivity, and hence the linear
ordering, in ZFC, without some of the machinery used by Dehornoy.
|
math
|
1,821 |
Embeddings of Iteration Trees
|
math.LO
|
This paper, dating from May 1991, contains preliminary (and unpublishable)
notes on investigations about iteration trees. They will be of interest only to
the specialist.
In the first two sections I define notions of support and embeddings for tree
iterations, proving for example that every tree iteration is a direct limit of
finite tree iterations. This is a generalization to models with extenders of
basic ideas of iterated ultrapowers using only ultrapowers.
In the final section (which is most of the paper) I sketch a proof that any
tree iteration can be embedded into a normal iteration, that is, a tree
iteration with the extenders in nondecreasing order of strength and with
strictly increasing critical points.
|
math
|
1,822 |
Reaping Numbers of Boolean Algebras
|
math.LO
|
A subset $A$ of a Boolean algebra $B$ is said to be $(n,m)$-reaped if there
is a partition of unity $P \subset B$ of size $n$ such that the cardinality of
$\{b \in P: b \wedge a \neq \emptyset\}$ is greater than or equal to $m$ for
all $a\in A$. The reaping number $r_{n,m}(B)$ of a Boolean algebra $B$ is the
minimum cardinality of a set $A \subset B\setminus \{0\}$ such which cannot be
$(n,m)$-reaped. It is shown that, for each $n \in \omega$, there is a Boolean
algebra $B$ such that $r_{n+1,2}(B) \neq r_{n,2}(B)$. Also, $\{r_{n,m}(B) :
\{n,m\}\subseteq\omega\}$ consists of at most two consecutive integers. The
existence of a Boolean algebra $B$ such that $r_{n,m}(B) \neq r_{n',m'}(B)$ is
equivalent to a statement in finite combinatorics which is also discussed.
|
math
|
1,823 |
Full reflection of stationary sets at regular cardinals
|
math.LO
|
A stationary subset S of a regular uncountable cardinal kappa reflects fully
at regular cardinals if for every stationary set T subseteq kappa of higher
order consisting of regular cardinals there exists an alpha in T such that S
cap alpha is a stationary subset of alpha. We prove that the Axiom of Full
Reflection which states that every stationary set reflects fully at regular
cardinals, together with the existence of n-Mahlo cardinals is equiconsistent
with the existence of Pi^1_n-indescribable cardinals. We also state the
appropriate generalization for greatly Mahlo cardinals.
|
math
|
1,824 |
The Cardinality of the second uniform indiscernible
|
math.LO
|
When the second uniform indiscernible is $\aleph_{2}$, the Martin-Solovay
tree only constructs countably many reals; this resolves a number of open
questions in descriptive set theory.
|
math
|
1,825 |
Critical points in an algebra of elementary embeddings
|
math.LO
|
Given two elementary embeddings from the collection of sets of rank less than
$\lambda$ to itself, one can combine them to obtain another such embedding in
two ways: by composition, and by applying one to (initial segments of) the
other. Hence, a single such nontrivial embedding $j$ generates an algebra of
embeddings via these two operations, which satisfies certain laws (for example,
application distributes over both composition and application). Laver has
shown, among other things, that this algebra is free on one generator with
respect to these laws.
The set of critical points of members of this algebra is the subject of this
paper. This set contains the critical point $\kappa_0$ of $j$, as well as all
of the other ordinals $\kappa_n$ in the critical sequence of $j$ (defined by
$\kappa_{n+1} = j(\kappa_n)$). But the set includes many other ordinals as
well. The main result of this paper is that the number of critical points below
$\kappa_n$ (which has been shown to be finite by Laver and Steel) grows so
quickly with $n$ that it dominates any primitive recursive function. In fact,
it grows faster than the Ackermann function, and even faster than a slow
iterate of the Ackermann function. Further results show that, even just below
$\kappa_4$, one can find so many critical points that the number is only
expressible using fast-growing hierarchies of iterated functions (six levels of
iteration beyond exponentials).
|
math
|
1,826 |
Many simple cardinal invariants
|
math.LO
|
For g < f in omega^omega we define c(f,g) be the least number of uniform
trees with g-splitting needed to cover a uniform tree with f-splitting. We show
that we can simultaneously force aleph_1 many different values for different
functions (f,g). In the language of Blass: There may be aleph_1 many distinct
uniform Pi^0_1 characteristics.
|
math
|
1,827 |
Covering games and the Banach-Mazur game: k-tactics
|
math.LO
|
Given a free ideal J of subsets of a set X, we consider games where player
ONE plays an increasing sequence of elements of the sigma completion of J, and
TWO tries to cover the union of this sequence by playing one set at a time from
J. We describe various conditions under which player TWO has has a winning
strategy that uses only information about the most recent k moves of ONE, and
apply some of these results to the Banach-Mazur game.
|
math
|
1,828 |
Donder's Version of Revised Countable Support
|
math.LO
|
Shelah introduced the revised countable support (RCS) iteration to iterate
semiproperness. This was an endpoint in the search for an iteration of a weak
condition, still implying that aleph1 is preserved.
Dieter Donder found a better manageable approach to this iteration, which is
presented here.
|
math
|
1,829 |
Remark on the Failure of Martin's Axiom
|
math.LO
|
Let m be the least cardinal k such that MA(k) fails. The only known model for
"m is singular" was constructed by Kunen. In Kunen's model cof(m)=omega_1. It
is unknown whether "omega_1 < cof(m) < m" is consistent. The purpose of this
paper is to present a proof of Kunen's result and to identify the difficulties
of generalizing this result to an arbitrary uncountable cofinality.
|
math
|
1,830 |
Non-existence of Universal Orders in Many Cardinals
|
math.LO
|
Our theme is that not every interesting question in set theory is independent
of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$
which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$
a covering theorem from the hypothesis of the existence of a universal model
for some theory; and we prove --- again in ZFC --- that for a large class of
cardinals there is no universal linear order (e.g. in every
$\aleph_1<\l<2^{\aleph_0}$). In fact, what we show is that if there is a
universal linear order at a regular $\l$ and its existence is not a result of a
trivial cardinal arithmetical reason, then $\l$ ``resembles'' $\aleph_1$ --- a
cardinal for which the consistency of having a universal order is known. As for
singular cardinals, we show that for many singular cardinals, if they are not
strong limits then they have no universal linear order. As a result of the non
existence of a universal linear order, we show the non-existence of universal
models for all theories possessing the strict order property (for example,
ordered fields and groups, Boolean algebras, p-adic rings and fields, partial
orders, models of PA and so on).
|
math
|
1,831 |
Finite left-distributive algebras and embedding algebras\endtitle
|
math.LO
|
We consider algebras with one binary operation $\cdot$ and one generator
({\it monogenic}) and satisfying the left distributive law $a\cdot (b\cdot
c)=(a\cdot b)\cdot (a\cdot c)$. One can define a sequence of finite
left-distributive algebras $A_n$, and then take a limit to get an infinite
monogenic left-distributive algebra~$A_\infty$. Results of Laver and Steel
assuming a strong large cardinal axiom imply that $A_\infty$ is free; it is
open whether the freeness of $A_\infty$ can be proved without the large
cardinal assumption, or even in Peano arithmetic. The main result of this paper
is the equivalence of this problem with the existence of a certain algebra of
increasing functions on natural numbers, called an {\it embedding algebra}.
Using this and results of the first author, we conclude that the freeness of
$A_\infty$ is unprovable in primitive recursive arithmetic.
|
math
|
1,832 |
Cardinal Characteristics and the Product of Countably Many Infinite Cyclic Groups
|
math.LO
|
Let P be the direct product of countably many copies of the additive group Z
of integers. We study, from a set-theoretic point of view, those subgroups of P
for which all homomorphisms to Z annihilate all but finitely many of the
standard unit vectors. Specifically, we relate the smallest possible size of
such a subgroup to several of the standard cardinal characteristics of the
continuum. We also study some related properties and cardinals, both
group-theoretic and set-theoretic. One of the set-theoretic properties and the
associated cardinal are combinatorially natural, independently of any
connection with algebra.
|
math
|
1,833 |
$μ$-complete Souslin trees on $μ^+$
|
math.LO
|
We prove that $\mu=\mu^{<\mu}$, $2^\mu=\mu^+$ and ``there is a non reflecting
stationary subset of $\mu^+$ composed of ordinals of cofinality $<\mu$'' imply
that there is a $\mu$-complete Souslin tree on $\mu^+$.
|
math
|
1,834 |
Perfect sets of random reals
|
math.LO
|
We discuss the relationship between perfect sets of random reals, dominating
reals, and the product of two copies of the random algebra B. Recall that B is
the algebra of Borel sets of 2^omega modulo the null sets. Also given two
models M subseteq N of ZFC, we say that g in omega^omega cap N is a dominating
real over M iff forall f in omega^omega cap M there is m in omega such that
forall n geq m (g(n) > f(n)); and r in 2^omega cap N is random over M iff r
avoids all Borel null sets coded in M iff r is determined by some filter which
is B-generic over M.
We show that there is a ccc partial order P which adds a perfect set of
random reals without adding a dominating real, thus answering a question asked
by the second author in joint work with T. Bartoszynski and S. Shelah some time
ago. The method of the proof of this result yields also that B times B does not
add a dominating real. By a different argument we show that B times B does not
add a perfect set of random reals (this answers a question that A. Miller asked
during the logic year at MSRI).
|
math
|
1,835 |
Amoeba-absoluteness and projective measurability
|
math.LO
|
We study the relationship between Amoeba forcing (the partial order which
generically adds a measure one set of random reals) and projective
measurability. Given a universe V of set theory and a forcing notion P in V we
say that V is Sigma^1_n - P - absolute iff for every Sigma^1_n-sentence phi
with parameters in V we have V models phi iff V^P models phi.
We show that Sigma^1_4-Amoeba-absoluteness implies that forall a in
omega^omega (omega_1^{L[a]} < omega_1^V), and hence Sigma^1_3-measurability.
This answers a question of Haim Judah (private communication).
|
math
|
1,836 |
Finite Combinations of Baire Numbers
|
math.LO
|
Let $\kappa$ be a regular cardinal. Consider the Baire numbers of the spaces
$(2^{\theta})_\kappa$ (functions from $\theta$ to 2 and the less than $\kappa$
topology) for various $\theta \geq \kappa$. Let l be the number of such
different Baire numbers. Models of set theory with l=1 or l=2 are known and it
is also known that l is finite. We show here that if $\kappa > \omega$, then l
could be any given finite number. We do not know whether the same is true for
$\kappa = \omega$.
|
math
|
1,837 |
Meager-nowhere dense games (III): Remainder strategies
|
math.LO
|
Player ONE chooses a meager set and player TWO, a nowhere dense set per
inning. They play $\omega$ many innings. ONE's consecutive choices must form a
(weakly) increasing sequence. TWO wins if the union of the chosen nowhere dense
sets covers the union of the chosen meager sets. A strategy for TWO which
depends on knowing only the uncovered part of the most recently chosen meager
set is said to be a remainder strategy. Theorem (among others): TWO has a
winning remainder strategy for this game played on the real line with its usual
topology.
|
math
|
1,838 |
Some Natural Internal Forcing Schemata Extending ZFC
|
math.LO
|
We give arguments for and prove the consistency of some internal forcing
axioms.
|
math
|
1,839 |
$^*$Forcing
|
math.LO
|
Let $M$ be a transitive model of $ZFC$ and let ${\bf B}$ be a $M$-complete
Boolean algebra in $M.$ (In general a proper class.) We define a generalized
notion of forcing with such Boolean algebras, $^*$forcing. (A $^*$ forcing
extension of $M$ is a transitive set of the form $M[{\bf G}]$ where ${\bf G}$
is an $M$-complete ultrafilter on ${\bf B}.$) We prove that
1. If ${\bf G}$ is a $^*$forcing complete ultrafilter on ${\bf B},$ then
$M[{\bf G}]\models ZFC.$
2. Let $H\sub M.$ If there is a least transitive model $N$ such that $H\in
M,$ $Ord^M=Ord^N,$ and $N\models ZFC,$ then we denote $N$ by $M[H].$ We show
that all models of $ZFC$ of the form $M[H]$ are $^*$forcing extensions of $M.$
As an immediate corollary we get that $L[0^{\#}]$ is a $^*$forcing extension
of $L.$
|
math
|
1,840 |
The Complexity of the Core Model
|
math.LO
|
We use the Sigma^1_3 absoluteness theorem to show that the complexity of the
statement "(omega,E)$ is isomorphic to an initial segment of the core model" is
Pi^1_4, and that the complexity of the statement "(omega,E)$ is isomorphic to a
member of the core model" is Delta^1_5.
|
math
|
1,841 |
Reflection and Weakly Collectionwise Hausdorff Spaces
|
math.LO
|
We show that square(theta) implies that there is a first countable
<theta-collectionwise Hausdorff space that is not weakly theta-collectionwise
Hausdorff. We also show that in the model obtained by Levy collapsing a weakly
compact (supercompact) cardinal to omega_2, first countable
aleph_1-collectionwise Hausdorff spaces are weakly aleph_2-collectionwise
Hausdorff (weakly collectionwise Hausdorff). In the last section we show that
assuming E^omega_theta, a certain theta-family of integer valued functions
exists and that in the model obtained by Levy collapsing a supercompact
cardinal to omega_2, these families do not exist.
|
math
|
1,842 |
Splitting number and the core model
|
math.LO
|
We can generalize the definition of {\it splitting number } $s(\kappa )$ for
$\kappa$ uncountable regular: $s(\kappa )=min\{ |\Cal S|:\Cal S\subset \Cal
P(\kappa ) \forall a\in \kappa ^\kappa \exists b\in \Cal S |a\cap
b|=|a\setminus b|=\kappa\}$ However,$\exists \kappa>\aleph_0$ $s(\kappa
)>\kappa ^+$ becomes a considerable hypothesis,shown consistent from a
supercompact.We show that it implies inner models of $\exists \alpha :o(\alpha
)=\alpha ^{++}$
|
math
|
1,843 |
Set-theoretic aspects of periodic $FC$-groups --- extraspecial p-groups and Kurepa trees
|
math.LO
|
Given a group G, we let Z(G) denote its center, G' its commutator subgroup,
and Phi (G) its Frattini subgroup (the intersection of all maximal proper
subgroups of G). Given U leq G, we let N_G (U) stand for the normalizer of U in
G. A group G is FC iff every element g in G has finitely many conjugates. A
p-group E is called extraspecial iff Phi (E) = E' = Z(E) cong Z_p, the cyclic
group with p elements.
When generalizing a characterization of centre-by-finite groups due to B. H.
Neumann, M. J. Tomkinson asked the following question. Is there an FC-group G
with vert G / Z(G) vert = kappa but [G:N_G(U)] < kappa for all (abelian)
subgroups U of G, where kappa is an uncountable cardinal. We consider this
question for kappa = omega_1 and kappa = omega_2. It turns out that the answer
is largely independent of ZFC, and that it differs greatly in the two cases.
More explicitly, for kappa = omega_1, it is consistent with, and independent
of, ZFC that there is a group G with vert G / Z(G) vert = omega_1 and [G:N_G
(A)] leq omega for all abelian A leq G. We do not know whether the same
statement is still consistent if we drop abelian. On the other hand, for kappa
= omega_2, the non-existence of groups G with vert G / Z(G) vert = omega_2 and
[G : N_G (A) ] leq omega_1 for all (abelian) A leq G is equiconsistent with the
existence of an inaccessible cardinal. In particular, there is an extraspecial
p-group with this property if there is a Kurepa tree.
|
math
|
1,844 |
Combinatorial properties of Hechler forcing
|
math.LO
|
In this work we use a notion of rank first introduced by James Baumgartner
and Peter Dordal and later developed independently by the third author to show
that adding a Hechler real has strong combinatorial consequences.
We prove:
1) assuming omega_1^V = omega_1^L, there is no real in V[d] which is
eventually different from the reals in L[d], where d is Hechler over V;
2) adding one Hechler real makes the invariants on the left-hand side of
Cicho'n's diagram equal omega_1 and those on the right-hand side equal 2^omega
and produces a maximal almost disjoint family of subsets of omega of size
omega_1;
3) there is no perfect set of random reals over V in V[r][d], where r is
random over V and d Hechler over V[r], thus answering a question of the first
and second authors.
As an intermediate step in the proof of 3) we show that given models M
subseteq N of ZFC such that there is a perfect set of random reals in N over M,
either there is a dominating real in N over M or mu (2^omega cap M) = 0 in N.
|
math
|
1,845 |
The Genericity Conjecture
|
math.LO
|
In this paper we produce a real r such that 0<r<0# in L-degree, yet R is NOT
generic over L (for a forcing amenable to L). This answers a question of
Beller-Jensen-Welch.
|
math
|
1,846 |
A simpler proof of Jensen's coding theorem
|
math.LO
|
We present a simplification of Jensen's proof of his Coding Theorem (even in
the case where 0# exists). The proof avoids Jensen's split into cases according
to whether or not 0# exists.
In addition, the paper contains self-contained proofs of the necessary forms
of Square and Diamond, based on an approach to fine structure using Jensen's
$\Sigma^*$ theory.
|
math
|
1,847 |
Minimal universes
|
math.LO
|
An inner model M is MINIMAL if there is a class A such that <M,A> is amenable
yet has no transitive proper elementary submodel. We study minimal universes in
the context of 0#. For example we prove: If 0# exists then there is an inner
model which is minimal and locally generic over L(i.e., every set in the inner
model is set-generic over L). This answers a question of Mack Stanley.
|
math
|
1,848 |
Measurable rectangles
|
math.LO
|
We give an example of a measurable set of reals E such that the set
E'={(x,y): x+y in E} is not in the sigma-algebra generated by the rectangles
with measurable sides. We also prove a stronger result that there exists an
analytic set E such that E' is not in the sigma-algebra generated by rectangles
whose horizontal side is measurable and vertical side is arbitrary. The same
results are true when measurable is replaced with property of Baire.
|
math
|
1,849 |
Baire property and Axiom of Choice
|
math.LO
|
We show that
(1) If ZF is consistent then the following theory is consistent "ZF +
DC(omega_{1}) + Every set of reals has Baire property" and
(2) If ZF is consistent then the following theory is consistent "ZFC + `every
projective set of reals has Baire property' + `any union of omega_{1} meager
sets is meager' ".
|
math
|
1,850 |
Planting Kurepa trees and killing Jech-Kunen trees in a model by using one inaccessible cardinal
|
math.LO
|
By an omega_1--tree we mean a tree of power omega_1 and height omega_1. Under
CH and 2^{omega_1}> omega_2 we call an omega_1--tree a Jech--Kunen tree if it
has kappa many branches for some kappa strictly between omega_1 and
2^{omega_1}. In this paper we prove that, assuming the existence of one
inaccessible cardinal,
(1) it is consistent with CH plus 2^{omega_1}> omega_2 that there exist
Kurepa trees and there are no Jech--Kunen trees,
(2) it is consistent with CH plus 2^{omega_1}= omega_4 that only Kurepa trees
with omega_3 many branches exist.
|
math
|
1,851 |
Jensen's Σ^* theory and the combinatorial content of V=L
|
math.LO
|
The purpose of this article is to indicate how a reformulation of Jensen's
$\Sigma^*$ theory (developed for the study of core models) can be used to
provide a more satisfactory treatment of uniformization, hulls and Skolem
functions for the $J_\alpha$'s. Then we use this approach to fine structure to
formulate a principle intended to capture the combinatorial content of the
axiom $V=L.$
|
math
|
1,852 |
A large Pi-1-2 set absolute for set forcing
|
math.LO
|
Let k be a definable L-cardinal. Then there is a set of reals X,
class-generic over L, such that L(X) and L have the same cardinals, X has size
k in L(X) and some pi-1-2 formula defines X in all set-generic extensions of
L(X). Two corollaries, both assuming the consistency of an inaccessible: It is
consistent for the Perfect Set Property to hold for boldface sigma-1-2 sets,
yet fail for some lightface pi-1-2 set. It is consistent that the Perfect Set
Property holds for boldface sigma-1-2 sets yet some lightface pi-1-2
wellordering of some set of reals has length aleph-1000.
|
math
|
1,853 |
Some compact logics --- results in ZFC
|
math.LO
|
We show that if we enrich first order logic by allowing quantification over
isomorphisms between definable ordered fields the resulting logic, L(Q_{Of}),
is fully compact. In this logic, we can give standard compactness proofs of
various results. Next, we attempt to get compactness results for some other
logics without recourse to diamond, i.e., all our results are in ZFC. We get
the full result for the language where we quantify over automorphisms
(isomorphisms) of ordered fields in Theorem 6.4. Unfortunately we are not able
to show that the language with quantification over automorphisms of Boolean
algebras is compact, but will have to settle for a close relative of that
logic. This is theorem 5.1. In section 4 we prove we can construct models in
which all relevant automorphism are somewhat definable: 4.1, 4.8 for BA, 4.13
for ordered fields. We also give a new proof of the compactness of another
logic -- the one which is obtained when a quantifier Q_{Brch} is added to first
order logic which says that a level tree (definitions will be given later) has
an infinite branch. This logic was previously shown to be compact, but our
proof yields a somewhat stronger result and provides a nice illustration of one
of our methods.
|
math
|
1,854 |
On the number of automorphism of uncontable models
|
math.LO
|
Let s(A) denote the number of automorphisms of a model A of power omega_1. We
derive a necessary and sufficient condition in terms of trees for the existence
of an A with omega_1 < s(A) < 2^{omega_1}. We study the sufficiency of some
conditions for s(A)=2^{omega_1}. These conditions are analogous to conditions
studied by D.Kueker in connection with countable models.
|
math
|
1,855 |
All meager filters may be null
|
math.LO
|
We show that it is consistent with ZFC that all filters which have the Baire
property are Lebesgue measurable. We also show that the existence of a
Sierpinski set implies that there exists a nonmeasurable filter which has the
Baire property.
|
math
|
1,856 |
Forcing isomorphism
|
math.LO
|
A forcing extension may create new isomorphisms between two models of a first
order theory. Certain model theoretic constraints on the theory and other
constraints on the forcing can prevent this pathology. A countable first order
theory is classifiable if it is superstable and does not have either the
dimensional order property or the omitting types order property. Shelah [Sh:c]
showed that if a theory T is classifiable then each model of cardinality lambda
is described by a sentence of L_{infty, lambda}. In fact this sentence can be
chosen in the L^*_{lambda}. (L^*_{lambda} is the result of enriching the
language L_{infty, beth^+} by adding for each mu < lambda a quantifier saying
the dimension of a dependence structure is greater than mu .) The truth of such
sentences will be preserved by any forcing that does not collapse cardinals <=
lambda and that adds no new countable subsets of lambda. Hence, if two models
of a classifiable theory of power lambda are non-isomorphic, they are
non-isomorphic after a lambda-complete forcing. Here we show that the
hypothesis of the forcing adding no new countable subsets of lambda cannot be
eliminated. In particular, we show that non-isomorphism of models of a
classifiable theory need not be preserved by ccc forcings.
|
math
|
1,857 |
Borel partitions of infinite subtrees of a perfect tree
|
math.LO
|
A theorem of Galvin asserts that if the unordered pairs of reals are
partitioned into finitely many Borel classes then there is a perfect set P such
that all pairs from P lie in the same class. The generalization to n-tuples for
n >= 3 is false. Let us identify the reals with 2^omega ordered by the
lexicographical ordering and define for distinct x,y in 2^omega, D(x,y) to be
the least n such that x(n) not= y(n). Let the type of an increasing n-tuple
{x_0, ... x_{n-1}}_< be the ordering <^* on {0, ...,n-2} defined by i<^*j iff
D(x_i,x_{i+1})< D(x_j,x_{j+1}). Galvin proved that for any Borel coloring of
triples of reals there is a perfect set P such that the color of any triple
from P depends only on its type. Blass proved an analogous result is true for
any n. As a corollary it follows that if the unordered n-tuples of reals are
colored into finitely many Borel classes there is a perfect set P such that the
n-tuples from P meet at most (n-1)! classes. We consider extensions of this
result to partitions of infinite increasing sequences of reals. We show, that
for any Borel or even analytic partition of all increasing sequences of reals
there is a perfect set P such that all strongly increasing sequences from P lie
in the same class.
|
math
|
1,858 |
On the existence of atomic models
|
math.LO
|
We give an example of a countable theory T such that for every cardinal
lambda >= aleph_2 there is a fully indiscernible set A of power lambda such
that the principal types are dense over A, yet there is no atomic model of T
over A. In particular, T(A) is a theory of size lambda where the principal
types are dense, yet T(A) has no atomic model.
|
math
|
1,859 |
Provable Pi-1-2 Singletons
|
math.LO
|
In this note I show that a pi-1-2 singleton R of L-degree strictly between 0
and 0# can be obtained so as to be the unique solution to a pi-1-2 formula
which provably has at most one solution, in the theory ZFC+(*) where (*) has
the approximate strength of an ineffable cardinal.
|
math
|
1,860 |
Full Reflection at a Measurable Cardinal
|
math.LO
|
A stationary subset $S$ of a regular uncountable cardinal $\kappa$ {\it
reflects fully} at regular cardinals if for every stationary set $T \subseteq
\kappa$ of higher order consisting of regular cardinals there exists an $\alpha
\in T$ such that $S \cap \alpha$ is a stationary subset of $\alpha$. {\it Full
Reflection} states that every stationary set reflects fully at regular
cardinals. We will prove that under a slightly weaker assumption than $\kappa$
having Mitchell order $\kappa^{++}$ it is consistent that Full Reflection holds
at every $\lambda \leq \kappa$ and $\kappa$ is measurable.
|
math
|
1,861 |
Combinatorics on Ideals and Axiom A
|
math.LO
|
Throughout this abstract let U be a fixed p-point ultrafilter and let I be
the dual ideal. Grigorieff forcing is P(U)={p:omega to 2|dom(p) is an element
of I} ordered by reverse inclusion. It is well known that Grigorieff forcing is
proper. The main result of this paper is the following:
THEOREM: Gregorieff forcing does not satisfy Axiom A. To prove this we use
the following game, denoted G(U), for two players playing alternatively:
Player I plays a partition of omega, {J_n| n<omega}, such that for all
n<omega, J_n is an element of I;
At the nth turn Player II plays a finite subset F_n of J_n. Player II wins
iff the union of the F_n is an element of U.
The following two Lemmas prove the Theorem:
LEMMA 1: If P(U) satisfies axiom A, then player II has a winning strategy in
the game G(U).
LEMMA 2:The game G(U) is undetermined.
|
math
|
1,862 |
The automorphism group of a saturated model has a large dense free subgroup
|
math.LO
|
We prove that for a stable theory $T,$ if $M$ is a saturated model of $T$ of
cardinality $\lambda$ where $\lambda > \big|T\big|,$ then $Aut(M)$ has a dense
free subgroup on $2^{\lambda}$ generators. This affirms a conjecture of Hodges.
|
math
|
1,863 |
Natural Internal Forcing Schemata Extending ZFC
|
math.LO
|
Let V be the universe of sets and V_{\alpha} the sets of rank \leq\alpha. We
develop some axiom schemata for set theory based on the following three
assumptions:
1. V \models ZFC
2. V is large with respect to the class of ordinals
3. V is large with respect to each of the V_{\alpha}
|
math
|
1,864 |
The Consistency of $ZFC+CIFS$
|
math.LO
|
This paper is a technical continuation of ``Natural Axiom Schemata Extending
ZFC. Truth in the Universe?'' In that paper we argue that $CIFS$ is a natural
axiom schema for the universe of sets. In particular it is a natural closure
condition on $V$ and a natural generalization of $IFS(L).$ Here we shall prove
the consistency of $ZFC\ +\ CIFS$ relative to the existence of a transitive
model of $ZFC$ using the compactness theorem together with a class forcing.
|
math
|
1,865 |
Bounding and dominating number of families of functions on N
|
math.LO
|
We pursue the study of families of functions on the natural numbers, with
emphasis here on the bounded families. The situation being more complicated
than the unbounded case, we attack the problem by classifying the families
according to their bounding and dominating numbers, the traditional scheme for
gaps. Many open questions remain.
|
math
|
1,866 |
Sums of Darboux and continuous functions
|
math.LO
|
It is shown that that for every Darboux function $F$ there is a non-constant
continuous function $f$ such that $F+f$ is still Darboux. It is shown to be
consistent --- the model used is iterated Sacks forcing --- that for every
Darboux function $F$ there is a nowhere constant continuous function $f$ such
that $F+f$ is still Darboux. This answers questions raised by B.~Kirchiem and
T.~Natkaniec who have shown that in various models of set theory there are
universally bad Darboux functions, Darboux functions whose sum with any nowhere
constant, continuous function fails to be Darboux.
|
math
|
1,867 |
Čech-Stone remainders of spaces that look like $[0,\infty)$
|
math.LO
|
We show that many spaces that look like the half~line~$\halfline=[0,\infty)$
have, under~$\CH$, a \v{C}ech-Stone-remainder that is homeomorphic to~$\Hstar$.
We also show that $\CH$ is equivalent to the statement that all standard
subcontinua of~$\Hstar$ are homeomorphic. The proofs use Model-theoretic tools
like reduced products and elementary equivalence; rather than constructing
homeomorphisms we show that the spaces in question have isomorphic bases for
the closed sets.
|
math
|
1,868 |
The Ehrenfeucht-Fraisse-game of length omega_1
|
math.LO
|
Let (A) and (B) be two first order structures of the same vocabulary. We
shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B
which we denote by G_{omega_1}(A,B). This game is like the ordinary
Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1
moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of
cardinality <= aleph_1. We prove the following results:
Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such
that the game G_{omega_1}(A,B) is non-determined.
Theorem B: If it is consistent that there is a measurable cardinal, then it
is consistent that G_{omega_1}(A,B) is determined for all A and B of
cardinality <= aleph_2.
Theorem C: For any kappa >= aleph_3 there are A and B of cardinality kappa
such that the game G_{omega_1}(A,B) is non-determined.
|
math
|
1,869 |
Strong measure zero sets without Cohen reals
|
math.LO
|
If ZFC is consistent, then each of the following are consistent with ZFC +
2^{{aleph_0}}= aleph_2 :
1.) X subseteq R is of strong measure zero iff |X| <= aleph_1 + there is a
generalized Sierpinski set.
2.) The union of aleph_1 many strong measure zero sets is a strong measure
zero set + there is a strong measure zero set of size aleph_2.
|
math
|
1,870 |
mu-complete Suslin trees on mu^+
|
math.LO
|
We prove that mu = mu^{< mu}, 2^mu = mu^+ and ``there is a non reflecting
stationary subset of mu^+ composed of ordinals of cofinality < mu'' imply that
there is a mu-complete Souslin tree on mu^+ .
|
math
|
1,871 |
An application of Shoenfield's absoluteness theorem to the theory of uniform distribution
|
math.LO
|
If (B_x: x in N) is a Borel family of sets, indexed by the Baire space N =
omega^omega, all B_x have measure zero, and the family is increasing, then the
union of all B_x also has measure zero. We give two proofs of this theorem: one
in the language of set theory, using Shoenfield's theorem on Sigma-1-2 sets,
the other in the language of probability theory, using von Neumann's selection
theorem, and we apply the theorem to a question on completely uniformly
distributed sequences.
|
math
|
1,872 |
Addendum to ``Maximal Chains in $\fomom$ and Ultrapowers of the Integers''
|
math.LO
|
Upon presenting the proof of Theorem 3.3 in "Maximal chains in $$ and
ultrapowers of the integers" I discovered that it is not entirely correct and
certainly some details should be added. I have therefore written an addendum to
the paper and made it available by ftp. Unfortunately the published version
will be somewhat wanting.
|
math
|
1,873 |
The structure of pleasant ideals
|
math.LO
|
Normal ideals on regular uncountable cardinals are familiar objects. We
investigate ideals that are pleasant--while a normal ideal is closed under
arbitrary diagonal unions, a pleasant ideal is closed only under diagonal
unions indexed by sets that are elements of the ideal. We show any selective
ideal extending the nonstationary ideal must be normal.
|
math
|
1,874 |
The canary tree
|
math.LO
|
A canary tree is a tree of cardinality the continuum which has no uncountable
branch, but gains a branch whenever a stationary set is destroyed (without
adding reals). Canary trees are important in infinitary model theory. The
existence of a canary tree is independent of ZFC + GCH.
|
math
|
1,875 |
On CH + 2^{aleph_1}-> (alpha)^2_2 for alpha < omega_2
|
math.LO
|
We prove the consistency of ``CH + 2^{aleph_1} is arbitrarily large +
2^{aleph_1} not-> (omega_1 x omega)^2_2''. If fact, we can get 2^{aleph_1}
not-> [omega_1 x omega]^2_{aleph_0}. In addition to this theorem, we give
generalizations to other cardinals.
|
math
|
1,876 |
Dominating functions and graphs
|
math.LO
|
A graph is called dominating if its vertices can be labelled with integers in
such a way that for every function f: omega-> omega the graph contains a ray
whose sequence of labels eventually exceeds f. We obtain a characterization of
these graphs by producing a small family of dominating graphs with the property
that every dominating graph must contain some member of the family.
|
math
|
1,877 |
A saturated model of an unsuperstable theory of cardinality greater than its theory has the small index property
|
math.LO
|
A model M of cardinality lambda is said to have the small index property if
for every G subseteq Aut(M) such that [Aut(M):G] <= lambda there is an A
subseteq M with |A|< lambda such that Aut_A(M) subseteq G. We show that if M^*
is a saturated model of an unsuperstable theory of cardinality > Th(M), then
M^* has the small index property.
|
math
|
1,878 |
A model in which there are Jech-Kunen trees but there are no Kurepa trees
|
math.LO
|
By an omega_1 --tree we mean a tree of power omega_1 and height omega_1. We
call an omega_1 --tree a Jech--Kunen tree if it has kappa --many branches for
some kappa strictly between omega_1 and 2^{omega_1}. In this paper we construct
the models of CH plus 2^{omega_1}> omega_2, in which there are Jech--Kunen
trees and there are no Kurepa trees.
|
math
|
1,879 |
Peano Arithmetic may not be interpretable in the monadic theory of orders
|
math.LO
|
Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in
the monadic second-order theory of short chains (hence, in the monadic
second-order theory of the real line). We show here that it is consistent that
there is no interpretation even in the monadic second-order theory of all
chains.
|
math
|
1,880 |
Consequences of arithmetic for set theory
|
math.LO
|
In this paper, we consider certain cardinals in ZF (set theory without AC,
the Axiom of Choice). In ZFC (set theory with AC), given any cardinals C and D,
either C <= D or D <= C. However, in ZF this is no longer so. For a given
infinite set A consider Seq(A), the set of all sequences of A without
repetition. We compare |Seq(A)|, the cardinality of this set, to |P(A)|, the
cardinality of the power set of A.
What is provable about these two cardinals in ZF? The main result of this
paper is that
ZF |- for all A: |Seq(A)| not= |P(A)|
and we show that this is the best possible result.
Furthermore, it is provable in ZF that if B is an infinite set, then
|fin(B)|<|P(B)|, even though the existence for some infinite set B^* of a
function f from fin(B^*) onto P(B^*) is consistent with ZF.
|
math
|
1,881 |
Universal graphs without large cliques
|
math.LO
|
We give some existence/nonexistence statements on universal graphs, which
under GCH give a necessary and sufficient condition for the existence of a
universal graph of size lambda with no K(kappa), namely, if either kappa is
finite or cf(kappa)>cf(lambda). (Here K(kappa) denotes the complete graph on
kappa vertices.) The special case when lambda^{< kappa}= lambda was first
proved by F. Galvin. Next, we investigate the question that if there is no
universal K(kappa)-free graph of size lambda then how many of these graphs
embed all the other. It was known, that if lambda^{< lambda}= lambda (e.g., if
lambda is regular and the GCH holds below lambda), and kappa = omega, then this
number is lambda^+. We show that this holds for every kappa <= lambda of
countable cofinality. On the other hand, even for kappa = omega_1, and any
regular lambda >= omega_1 it is consistent that the GCH holds below lambda,
2^{lambda} is as large as we wish, and the above number is either lambda^+ or
2^{lambda}, so both extremes can actually occur.
|
math
|
1,882 |
On uniformly antisymmetric functions
|
math.LO
|
We show that there is always a uniformly antisymmetric f:A-> {0,1} if A
subset R is countable. We prove that the continuum hypothesis is equivalent to
the statement that there is an f:R-> omega with |S_x| <= 1 for every x in R. If
the continuum is at least aleph_n then there exists a point x such that S_x has
at least 2^n-1 elements. We also show that there is a function f:Q-> {0,1,2,3}
such that S_x is always finite, but no such function with finite range on R
exists
|
math
|
1,883 |
Ultrafilters: Where topological dynamics = algebra = combinatorics
|
math.LO
|
We survey some connections between topological dynamics, semigroups of
ultrafilters, and combinatorics. As an application, we give a proof, based on
ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem.
|
math
|
1,884 |
Evasion and prediction --- the Specker phenomenon and Gross spaces
|
math.LO
|
We study the set--theoretic combinatorics underlying the following two
algebraic phenomena.
(1) A subgroup G leq Z^omega exhibits the Specker phenomenon iff every
homomorphism G to Z maps almost all unit vectors to 0. Let se be the size of
the smallest G leq Z^omega exhibiting the Specker phenomenon.
(2) Given an uncountably dimensional vector space E equipped with a symmetric
bilinear form Phi over an at most countable field KK, (E,Phi) is strongly Gross
iff for all countably dimensional U leq E, we have dim(U^perp) leq omega.
Blass showed that the Specker phenomenon is closely related to a
combinatorial phenomenon he called evading and predicting. We prove several
additional results (both theorems of ZFC and independence proofs) about evading
and predicting as well as se, and relate a Luzin--style property associated
with evading to the existence of strong Gross spaces.
|
math
|
1,885 |
Combinatorial properties of classical forcing notions
|
math.LO
|
We discuss the effect of adding a single real (for various forcing notions
adding reals) on cardinal invariants associated with the continuum (like the
unbounding or the dominating number or the cardinals related to measure and
category on the real line). For random and Cohen forcing, this question was
investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah
and myself. We show here:
(1) adding an eventually different or a localization real adjoins a Luzin set
of size continuum and a mad family of size omega_1;
(2) Laver and Mathias forcing collapse the dominating number to omega_1 ---
consequences:
(A) CON(d=omega_1 + unif(L) = unif (M) = kappa = 2^omega) for any regular
uncountable kappa;
(B) Two Laver or Mathias reals added iteratively always force CH (even
diamond);
(C) Sigma^1_4-Mathias-absoluteness implies the Sigma^1_3- Ramsey property;
(3) Miller's rational perfect set forcing preserves the axiom
MA(sigma-centered).
|
math
|
1,886 |
The additivity of porosity ideals
|
math.LO
|
We show that several sigma-ideals related to porous sets have additivity
omega_1 and cofinality 2^omega. This answers a question addressed by Miroslav
Repick'y.
|
math
|
1,887 |
A New Proof of Kunen's Inconsistency
|
math.LO
|
Using elementary pcf, we show that there is no $j:V\to M,$ $M$ transitive,
$j\lambda =\lambda >crit(j),$ $j^{\prime \prime}\lambda \in M.$
|
math
|
1,888 |
Questions and answers -- a category arising in linear logic, complexity theory, and set theory
|
math.LO
|
A category used by de Paiva to model linear logic also occurs in Vojtas's
analysis of cardinal characteristics of the continuum. Its morphisms have been
used in describing reductions between search problems in complexity theory. We
describe this category and how it arises in these various contexts. We also
show how these contexts suggest certain new multiplicative connectives for
linear logic. Perhaps the most interesting of these is a sequential composition
suggested by the set-theoretic application.
|
math
|
1,889 |
Possible Behaviours of the Reflection Ordering of Stationary Sets
|
math.LO
|
If $S,T$ are stationary subsets of a regular uncountable cardinal $\kappa$,
we say that $S$ reflects fully in $T$, $S<T$, if for almost all $\alpha \in T$
(except a nonstationary set) $S \cap \alpha$ is stationary in $\alpha .$ This
relation is known to be a well founded partial ordering. We say that a given
poset $P$ is realized by the reflection ordering if there is a maximal
antichain $\langle X_p ; p \in P \rangle$ of stationary subsets of
$Reg(\kappa)$ so that
$$\forall p,q \in P \; \forall S\subseteq X_p, T\subseteq X_q \text{
stationary}:(S<T \leftrightarrow p<_P q ) .$$
We prove that if $\kappa$ is $\Cal P _2 \kappa -$strong and $P$ an arbitrary
well founded poset of cardinality $\leq \k^+$ then there is a generic extension
where P is realized by the reflection ordering on $\kappa .$
|
math
|
1,890 |
On the divisible parts of quotient groups
|
math.LO
|
Techniques of combinatorial set theory are applied to the following algebraic
problem. Suppose G is an abelian group such that, for all countable subgroups
C, the divisible part of the quotient G/C is countable. What can one conclude
about the size of the divisible part of G/K when the cardinality of the
subgroup K is a given uncountable cardinal?
|
math
|
1,891 |
Is game semantics necessary?
|
math.LO
|
We discuss the extent to which game semantics is implicit in the formalism of
linear logic and in the intuitions underlying linear logic.
|
math
|
1,892 |
How to win some simple iteration games
|
math.LO
|
We introduce two new iteration games: the game G, which is a strengthening of
the weak iteration game, and the game G+, which is somewhat stronger than G but
weaker than the full iteration game of length omega_1.
For a countable M elementarily embeddable in some V_{eta}, we can show that
II wins G(M,omega_1) and that I does not win the G+(M).
|
math
|
1,893 |
IST is more than an algorithm to prove ZFC theorems
|
math.LO
|
There is a sentence in the language of IST, Nelson's internal set theory,
which is not equivalent in IST to a sentence in the ZFC language. Thus the
Reduction algorithm of Nelson, that converts bounded IST formulas with standard
parameters to provably (in IST) equivalent formulas in the ZFC language, cannot
be extended to all formulas of the IST language. Therefore, certain IST
sentences are "meaningless" from the point of view of the standard universe.
|
math
|
1,894 |
Examples for Souslin forcing
|
math.LO
|
We give a model where there is a ccc Souslin forcing which does not satisfy
the Knaster condition. Next, we present a model where there is a sigma-linked
not sigma-centered Souslin forcing such that all its small subsets are
sigma-centered but Martin Axiom fails for this order. Furthermore, we construct
a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which
is proper but not ccc that does not contain a perfect set of mutually
incompatible conditions. Finally we show that ccc Sigma^1_2-notions of forcing
may not be indestructible ccc.
|
math
|
1,895 |
On hidden extenders
|
math.LO
|
A model with a sequence of indiscernibles depending on a particular
precovering set is constructed.The initial assumption is as follows: for every
n<omega the set {alpha | o(alpha)=alpha^+n } is unbounded in kappa.
|
math
|
1,896 |
Iterated Class Forcing
|
math.LO
|
In this paper we isolate the notion of Stratified class forcing and show that
Stratification implies cofinality-preservation and is preserved by iterations
with the appropriate support. Many familiar class forcings are stratified and
therefore can be simultaneously iterated without changing cofinalities,
provided the proper support is used. Easton forcing, Backward Easton forcings
and some modifications of Jensen coding are stratified. Jensen coding is not
stratified but instead obeys a related property, Delta-Stratification, which is
also preservedby iteration with an appropriate larger support.
|
math
|
1,897 |
A combinatorial forcing for coding the universe by a real when there are no sharps
|
math.LO
|
Assuming 0# does not exist, we present a combinatorial approach to Jensen's
method of coding by a real. The forcing uses combinatorial consequences of fine
structure (including the Covering Lemma, in various guises), but makes no
direct appeal to fine structure itself.
|
math
|
1,898 |
Applications of cohomology to questions in set theory i: hausdorff gaps
|
math.LO
|
We explore an application of homological algebra to set theoretic objects by
developing a cohomology theory for Hausdorff gaps. The cohomology theory is
introduced with enough generality to be applicable to other questions in set
theory. For gaps, this leads to a natural equivalence notion about which we
answer questions by constructing many simultaneous gaps.
The first result is proved by constructing countably many gaps such that the
union of any combination corresponding to a subset of $\omega$ is again a gap.
This is done in ZFC. New combinatorial hypotheses related to club ($\clubsuit$)
are introduced to prove the second result which increases the number of gaps to
$\aleph_1$. The presentation of these results does not depend on the
development of the cohomology theory.
Additionally, the notion of an incollapsible gap is introduced and the
existence of such a gap is shown to be independent of ZFC.
|
math
|
1,899 |
Ultrafilters on omega
|
math.LO
|
A variety of classes of naturally arising ultrafilters on omega is discussed,
and the question is raised whether it is consistent that the classes are empty.
Since all the classes contain the P-point ultrafilters, a negative answer would
greatly extend the famous theorem of Shelah.
|
math
|
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