1.
Dialectica Categories and
Cardinalities of the Continuum
Valeria de Paiva and Samuel Gomes da Silva
EBL2014
March, 2014
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 1 / 31
2.
Outline
Outline
1 Introduction
2 Dialectica Categories
3 Cardinal characteristics
4 Computational Complexity
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 2 / 31
3.
Introduction
Introduction
[...]It often happens that there are similarities between the
solutions to problems, or between the structures that are thrown
up as part of the solutions. Sometimes, these similarities point to
more general phenomena that simultaneously explain several
diﬀerent pieces of mathematics. These more general phenomena
can be very diﬃcult to discover, but when they are discovered,
they have a very important simplifying and organizing role, and
can lead to the solutions of further problems, or raise new and
fascinating questions.
– T. Gowers, The Importance of Mathematics, 2000
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 3 / 31
4.
Dialectica Categories
Hilbert’s Program
Main goal of Hilbert’s program:
provide secure foundations for all mathematics.
How? Formalization
all mathematical statements should be written in a precise formal language,
and manipulated according to well deﬁned rules.
Sounds good, doesn’t it?
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 4 / 31
5.
Dialectica Categories
Hilbert’s Program: there is no ignorabimus in mathematics...
Wir müssen wissen, wir werden wissen.
Consistent: no contradiction can be obtained in the formalism of
mathematics.
Complete: all true mathematical statements can be proven in the
formalism. Consistency proof use only "ﬁnitistic" reasoning about ﬁnite mathematical objects.
Conservative: any result about "real objects" obtained using reasoning
about "ideal objects" (such as uncountable sets) can be proved without
ideal objects.
Decidable: an algorithm for deciding the truth or falsity of any
mathematical statement.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 5 / 31
6.
Dialectica Categories
Gödel’s Incompleteness
Gödel’s incompleteness theorems (1931) showed that Hilbert’s program was
impossible to achieve, at least if interpreted in the most obvious way. BUT:
The development of proof theory itself is an outgrowth of
Hilbert’s program. Gentzen’s development of natural deduction
and the sequent calculus [too]. Gödel obtained his incompleteness
theorems while trying to prove the consistency of analysis. The
tradition of reductive proof theory of the Gentzen-Schütte school
is itself a direct continuation of Hilbert’s program.
R. Zach 2005
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 6 / 31
7.
Dialectica Categories
Gödel’s Dialectica Interpretation
If we cannot do Hilbert’s program with ﬁnitistic means, can we do it some
other way? Can we, at least, prove consistency of arithmetic?
Try: liberalized version of Hilbert’s programme – justify classical systems in
terms of notions as intuitively clear as possible.
G’s approach: computable (or primitive recursive) functionals of ﬁnite type,
using the Dialectica Interpretation (named after the Swiss journal
Dialectica, special volume dedicated to Paul Bernays 70th birthday) in
1958.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 7 / 31
8.
Dialectica Categories
Dialectica Categories
Hyland suggested that to provide a categorical model of the Dialectica
Interpretation, one should look at the functionals corresponding to the
interpretation of logical implication. The categories de Paiva came up with
proved to be a model of Linear Logic...
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 8 / 31
9.
Dialectica Categories
Dialectica categories are models of Linear Logic
Linear Logic was created by Girard (1987) as a proof-theoretic tool: the
dualities of classical logic plus the constructive content of proofs of
intuitionistic logic.
Linear Logic: a tool for semantics of Computing.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 9 / 31
10.
Dialectica Categories
Dialectica Interpretation: motivations...
For Gödel (in 1958) the interpretation was a way of proving consistency of
arithmetic.
For me (in 1988) an internal way of modelling Dialectica that turned out to
produce models of Linear Logic instead of models of Intuitionistic Logic,
which were expected...
For Blass (in 1995) a way of connecting work of Votjás in Set Theory with
mine and also his own work on Linear Logic and cardinalities of the
continuum.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 10 / 31
11.
Dialectica Categories
Dialectica categories: useful for proving what...?
Blass (1995) Dialectica categories as a tool for proving inequalities between
cardinalities of the continuum.
Questions and Answers – A Category Arising in Linear Logic, Complexity
Theory, and Set Theory in Advances in Linear Logic (ed. J.-Y. Girard, Y.
Lafont, and L. Regnier) London Math. Soc. Lecture Notes 222 (1995).
Also Propositional connectives and the set theory of the continuum (1995)
and the survey Nearly Countable Cardinals.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 11 / 31
12.
Dialectica Categories
Questions and Answers: The category GSets
Is this simply a coincidence?
The second dialectica category in my thesis GSets (for Girard’s sets) is the
dual of GT the Galois-Tukey connections category in Votjás work.
Blass calls this category PV.
The objects of PV are triples A = (U, X, α) where U, X are sets and
α ⊆ U × X is a binary relation, which we usually write as uαx or α(u, x).
Blass writes it as (A−, A
+
, A) but I get confused by the plus and minus signs.
Two conditions on objects in PV (not the case in GSets):
1. U, X are sets of cardinality at most |R|.
2. The condition in Moore, Hrusák and Dzamonja (MHD) holds:
∀u ∈ U, ∃x ∈ X such that α(u, x) and ∀x ∈ X∃u ∈ U such that ¬α(u, x)
We work with sets 0, 1, 2, N and R and possible relations on those, mostly.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 12 / 31
13.
Dialectica Categories
Questions and Answers: The category PV
In Category Theory morphisms are more important than objects.
Given objects A = (U, X, α) and B = (V, Y, β)
a map from A to B in GSets is a pair of functions f : U → V and
F : Y → X such that α(u, Fy) implies β(fu, y).
Usually write maps as
U α X
⇓ ∀u ∈ U, ∀y ∈ Y α(u, Fy) implies β(fu, y)
V
f
?
β Y
6
F
But a map in PV satisﬁes the dual condition that is β(fu, y) → α(u, Fy).
Trust set-theorists to create morphisms A → B where the relations go in
the opposite direction!...
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 13 / 31
14.
Dialectica Categories
Questions and Answers
Objects in PV are only certain objects of GSets, as they have to satisfy the
two extra conditions. What happens to the structure of the category when
we restrict ourselves to this subcategory?
Fact: GSets has products and coproducts, as well as a terminal and an
initial object.
Given objects of PV, A = (U, X, α) and B = (V, Y, β)
the product A × B in GSets is the object (U × V, X + Y, choice)
where choice: U × V × (X + Y ) → 2 is the relation sending
(u, v, (x, 0)) to α(u, x) and (u, v, (y, 1)) to β(v, y).
The terminal object is T = (1, 0, e) where e is the empty relation,
e: 1 × 0 → 2 on the empty set.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 14 / 31
15.
Dialectica Categories
Questions and Answers
Similarly the coproduct of A and B in GSets is the object
(U + V, X × Y, choice)
where choice: U + V × (X × Y ) → 2 is the relation sending ((u, 0), x, y)
to α(u, x) and ((v, 1), x, y) to β(v, y)
The initial object is 0 = (0, 1, e) where e is the empty relation.
Now if the basic sets U, V, X, Y have all cardinality up to |R| then
(co)products will do the same.
But the MHD condition is a diﬀerent story.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 15 / 31
16.
Dialectica Categories
The structure of PV
Note that neither T or 0 are objects in PV,
as they don’t satisfy the MHD condition.
∀u ∈ U, ∃x ∈ X such that α(u, x) and ∀x ∈ X∃u ∈ U such that ¬α(u, x)
To satisfy the MHD condition neither U nor X can be empty.
Also the object I = (1, 1, id) is not an object of PV, as it satisﬁes the ﬁrst
half of the MHD condition, but not the second. And the object
⊥ = (1, 1, ¬id) satisﬁes neither of the halves. The constants of Linear
Logic do not fare too well in PV.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 16 / 31
17.
Dialectica Categories
The structure of PV
Back to morphisms of PV, the use that is made of the category in this
application is simply of the pre-order induced by the morphisms.
It is somewhat perverse that here, in contrast to usual categorical logic,
A ≤ B ⇐⇒ There is a morphism from B to A
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 17 / 31
18.
Dialectica Categories
Examples of objects in PV
1. The object (N, N, =) where n is related to m iﬀ n = m.
To show MHD is satisﬁed we need to know that ∀n ∈ N∃m ∈ N(n = m),
can take m = n. But also that ∀m ∈ N∃k ∈ N such that ¬(m = k). Here
we can take k = suc(m).
2. The object (N, N, ≤) where n is related to m iﬀ n ≤ m.
3. The object (R, R, =) where r1 and r2 are related iﬀ r1 = r2, same
argument as 1 but equality of reals is logically much more complicated.
4. The objects (2, 2, =) and (2, 2, =), with usual equality.
5.The object (NN
, NN
, ≤) where ≤ is the eventual majorization ordering in
NN
, the set of all functions from N to N. The eventual majorization order
says that given f, g in NN
, g eventually majorizes f iﬀ f(n) ≤ g(n) for all
but ﬁnitely many ns.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 18 / 31
19.
Cardinal characteristics
What have Set Theorists done with PV?
Cardinal Characteristics of the Continuum
Write N for the natural numbers and R for the reals.
All the cardinal characteristics considered will be smaller or equal to the
cardinality of the reals, or the continuum.
They are of little interest if the Continuum Hypothesis holds, as then there
are no cardinalities between the cardinality of the integers ω and the
cardinality of the reals 2ω
.
But if the continuum hypothesis does NOT hold there are many interesting
connections (in general in the form of inequalities) between various
characteristics that Votjás discusses.
Votjás work can be characterized as a way of extracting from the inequality
proofs information that is of interest even if the Continuum Hypothesis
holds.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 19 / 31
20.
Cardinal characteristics
Cardinals from Analysis
- If X and Y are two subsets of N we say that X splits Y if both
X ∩ Y and Y X are inﬁnite.
- The splitting number s is the smallest cardinality of any family S of
subsets of N such that every inﬁnite subset of N is split by some
element of S.
Recall the Bolzano-Weierstrass Theorem: Any bounded sequence of
real numbers has a convergent subsequence.
Recall that we can extend the Bolzano-Weierstrass theorem to say: For any
many countably bounded sequence of real numbers xk = (xkn)n ∈ N there
is a single inﬁnite set A ⊆ N such that the subsequences indexed by A,
(xkn)n ∈ A all converge.
If one tries to extend the result for uncountably many sequences s above is
the ﬁrst cardinal for which the analogous result fail.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 20 / 31
21.
Cardinal characteristics
The structure of PV
Proposition[Rangel] The object (R, R, =) is maximal amongst objects of
PV.
Given any object A = (U, X, α) of PV we know both U and X have
cardinality small than |R|. In particular this means that there is an injective
function ϕ: U → R. (let ψ be its left inverse, i.e ψ(ϕu) = u)
Since α is a relation α ⊆ U × X over non-empty sets, if one accepts the
Axiom of Choice, then for each such α there is a map f : U → X such
that for all u in U, uαf(u).
Need a map Φ: R → X such that
U α X
⇑ ∀u ∈ U, ∀r ∈ R (ϕu = r) → α(u, Φr)
R
ϕ
?
= R
6
Φ
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 21 / 31
22.
Cardinal characteristics
The structure of PV
Axiom of Choice is essential
Given cardinality fn ϕ: U → R, let ψ be its left inverse, ψ(ϕu) = u, for all
u ∈ U. Need a map Φ: R → X such that
U α X
⇑ ∀u ∈ U, ∀r ∈ R (ϕu = r) → α(u, Φr)
R
ϕ
?
= R
6
Φ
Let u and r be such that ϕu = r and deﬁne Φ as ψ ◦ f.
Since ϕu = r can apply Φ to both sides to obtain Φ(ϕ(u)) = Φ(r).
Substituting Φ’s deﬁnition get f(ψ(ϕu))) = Φ(r). As ψ is left inverse of ϕ
(ψ(ϕu) = u) get fu = Φ(r).
Now the deﬁnition of f says for all u in U uαfu, which is uαΦr holds, as
desired. (Note that we did not need the cardinality function for X.)
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 22 / 31
23.
Cardinal characteristics
The structure of PV
Proposition[Rangel] The object (R, R, =) is minimal amongst objects of
PV.
This time we use the cardinality function ϕ: X → R for X. We want a
map in PV of the shape:
R = R
⇑ ∀r ∈ R, ∀x ∈ X α(Φr, x) → (r = ϕx)
U
Φ
?
α X
6
ϕ
Now using Choice again, given the relation α ⊆ U × X we can ﬁx a
function g: X → U such that for any x in X g(x) is such that ¬g(x)αx.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 23 / 31
24.
Cardinal characteristics
The structure of PV
Axiom of Choice is essential
Given cardinality fn ϕ: X → R, let ψ: R → X be its left inverse,
ψ(ϕx) = x, for all x ∈ X. Need a map Φ: R → U such that
R = R
⇑ ∀r ∈ R, ∀x ∈ X α(Φr, x) → ¬(ϕx = r)
U
Φ
?
α X
6
ϕ
Exactly the same argument goes through. Let r and x be such that
ϕx = r and deﬁne Φ as ψ ◦ g.
Since ϕx = r can apply Φ to both sides to obtain Φ(ϕ(x)) = Φ(r).
Substituting Φ’s deﬁnition get g(ψ(ϕx))) = Φ(r). As ψ is left inverse of ϕ,
(ψ(ϕx) = x) get gx = Φ(r).
But the deﬁnition of g says for all x in X ¬gxαx, which is ¬α(Φr, x)
holds. Not quite the desired, unless you’re happy with RAA.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 24 / 31
25.
Cardinal characteristics
More structure of GSets
Given objects A and B of GSets we can consider their tensor products.
Actually two diﬀerent notions of tensor products were considered in GSets,
but only one has an associated internal-hom.
(Blass considered also a mixture of the two tensor products of GSets, that
he calls a sequential tensor product.)
Having a tensor product with associated internal hom means that we have
an equation like:
A ⊗ B → C ⇐⇒ A → (B → C)
Can we do the same for PV? Would it be useful?
The point is to check that the extra conditions on PV objects are satisﬁed.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 25 / 31
26.
Computational Complexity
Computational Complexity 101
A decision problem is given by a set of instances of the problem, together
with a set of positive instances.
The problem is to determine, given an instance whether it is a positive one
or not.
Examples:
1. Instances are graphs and positive instances are 3-colorable graphs;
2. Instances are Boolean formulas and positive instances are satisﬁable
formulas.
A many-one reduction from one decision problem to another is a map
sending instances of the former to instances of the latter, in such a way
that an instance of the former is positive if and only its image is positive.
An algorithm computing a reduction plus an algorithm solving the latter
decision problem can be combined in an algorithm solving the former.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 26 / 31
27.
Computational Complexity
Computational Complexity 101
A search problem is given by a set of instances of the problem, a set of
witnesses and a binary relation between them.
The problem is to ﬁnd, given an instance, some witness related to it.
The 3-way colorability decision problem (given a graph is it 3-way
colorable?) can be transformed into the 3-way coloring search problem:
Given a graph ﬁnd me one 3-way coloring of it.
Examples:
1. Instances are graphs, witnesses are 3-valued functions on the vertices of
the graph and the binary relation relates a graph to its proper 3-way
colorings
2. Instances are Boolean formulas, witnesses are truth assignments and the
binary relation is the satisﬁability relation.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 27 / 31
28.
Computational Complexity
Computational Complexity 101
A many-one reduction from one search problem to another is a map
sending instances of the former to instances of the latter, in such a way
that an instance of the former is positive if and only its image is positive.
An algorithm computing a reduction plus an algorithm solving the latter
decision problem can be combined in an algorithm solving the former.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 28 / 31
29.
Computational Complexity
Cardinals from Analysis
- If f and g are functions N → N, we say that f dominates g if for all
except ﬁnitely many n’s in N, f(n) ≤ g(n).
- The dominating number d is the smallest cardinality of any family D
contained in NN
such that every g in NN
is dominated by some f in D.
- The bounding number b is the smallest cardinality of any family
B ⊆ NN
such that no single g dominates all the members of B.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 29 / 31
30.
Computational Complexity
Connecting Cardinals to the Category
Have the following inequalities:
ω ≤ s ≤ d ≤ 2ω
ω ≤ b ≤ r ≤ rσ ≤ 2ω
The proofs of these inequalities use the category of Galois-Tukey
connections.
Valeria de Paiva and Samuel Gomes da Silva (EBL2014) March, 2014 30 / 31
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