A bunch of people have been asking me to take a look at
href="http://arxiv.org/abs/1002.4433">yet another piece of Cantor crankery
recently posted to Arxiv. In general, I’m sick and tired of Cantor crankery –
it’s been occupying much too much space on this blog lately. But this one is a
real prize. It’s an approach that I’ve never seen before: instead of the usual
weaseling around, this one goes straight for Cantor’s proof.
But it does much, much more than that. In terms of ambition, this thing
really takes the cake. According to the author, one J. A. Perez, he doesn’t
just refute Cantor. No, that would be trivial! Every run-of-the-mill crackpot
claims to refute cantor! Perez claims to refute Cantor, Gödel, Church, and
Turing. Among others. He claims to reform the axiom of infinity in set theory
to remove the problems that it supposedly causes. He claims to be able to use
his reformed axiom of infinity together with his refutation of Cantor to get
rid of the continuum hypothesis, and to eliminate any strange results proved
by the axiom of choice.
Yes, Mr. (Dr? Professor? J. Random Schmuck?) Perez is nothing if not
a true mastermind, a mathematical genius of utterly epic proportions! The
man who single-handedly refutes pretty much all of 20th century mathematics!
The man who has determined that now we must throw away Cantor and Gödel, and
reinstate Hilbert’s program. The perfect mathematics is at hand, if we will only
listen to his utter brilliance!
To give you a bit of background: towards the beginning of the 20th
century, mathematics was in a confused state. They’d just started to encounter
problems with paradoxical statements in foundational things like set theory.
David Hilbert, one of the great mathematicians of the time, set out a grand
goal: to create the perfect mathematical foundation. The goal was to carefully
start from first principles, and:
- create a precise formal system in which everything followed a set of strict,
- define the rules to prevent inconsistency by strictly
preventing anything from reasoning about itself. This was done by
by separating everything strictly into orders. A first order statement
could talk about mathematical objects, but not about predicates. A second
order statement could talk about objects and first-order predicates, but
not about second-order predicates. A third order statement could talk
about primitive objects, first order predicates, and second order predicates – but
not about third. But within the system, you could
never even formulate a statement in which a second-order predicate talked
about a second-order predicate.
- the system was supposed to be complete: in it, all true statements could
be proved to be true; all false statements could be proved
to be false; and all statements were either true or false.
- the system was supposed to be consistent: no contradictions could ever
be derived from valid applications of the system. Every statement
was either true or false – there would be no statements that could be
proved to be both true and false; and there would be no statements that
could not be proved to be either true or false.
- the system was supposed to be decidable: there would be
a mechanical process which, given any statement, would be able to tell you
after a finite period of time whether the statement was true or false.
ere would be an algorithm which would, after a finite amount of time,
Hilbert’s program was a huge deal. It consumed a huge amount of time and
resources from the best mathematicians of the time. It’s well known for
producing a major work by Russell and Whitehead called the Principia, which
took hundreds of pages to work its way up to href="http://scienceblogs.com/goodmath/2006/06/extreme_math_1_1_2.php">being
able to prove that 1 + 1 = 2. It was a truly epic project; but if it were
successful, it would have put all of mathematics on a sound basis. It would,
in some deep and important sense, have completed mathematics.
Alas, Gödel’s work blew Hilbert’s program right out of the water.
You can’t built a complete, consistent system of mathematics. You can’t define
a complete system of math with a decision procedure. But our intrepid genius
claims that all of this is wrong: Cantor’s set theory is bogus, the set of
reals is the same size as the set of natural numbers, Gödel’s
incompleteness theorems are wrong, Turing was wrong about computability, and
we can re-instate Hilbert’s program, finish the Principia, and happily have
the perfect mathematical system, with J. A. Perez as the most brilliant
mathematician of all time.
Except for the minor problem that he’s completely wrong.
What he does is interesting, in a brain-dead sort of way. He basically
tries to find a problem with the basic proof technique used by both Cantor and
Gödel. Of course, proof by contradiction is an old, well-respected proof
system. You can’t just say that all proofs by contradiction are wrong.
There are lots of really good proofs that are done by contradiction. But
Cantor and Gödel are obviously wrong, and yet both proofs look
bulletproof – so there’s got to be something wrong with how they used
proofs by contradiction.
What he homed in on is what he calls internal contradiction versus
He claim that the classical uses of proof by contradiction are what he
calls external contradiction. In proof by external contradiction, you start
with a statement P, which you want to prove is true. So you assume that it’s
false, and then you use that to derive a contradiction. So you assume not-P,
and then use that to derive a proof of some other statement, R, which
is known to be false. So by the process of proof from not-P, you’ve derived
the contradiction R∧¬R. By simple logic, that means that not-P
allowed you to prove a false statement, which in turn means that not-P is
false, and that means that P is true. Presto, you’ve got your proof.
In contrast, Cantor and Gödel are what he calls proof by
internal contradiction. He claims that in these proofs, you start
with the statement you want to prove – P. Then you negate P; and by using
not-P, you can derive a proof that P is true. So not-P is used to derive P.
Instead of ¬P deriving R∧¬R, ¬P is used to derive
It’s an interesting distinction in its way – it’s fundamentally a
difference between a proof built on self-reference, and one that isn’t.
Self-reference is the source of many problems – the fundamental paradoxical
statements that have plagued modern math are all built on self reference:
Cantor’s naive set theory was shown inconsistent by the use of a
self-referential object: the set of all sets that do not contain themselves;
Gödel’s proof is build around the self referential statement “There is no
proof that this statement is true”.
But there’s no logical flaw with a proof that has a self-referential
aspect. Or is there? Perez claims that there is.
Let’s start with his version of a proof by external contradiction. He says
that every proof is, essentially, a sequence of statements, Q1,
Q2, …, Qn, where each statement Qi+1 is an
implication of Qn. So you can state a proof as Q1 ⇒
Q2 ⇒ … ⇒ Qn, where Qn is the
conclusion. In this formulation, Q1 is the statement ¬P – that
is, the negation of the statement you want to prove; and
Qn is the statement R∧¬R, where R is some statement that is
neither P nor ¬P. (That last line is, according to Perez, very important.
He repeats it several times in different ways.) So the entire proof can be
reduced to ¬P⇒(R∧¬R).
In a proof by internal contradiction, you’ve got the same basic
setup, except that you start with ¬P, and you end up with
¬P⇒(¬P∧P). That is, the final contradiction that breaks the
proof contains the initial false assumption.
According to Perez, that’s potentially a problem. It isn’t necessarily
a problem, but it might be. See, if you can reason backwards from the
contradiction to cast doubt on any of the earlier statements, then according
to him, the proof will fail.
This is where things start to get a bit goofy. Suppose that in your basic
proof schema, all of the implications were not just implications, but logical
equivalences. That is, your proof schema for the proof by internal
contradiction was: (¬P)⇔Q2⇔…⇔Qn-1⇔(¬P∧P).
Then you’ve got a known-falsehood at the end – and you can also use that
backwards. So (¬P∧P) ⇔ Qn-1. And, in fact, (¬P∧P)
imply each any every one of the Qi. So the invalid conclusion
implies each of the statements in the proof – which means, according
to Perez, that the truth of every statement is thrown into question; in fact,
the proof by internal contradiction contains an implicit disproof of every true
statement used in the proof. Since the entire proof technique is doing something
that we know to be incorrect, that demonstrates that the proof by internal
contradiction, when the logical connectives are equivalences, is invalid.
If that proof technique is invalid, then every proof that was built using
it is also invalid. So Cantor’s diagonalization goes out the window. And so
does Gödel. And so does most of Chaitin. And, well, lots of other
The problem for Perez is that this argument is bogus in so
Let’s take one nice one, which he’s absolutely explicit about. His
proof schema is based on a proof as a sequence of atomic statements,
each of which implies the following statement. That’s not the
structure of a proof. A proof does have a series of implications leading to
a conclusion, but it’s not linear. Proofs are graphs of implications, not
A perfect demonstration of how bogus this is comes from Perez’s discussion
of Cantor’s diagonalization. According to Perez, you start with
the statement P that you want to prove. In Cantor’s diagonalization,
P is the statement: “The unit interval I = [0, 1)” is an uncountable set.”.
The proof is sketched as:
- Take ¬P to use as the foundational step of the proof by contradiction: ¬P=”The unit interval
I := [0, 1) is a countable set”.
- Q1 = “The set I can be written out as an array of decimal expansions, …”
- Q2 = “It is possible to define a real number … such that it belongs to
the interval, but …”
- Q3 = “The array is not a complete listing of the elements of I.”
According to Perez, ¬P⇔Q1. That is not true:
assuming that the interval is a countable set does not give you the
fact that the set can be written out as an array of decimal expansions. The
decimal expansions comes from an entirely different starting point. So the
proof actually has two branches from the beginning. The proof isn’t a straight
line of implication; it’s a branched structure. So the proof schema isn’t
actually “¬P. Q1. ¬P∧Q1∧…↔Q3.” (I left the dots, because
there are still missing steps; ¬P∧Q1 isn’t enough to get
get to Q2.)
The branching is a fundamental problem for Perez. Because what he wants to
say is that every statement used in the proof is invalidated by the
contradiction. But with logical connectives like “∧”, that’s not true.
Perez wants to say that since Q3 provides a contradiction, that it
invalidates Q2; which invalidates Q1. And, in fact,
according to Perez’s reasoning, it does worse than that: it invalidates every
axiom used in the proof – because the axioms are used as steps.
But axioms aren’t used as discrete atomic steps of the proof implied by
prior steps. They’re used as parts of complex statements. Axioms are used in
the form “Qi ∧ Axiom ⇔ Qi+1.” The negation of
that doesn’t imply the negation of the axiom; it implies (¬Qi)
And boom, there goes Perez’s proof; the structure of a proof by internal
contradiction doesn’t imply the falsehood of every statement used in
the proof. It doesn’t imply the falsehood of any axioms.
But there’s more stupidity. One of the most basic rules of logic is that
you can’t reason from contradiction. That is, given a statement like (A ∧
¬A), the statement is always false. Given any false statement,
you can arbitrarily construct implications: for any possible
statement S, no matter whether S is true or false, (A∧¬A)⇔S. The
fact that a contradiction implies something is utterly worthless. You
can never reason from that implication, because the statement (A∧¬A)
is always false, so you can’t infer anything about the conclusion. The
entire process that Perez uses, trying to trace backwards from a contradiction
is fundamentally meaningless.
From this point, things go downhill. Perez then verges into classic
anti-Cantor-crackpot material, providing his version of several different
constructions that allegedly demonstrate a correspondence between the natural
numbers and the reals. And then he really goes off the deep end.
Seriously, this is one of the most ridiculously grandiose works of crankery
that I’ve ever seen. This guy seriously believes that he’s pretty much the
most brilliant mathematician in the history of mathematics, and that this paper
is the ultimate mathematical tour-de-force. He’s probably sitting in his
office waiting for his Fields medal.