Inconsistent Mathematics

Inconsistent mathematics is the study of commonplace mathematical
objects, like sets, numbers, and functions, where some contradictions
are allowed. Tools from formal logic are used to make sure any
contradictions are contained and that the overall theories remain
coherent. Inconsistent mathematics began as a response to the set
theoretic and semantic paradoxes such as Russell's Paradox and the
Liar Paradox—the response being that these are interesting facts to
study rather than problems to solve—and has so far been of interest
primarily to logicians and philosophers. More recently, though, the
techniques of inconsistent mathematics have been extended into wider
mathematical fields, such as vector spaces and topology, to study
inconsistent structure for its own sake.

To be precise, a mathematical theory is a collection of sentences, the
theorems, which are deduced through logical proofs. A contradiction is
a sentence together with its negation, and a theory is inconsistent if
it includes a contradiction. Inconsistent mathematics considers
inconsistent theories. As a result, inconsistent mathematics requires
careful attention to logic. In classical logic, a contradiction is
always absurd: a contradiction implies everything. A theory containing
every sentence is trivial. Classical logic therefore makes nonsense of
inconsistency and is inappropriate for inconsistent mathematics.
Classical logic predicts that the inconsistent has no structure. A
paraconsistent logic guides proofs so that contradictions do not
necessarily lead to triviality. With a paraconsistent logic,
mathematical theories can be both inconsistent and interesting.

This article discusses inconsistent mathematics as an active research
program, with some of its history, philosophy, results and open
questions.

1. Introduction

Inconsistent mathematics arose as an independent discipline in the
twentieth century, as the result of advances in formal logic. In the
nineteenth century, a great deal of extra emphasis was placed on
formal rigor in proofs, because various confusions and contradictions
had appeared in the analysis of real numbers. To remedy the situation
required examining the inner workings of mathematical arguments in
full detail. Mathematics had always been conducted through
step-by-step proofs, but formal logic was intended to exert an extra
degree of control over the proofs, to ensure that all and only the
desired results would obtain. Various reconstructions of mathematical
reasoning were advanced.

One proposal was classical logic, pioneered by Giuseppe Peano, Gottlob
Frege, and Bertrand Russell. Another was paraconsistent logic, arising
out of the ideas of Jan Łukasiewicz and N. A. Vasil'év around 1910,
and first realized in full by Jakowski in 1948. The first to suggest
paraconsistency as a ground for inconsistent mathematics was Newton da
Costa in Brazil in 1958. Since then, his school has carried on a study
of paraconsistent mathematics. Another school, centered in Australia
and most associated with the name of Graham Priest, has been active
since the 1970s. Priest and Richard Routley have forwarded the thesis
that some inconsistent theories are not only interesting, but true;
this is dialetheism.

Like any branch of mathematics, inconsistent mathematics is the study
of abstract structures using proofs. Paraconsistent logic offers an
unusually exacting proof guide that makes sure inconsistency does not
get out of hand. Paraconsistency is not a magic wand or panacea. It is
a methodology for hard work. Paraconsistency only helps us from
getting lost, or falling into holes, when navigating through rough
terrain.
a. An Example

Consider a collection of objects. The collection has some size, the
number of objects in the collection. Now consider all the ways that
these objects could be recombined. For instance, if we are considering
the collection {a, b}, then we have four possible recombinations: just
a, just b, both a and b, or neither a nor b. In general, if a
collection has κ members, it has 2κ recombinations. It is a theorem
from the nineteenth century that, even if the collections in question
are infinitely large, still κ < 2κ, that is, the number of
recombinations is always strictly larger than the number of objects in
the original collection. This is Georg Cantor's theorem.

Now consider the collection of all objects, the universe, V. This
collection has some size,
|V|, and quite clearly, being by definition the collection of
everything, this size is the absolutely largest size any collection
can be. (Any collection is contained in the universe by definition,
and so is no bigger than the universe.) By Cantor's theorem, though,
the number of recombinations of all the objects exceeds the original
number of objects. So the size of the recombinations is both larger
than, and cannot be larger than, the universe,

This is Cantor's paradox. Inconsistent mathematics is unique in that,
if rigorously argued, Cantor's paradox is a theorem.
2. Background
a. Motivations

There are at least two reasons to take an interest in inconsistent
mathematics, which roughly fall under the headings of pure and
applied. The pure reason is to study structure for its own sake.
Whether or not it has anything to do with physics, for example,
Reimann geometry is beautiful. If the ideas displayed in inconsistent
mathematics are rich and elegant and support unexpected developments
that make deep connections, then people will study it. G. H. Hardy's A
Mathematician's Apology (1940) makes a stirring case that pure
mathematics is inherently worth doing, and inconsistent mathematics
provides some panoramic views not available anywhere else.

The applied reasons derive from a longstanding project at the
foundations of mathematics. Around 1900, David Hilbert proposed a
program to ensure mathematical security. Hilbert wanted:

* to formalize all mathematical reasoning into an exact notation
with algorithmic rules;
* to provide axioms for all mathematical theories, such that no
contradictions are provable (consistency), and all true facts are
provable (completeness).

Hilbert's program was (in part) a response to a series of conceptual
crises and responses from ancient Greece through Issac Newton and G.
W. Leibniz (see section 6 below) to Cantor. Each crisis arose due to
the imposition of some objects that did not behave well in the
theories of the day—most dramatically in Russell's paradox, which
seems to be about logic itself.

The inconsistency would not have been such trouble, except the logic
employed at that time was explosive: From a contradiction, anything at
all can be proved, so Russell's paradox was a disaster. In 1931, Kurt
Gödel's theorems showed that consistency is incompatible with
completeness, that any complete foundation for mathematics will be
inconsistent. Hilbert's program as stated is dead, and with it even
more ambitious projects like Frege-Russell logicism.

The failure of completeness was hard to understand. Hilbert and many
others had felt that any mathematical question should be amenable to a
mathematical answer. The motive to inconsistency, then, is that an
inconsistent theory can be complete. In light of Gödel's result, an
inconsistent foundation for mathematics is the only remaining
candidate for completeness.
b. Perspectives

There are different ways to view the place of inconsistent
mathematics, ranging from the ideological to the pragmatic.

The most extreme view is that inconsistent mathematics is a rival to,
or replacement for, classical consistent mathematics. This seems to
have been Routley's intent. Routley wanted to perfect an "ultramodal
universal logic," which would be a flexible and powerful reasoning
tool applicable to all subjects and in all situations. Routley argued
that some subjects and situations are intractably inconsistent, and so
the universal logic would be paraconsistent. He wanted such a logic to
underly not only set theory and arithmetic, but metaphysics, ecology
and economics. (For example, Routley and Meyer [1976] suggest that our
economic woes are caused by using classical logic in economic theory.)
Rotuley (1980, p.927) writes:

There are whole mathematical cities that have been closed off and
partially abandoned because of the outbreak of isolated
contradictions. They have become like modern restorations of ancient
cities, mostly just patched up ruins visited by tourists.

In order to sustain the ultramodal challenge to classical logic it
will have to be shown that even though leading features of classical
logic and theories have been rejected, … by going ultramodal one does
not lose great chunks of the modern mathematical megalopolis. … The
strong ultramodal claim—not so far vindicated—is the expectedly brash
one: we can do everything you can do, only better, and we can do more.

A more restrained, but still unorthodox, view is of inconsistency as a
non-revisionary extension of classical theory. There is nothing wrong
with the classical picture of mathematics, says a proponent of this
position, except if we think that the classical picture exhausts all
there is to know. A useful analogy is the extension of the rational
numbers by the irrational numbers, to get the real numbers. Rational
numbers are not wrong; they are just not all the numbers. This
moderate line is found in Priest's work. As articulated by da Costa
(1974, p.498):

It would be as interesting to study the inconsistent systems as,
for instance, the non-euclidean geometries: we would obtain a better
idea of the nature of certain paradoxes, could have a better insight
on the connections amongst the various logical principles necessary to
obtain determinate results, etc.

In a similar vein, Chris Mortensen argues that many important
questions about mathematics are deeper than consistency or
completeness.

A third view is even more open-minded. This is to see all theories
(within some basic constraints) as genuine, interesting and useful for
different purposes. Jc Beall and Greg Restall have articulated a
version of this view at length, which they call logical pluralism.
c. Methods

There are at least two ways to go about mathematical research in this
field. The first is axiomatic. The second is model theoretic. The
axiomatic approach is very pure. We pick some axioms and inference
rules, some starting assumptions and a logic, and try to prove some
theorems, with the aim of producing something on the model of Euclid,
or Russell and A. N. Whitehead's Principia Mathematica. This would be
a way of obtaining results in inconsistent mathematics independently,
as if we were discovering mathematics for the first time. On the
axiomatic approach there is no requirement that the same theorems as
classical mathematics be proved. The hardest work goes into choosing a
logic that is weak enough to be paraconsistent, but strong enough to
get results, and formulating the definitions and starting assumptions
in a way that is compatible with the logic. Little work has so far
been done using axiomatics.

By far more attention has been given to the model theoretic approach,
because it allows inconsistent theories to "ride on the backs" of
already developed consistent theories. The idea here is to build up
models—domains of discourse, along with some relations between the
objects in the domain, and an interpretation—and to read off facts
about the attached theory. A way to do this is to take a model from
classical mathematics, and to tinker with the interpretation, as in
collapsed models of arithmetic (section 5 below). The model theoretic
approach shows how different logics interact with different
mathematical structures. Mortensen has followed through on this in a
wide array of subjects, from the differential calculus to vector
spaces to topology to category theory, always asking: Under what
conditions is identity well-behaved? Let Φ(a) be some sentence about
an object a. Mortensen's question is, if a = b holds in a theory, then
is it the case that Φ(a) exactly when Φ(b)? It turns out that the
answer to this question is extremely sensitive to small changes in
logic and interpretations, and the answer can often be "no."

Most of the results obtained to date have been through the model
theoretic approach, which has the advantage of maintaining a
connection with classical mathematics. The model theory approach has
the same disadvantage, since it is unlikely that radically new or
robustly inconsistent ideas will arise from always beginning at
classical ideas.
d. Proofs

It is often thought that inconsistent mathematics faces a grave
problem. A very common mathematical proof technique is reductio ad
absurdum. The concern, then, is that if contradictions are not
absurd—a fortiori, if a theory has contradictions in it—then reductio
is not possible. How can mathematics be done without the most common
sort of indirect proof?

The key to working inconsistent mathematics is its logic. Much hinges
on which paraconsistent logic we are using. For instance, in da
Costa's systems, if a proposition is marked as "consistent," then
reductio is allowed. Similarly, in most relevance logics,
contraposition holds. And so forth. The reader is recommended to the
bibliography for information on paraconsistent logic. Independently of
logic, the following may help.

In classical logic, all contradictions are absurd; in a paraconsistent
logic this is not so. But some things are absurd nevertheless.
Classically, contradiction and absurdity play the same role, of being
a rejection device, a reason to rule out some possibility. In
inconsistent mathematics, there are still rejection devices. Anything
that leads to a trivial theory is to be rejected. More, suppose we are
doing arithmetic and hypothesize that Φ. But we find that Φ has as a
consequence that j=k for every number j, k. Now, we are looking for
interesting inconsistent structure. This may not be full triviality,
but 0 = 1 is nonsense. Reject Φ.

There are many consistent structures that mathematicians do not, and
will never, investigate, not by force of pure logic but because they
are not interesting. Inconsistent mathematicians, irrespective of
formal proof procedures, do the same.
3. Geometry

Intuitively, M. C. Escher's "Ascending, Descending" is a picture of an
impossible structure—a staircase that, if you walked continuously
along it, you would be going both up and down at the same time. Such a
staircase may be called impossible. The structure as a whole seems to
present us with an inconsistent situation; formally, defining down as
not up, then a person walking the staircase would be going up and not
up, at the same time, in the same way, a contradiction. Nevertheless,
the picture is coherent and interesting. What sorts of mathematical
properties does it have? The answers to this and more would be the
start of an inconsistent geometry.

So far, the study has focused on the impossible pictures themselves. A
systematic study of these pictures is being carried out by the
Adelaide school. Two main results have been obtained. First, Bruno
Ernst conjectured that one cannot rotate an impossible picture. This
was refuted in 1999 by Mortensen; later, Quigley designed computer
simulations of rotating impossible Necker cubes. Second, all
impossible pictures have been given a preliminary classification of
four basic forms: Necker cubes, Reutersvärd triangles, Schuster pipes
or fork, and Ernst stairs. It is thought that these forms exhaust the
universe of impossible pictures. If so, an important step towards a
fuller geometry will have been taken, since, for example, a central
theme in surface geometry is to classify surfaces as either convex,
flat, or concave.

Most recently, Mortensen and Leishman (2009) have characterized Necker
cubes, including chains of Neckers, using linear algebra. Otherwise,
algebraic and analytic methods have not yet been applied in the same
way they have been in classical geometry. Inconsistent equational
expressions are not at the point where a robust answer can be given to
questions of length, area, volume etc. On the other hand, as the
Adelaide school is showing, the ancient Greeks do not have a monopoly
on basic "circles drawn in sand" geometric discoveries.
4. Set Theory

Set theory is one of the most investigated areas in inconsistent
mathematics, perhaps because there is the most consensus that the
theories under study might be true. It is here we have perhaps the
most important theorem for inconsistent mathematics, Ross Brady's
(2006) proof that inconsistent set theory is non-trivial.

Set theory begins with two basic assumptions, about the existence and
uniqueness of sets:

* A set is any collection of objects all sharing some property Φ;
* Sets with exactly the same members are identical.

These are the principles of comprehension (a.k.a. abstraction) and
extensionality, respectively. In symbols,

x ∈ {z : Φ(z)} ↔ Φ(x);
x = y ↔ ∀z (z ∈ x ↔ z ∈ y).

Again, these assumptions seem true. When the first assumption, the
principle of comprehension, was proved to have inconsistent
consequences, this was felt to be highly paradoxical. The inconsistent
mathematician asserts that a theory implying an inconsistency is not
automatically equivalent to a theory being wrong.

Newton da Costa was the first to develop an openly inconsistent set
theory in the 1960s, based on Alonzo Church's set theory with a
universal set, or what is similar, W. V. O. Quine's new foundations.
In this system, axioms like those of standard set theory are assumed,
along with the existence of a Russell set

R = {x : x ∉ x}

and a universal set

V = {x : x = x}.

Da Costa has defined "russell relations" and extended this foundation
to model theory, arithmetic and analysis.

Note that V ∈ V, since V = V. This shows that some sets are
self-membered. This also means that V ≠ R, by the axiom of
extensionality. On the other hand, in perhaps the first truly
combinatorial theorem of inconsistent mathematics, Arruda and Batens
(1982) proved

where ∪R is the union of R, the set of all the members of members of
R. This says that every set is a member of a non-self-membered set.
The Arruda-Batens result was obtained with a very weak logic, and
shows that there are real set theoretical theorems to be learned about
inconsistent objects. Arruda further showed that

where P (X) denotes all the subsets of X and ⊆ is the subset relation.

Routley, meanwhile, in 1977 took up his own dialetheic logic and used
it on a full comprehension principle. Routley went as far as to allow
a comprehension principle where the set being defined could appear in
its own definition. A more mundane example of a set appearing in its
own defining condition could be the set of "critics who only criticize
each other." One of Routley's examples is the ultimate inconsistent
set,

x ∈ Z ↔ x ∉ Z.

Routley indicated that the usual axioms of classical set theory can be
proven as theorems—including a version of the axiom of choice—and
began work towards a full reconstruction of Cantorian set theory.

The crucial step in the development of Routley's set theory came in
1989 when Brady adapted an idea from 1971 to produce a model of
dialetheic set theory, showing that it is not trivial. Brady proves
that there is a model in which all the axioms and consequences of set
theory are true, including some contradictions like Russell's, but in
which some sentences are not true. By the soundness of the semantics,
then, some sentences are not provable, and the theory is decidedly
paraconsistent. Since then Brady has considerably refined and expanded
his result.

A stream of papers considering models for paraconsistent set theory
has been coming out of Europe as well. Olivier Esser has determined
under what conditions the axiom of choice is true, for example. See
Hinnion and Libert (2008) for an opening into this work.

Classical set theory, it is well known, cannot answer some fundamental
questions about infinity, Cantor's continuum hypothesis being the most
famous. The theory is incomplete, just as Gödel predicted it would be.
Inconsistent set theory, on the other hand, appears to be able to
answer some of these questions. For instance, consider a large
cardinal hypothesis, that there are cardinals λ such that for any κ <
λ, also 2κ < λ. The existence of large cardinals is undecidable by
classical set theory. But recall the universe, as we did in the
introduction (section 1), and its size |V|. Almost obviously, |V| is
such large a cardinal, just because everything is smaller than it.
Taking the full sweep of sets into account, the hypothesis is true.

Set theory is the lingua franca of mathematics and the home of
mathematical study of infinity. Since Zeno's paradoxes it has been
obvious that there is something paradoxical about infinity. Since
Russell's paradox, it has been obvious that there is something
paradoxical about set theory. So a rigorously developed paraconsistent
set theory serves two purposes. First, it provides a reliable
(inconsistent) foundation for mathematics, at least in the sense of
providing the basic toolkit for expressing mathematical ideas. Second,
the mathematics of infinity can be refined to cover the inconsistent
cases like Cantor's paradox, and cases that have yet to be considered.
See the references for what has been done in inconsistent set theory
so far; what can be still be done in remains one of the discipline's
most exciting open questions.
5. Arithmetic

An inconsistent arithmetic may be considered an alternative or variant
on the standard theory, like a non-euclidean geometry. Like set
theory, though, there are some who think that an inconsistent
arithmetic may be true, for the following reason.

Gödel, in 1931, found a true sentence G about numbers such that, if G
can be decided by arithmetic, then arithmetic is inconsistent. This
means that any consistent theory of numbers will always be an
incomplete fragment of the whole truth about numbers. Gödel's second
incompleteness theorem states that, if arithmetic is consistent, then
that very fact is unprovable in arithmetic. Gödel's incompleteness
theorems state that all consistent theories are terminally unable to
process everything that we know is true about the numbers. Priest has
argued in a series of papers that this means that the whole truth
about numbers is inconsistent.

The standard axioms of arithmetic are Peano's, and their
consequences—the standard theory of arithmetic—is called P A. The
standard model of arithmetic is N = {0, 1, 2, …}, zero and its
successors. N is a model of arithmetic because it makes all the right
sentences true. In 1934 Skolem noticed that there are other
(consistent) models that make all the same sentences true, but have a
different shape—namely, the non-standard models include blocks of
objects after all the standard members of N. The consistent
non-standard models are all extensions of the standard model, models
containing extra objects. Inconsistent models of arithmetic are the
natural dual, where the standard model is itself an extension of a
more basic structure, which also makes all the right sentences true.

Part of this idea goes back to C. F. Gauss, who first introduced the
idea of a modular arithmetic, like that we use to tell the time on
analog clocks: On a clock face, 11 + 2 = 1, since the hands of the
clock revolve around 12. In this case we say that 11 + 2 is congruent
to 1 modulo 12. An important discovery in the late 19th century was
that arithmetic facts are reducible to facts about a successor
relation starting from a base element. In modular arithmetic, a
successor function is wrapped around itself. Gauss no doubt saw this
as a useful technical device. Inconsistent number theorists have
considered taking such congruences much more seriously.

Inconsistent arithmetic was first investigated by Robert Meyer in the
1970's. There he took the paraconsistent logic R and added to it
axioms governing successor, addition, multiplication, and induction,
giving the system R#. In 1975 Meyer proved that his arithemtic is
non-trivial, because R# has models. Most notably, R# has finite models
with a two element domain {0, 1}, with the successor function moving
in a very tight circle over the elements. Such models make all the
theorems of R# true, but keep equations like 0 = 1 just false.

The importance of such finite models is just this: The models can be
represented within the theory itself, showing that a paraconsistent
arithmetic can prove its own non-triviality. In the case of Meyer's
arithemetic, R# has a finitary consistency proof, formalizable in R#.
Thus, in non-classical contexts, Gödel's second incompleteness theorem
loses its bite. Since 1976 relevance logicians have studied the
relationship between R# and PA. Their hope was that R# contains PA as
a subtheory and could replace PA as a stronger, more genuine
arithmetic. The outcome of that project for our purposes is the
development of inconsistent models of arithmetic. Following Dunn,
Meyer, Mortensen, and Friedman, these models have now been extensively
studied by Priest, who bases his work not on the relevant logic R but
on the more flexible logic LP.

Priest has found inconsistent arithmetic to have an elegant general
structure. Rather than describe the details, here is an intuitive
example. We imagine the standard model of arithmetic, up to an
inconsistent element

n = n + 1.

This n is suspected to be a very, very large number, "without physical
reality or psychological meaning." Depending on your tastes, it is the
greatest finite number or the least inconsistent number. We further
imagine that for j, k > n, we have j=k. If in the classical model j≠
k, then this is true too; hence we have an inconsistency, j=k and j≠
k. Any fact true of numbers greater than n are true of n, too, because
after n, all numbers are identical to n. No facts from the consistent
model are lost. This technique gives a collapsed model of arithmetic.

Let T be all the sentences in the language of arithmetic that are true
of N; then let T(n) similarly be all the sentences true of the numbers
up to n, an inconsistent number theory. Since T(n) does not contradict
T about any numbers below n, if n > 0 then T(n) is non-trivial. (It
does not prove 0 = 1, for instance.) The sentences of T(n) are
representable in T(n), and its language contains a truth predicate for
T(n). The theory can prove itself sound. The Gödel sentence for T(n)
is provable in T(n), as is its negation, so the theory is
inconsistent. Yet as Meyer proved, the non-triviality of T(n) can be
established in T(n) by a finite procedure.

Most striking with respect to Hilbert's program, there is a way, in
principle, to figure out for any arithmetic sentence Φ whether or not
Φ holds, just by checking all the numbers up to n. This means that
T(n) is decidable, and that there must be axioms guaranteed to deliver
every truth about the collapsed model. This means that an inconsistent
arithmetic is coherent and complete.
6. Analysis

Newton and Leibniz independently developed the calculus in the 17th
century. They presented ingenious solutions to outstanding problems
(rates of change, areas under curves) using infinitesimally small
quantities. Consider a curve and a tangent to the curve. Where the
tangent line and the curve intersect can be though of as a point. If
the curve is the trajectory of some object in motion, this point is an
instant of change. But a bit of thought shows that it must be a little
more than a point—otherwise, as a measure a rate of change, there
would be no change at all, any more than a photograph is in motion.
There must be some smudge. On the other hand, the instant must be less
than any finite quantity, because there are infinitely many such
instants. An infinitesimal would respect both these concerns, and with
these provided, a circle could be construed as infinitely many
infinitesimal tangent segments.

Infinitesimals were essential, not only for building up the conceptual
steps to inventing calculus, but in getting the right answers. Yet it
was pointed out, most famously by Bishop George Berkeley, that
infinitesimals were poorly understood and were being used
inconsistently in equations. Calculus in its original form was
outright inconsistent. Here is an example. Suppose we are
differentiating the polynomial f(x) =ax2+bx+c. Using the original
definition of a derivative,

In the example, ε is an infinitesimal. It marks a small but
non-trivial neighborhood around x, and can be divided by, so it is not
zero. Nevertheless, by the end ε has simply disappeared. This example
suggests that paraconsistent logic is more than a useful technical
device. The example shows that Leibniz was reasoning with
contradictory information, and yet did not infer everything. On the
contrary, he got the right answer. Nor is this an isolated incident.
Mathematicians seem able to sort through "noise" and derive
interesting truths, even out of contradictory data sets. To capture
this, Brown and Priest (2004) have developed a method they call "chunk
and permeate" to model reasoning in the early calculus. The idea is to
take all the information, including say ε = 0 and ε ≠ 0, and break it
into smaller chunks. Each chunk is consistent, without conflicting
information, and one can reason using classical logic inside of a
chunk. Then a permeation relation is defined which controls the
information flow between chunks. As long as the permeation relation is
carefully defined, conclusions reached in one chunk can flow to another
chunk and enter into reasoning chains there. Brown and Priest propose
this as a model, or rational reconstruction, of what Newton and
Leibniz were doing.

Another, more direct tack for inconsistent mathematics is to work with
infinitesimal numbers themselves. There are classical theories of
infinitesimals due to Abraham Robinson (the hyperreals), and J. H.
Conway (the surreals). Mortensen has worked with differential
equations using hyperreals. Another approach is from category theory.
Tiny line segments ("linelets") of length ϵ are considered, such that
ϵ2 = 0 but it is not the case that ϵ = 0. In this theory, it is also
not the case that ϵ ≠ 0, so the logical law of excluded middle fails.
The category theory approach is the most like inconsistent
mathematics, then, since it involves a change in the logic. However,
the most obvious way to use linelets with paraconsistent logics, to
say that both ϵ = 0 and ϵ ≠ 0 are true, means we are dividing by 0 and
so is probably too coarse to work.

In general the concept of continuity is rich for inconsistent
developments. Moments of change, the flow of time, and the very
boundaries that separate objects have all been considered from the
standpoint of inconsistent mathematics.
7. Computer Science

The questions posed by David Hilbert can be stated in very modern language:

Is there a computer program to decide, for any arithmetic statement,
whether or not the statement can be proven? Is there a program to
decide, for any arithmetic statement, whether or not the statement is
true? We have already seen that Gödel's theorems devastated Hilbert's
program, answering these questions in the negative. However, we also
saw that inconsistent arithmetic overcomes Gödel's results and can
give a positive answer to these questions. It is natural to extend
these ideas into computer science.

Hilbert's program demands certain algorithms—a step-by-step procedure
that can be carried out without insight or creativity. A Turing
machine runs programs, some of which halt after a finite number of
steps, and some of which keep running forever. Is there a program E
that can tell us in advance whether a given program will halt or not?
If there is, then consider the program E*, which exists if E does by
defining it as follows. When considering some program x, E* halts if
and only if x keeps running when given input x. Then

E* halts for E*
if and only if
E* does not halt for E*,

which implies a contradiction. Turing concluded that there is no E*,
and so there is no E—that there cannot be a general decision
procedure.

Any program that can decide in advance the behavior of all other
programs will be inconsistent.

A paraconsistent system can occasionally produce contradictions as an
output, while its procedure remains completely deterministic. (It is
not that the machine occasionally does and does not produce an
output.) There is, in principle, no reason a decision program cannot
exist. Richard Sylvan identifies as a central idea of paraconsistent
computability theory the development of machines "to compute diagonal
functions that are classically regarded as uncomputable." He discusses
a number of rich possibilities for a non-classical approach to
algorithms, including a fixed-point result on the set of all
algorithmic functions, and a prototype for dialetheic machines.

Important results have been obtained by the paraconsistent school in
Brazil—da Costa and Doria in 1994, and Agudelo and Carnielli in 2006.
Like quantum computation, though, at present the theory of
paraconsistent machines outstrips the hardware. Machines that can
compute more than Turing machines await advances in physics.
8. References and Further Reading
a. Further Reading

Priest's In Contradiction (2006) is the best place to start. The
second edition contains material on set theory, continuity, and
inconsistent arithmetic (summarizing material previously published in
papers). A critique of inconsistent arithmetic is in Shapiro (2002).
Franz Berto's book, How to Sell a Contradiction (2007), is harder to
find, but also an excellent and perhaps more gentle introduction.

Some of da Costa's paraconsistent mathematics is summarized in the
interesting collection Frontiers of Paraconsistency (2000)—the
proceedings of a world congress on paraconsistency edited by Batens et
al. More details are in Jacquette's Philosophy of Logic (2007)
handbook; Beall's paper in that volume covers issues about truth and
inconsistency.

Those wanting more advanced mathematical topics should consult
Mortensen's Inconsistent Mathematics (1995). For impossible geometry,
his recent pair of papers with Leishman are a promising advance. His
school's website is well worth a visit. Brady's Universal Logic (2006)
is the most worked-out paraconsistent set theory to date, but not for
the faint of heart.

If you can find it, read Routley's seminal paper, "Ultralogic as
Universal?", reprinted as an appendix to his magnum opus, Exploring
Meinong's Jungle (1980). Before too much confusion arises, note that
Richard Routley and Richard Sylvan, whose posthumous work is collected
by Hyde and Priest in Sociative Logics and their Applications (2000),
in a selfless feat of inconsistency, are the same person.

For the how-to of paraconsistent logics, consult both the entry on
relevance and paraconsistency in Gabbay & Günthner's Handbook of
Philosophical Logic volume 6 (2002), or Priest's textbook An
Introduction to Non-Classical Logic (2008). For paraconsistent logic
and its philosophy more generally see Routley, Priest and Norman's
1989 edited collection. The collection The Law of Non-Contradiction
(Priest et al. 2004) discusses the philosophy of paraconsistency, as
does Priest's Doubt Truth be a Liar (2006).

For the broader philosophical issues associated with inconsistent
mathematics, especially in applications (for example, consequences for
realism and antirealism debates), see Mortensen's recent entry in the
Handbook of the Philosophy of Science (2009, volume 9).
b. References

* Arruda, A. I. & Batens, D. (1982). "Russell's set versus the
universal set in paraconsistent set theory." Logique et Analyse, 25,
pp. 121-133.
* Batens, D., Mortensen, C. , Priest, G., & van Bendegem, J-P.,
eds. (2000). Frontiers of Paraconsistency. Kluwer Academic Publishers.
* Berto, Francesco (2007). How to Sell a Contradiction. Studies in
Logic v. 6. College Publications.
* Brady, Ross (2006). Universal Logic. CSLI Publications.
* Brown, Bryson & Priest, G. (2004). "Chunk and permeate i: the
infinitesimal calculus." Journal of Philosophical Logic, 33, pp.
379–88.
* Colyvan, Mark (2008). "The ontological commitments of
inconsistent theories." Philosophical Studies, 141(1):115 – 23,
October.
* da Costa, Newton C. A. (1974). "On the theory of inconsistent
formal systems." Notre Dame Journal of Formal Logic, 15, pp. 497– 510.
* da Costa, Newton C. A. (2000). Paraconsistent mathematics. In
Batens et al. 2000, pp. 165–180.
* da Costa, Newton C. A., Sylvester, JJ., Krause, D´ecio & Bueno,
Ot´avio (2004). "Paraconsistent logics and paraconsistency: Technical
and philosophical developments."
* da Costa, Newton C.A., Krause, D´ecio & Bueno, Ot´avio (2007).
"Paraconsistent logics and paraconsistency." In Jacquette 2007, pp.
791 – 912.
* Gabbay, Dov M. & Günthner, F. eds. (2002). Handbook of
Philosophical Logic, 2nd Edition, volume 6, Kluwer.
* Hinnion,Roland & Libert, Thierry (2008). "Topological models for
extensional partial set theory." Notre Dame Journal of Formal Logic,
49(1).
* Hyde, Dominic & Priest, G., eds. (2000). Sociative Logics and
their Applications: Essays by the Late Richard Sylvan. Ashgate.
* Jacquette, Dale, ed. (2007). Philosophy of Logic. Elsevier: North Holland.
* Libert, Thierry (2004). "Models for paraconsistent set theory."
Journal of Applied Logic, 3.
* Mortensen, Chris (1995). Inconsistent Mathematics. Kluwer
Academic Publishers.
* Mortensen, Chris (2009). "Inconsistent mathematics: Some
philosophical implications." In A.D. Irvine, ed., Handbook of the
Philosophy of Science Volume 9: Philosophy of Mathematics. North
Holland/Elsevier.
* Mortensen, Chris (2009). "Linear algebra representation of
necker cubes II: The routley functor and necker chains." Australasian
Journal of Logic, 7.
* Mortensen, Chris & Leishman, Steve (2009). "Linear algebra
representation of necker cubes I: The crazy crate." Australasian
Journal of Logic, 7.
* Priest, Graham, Beall, J.C. & Armour-Garb, B., eds. (2004). The
Law of Non-Contradiction. Oxford: Clarendon Press.
* Priest, Graham (1994). "Is arithmetic consistent?" Mind, 103.
* Priest, Graham (2000). "Inconsistent models of arithmetic, II:
The general case." Journal of Symbolic Logic, 65, pp. 1519–29.
* Priest, Graham (2002). "Paraconsistent logic." In Gabbay and
Günthner, eds. 2002, pp. 287–394.
* Priest, Graham (2006a). Doubt Truth Be A Liar. Oxford University Press.
* Priest, Graham (2006b). In Contradiction: A Study of the
Transconsistent. Oxford University Press. second edition.
* Priest, Graham (2008). An Introduction to Non-Classical Logic.
Cambridge University Press, second edition.
* Priest, Graham, Routley, R. & Norman, J. eds. (1989).
Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag.
* Routley, Richard (1977). "Ultralogic as universal?" Relevance
Logic Newsletter, 2, pp. 51–89. Reprinted in Routley 1980.
* Routley, Richard (1980). "Exploring Meinong's Jungle and
Beyond." Philosophy Department, RSSS, Australian National University,
1980. Interim Edition, Departmental Monograph number 3.
* Routley, Richard & Meyer, R. K. (1976). "Dialectical logic,
classical logic and the consistency of the world." Studies in Soviet
Thought, 16, pp. 1–25.
* Shapiro, Stewart (2002). "Incompleteness and inconsistency."
Mind, 111, pp. 817 – 832.