A bunch of people have been asking me to take a look at 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,

well-defined rules; - 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 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

external contradiction.

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

¬P∧P.

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, Q_{1},

Q_{2}, …, Q_{n}, where each statement Q_{i+1} is an

implication of Q_{n}. So you can state a proof as Q_{1} ⇒

Q_{2} ⇒ … ⇒ Q_{n}, where Q_{n} is the

conclusion. In this formulation, Q_{1} is the statement ¬P – that

is, the *negation* of the statement you want to prove; and

Q_{n} 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)⇔Q_{2}⇔…⇔Q_{n-1}⇔(¬P∧P).

Then you’ve got a known-falsehood at the end – and you can also use that

backwards. So (¬P∧P) ⇔ Q_{n-1}. And, in fact, (¬P∧P)

imply each any every one of the Q_{i}. 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

stuff.

The problem for Perez is that this argument is bogus in *so
many ways*.

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

lines.

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”. - Q
_{1}= “The set I can be written out as an array of decimal expansions, …” - Q
_{2}= “It is possible to define a real number … such that it belongs to

the interval, but …” - Q
_{3}= “The array is not a complete listing of the elements of I.”

According to Perez, ¬P⇔Q_{1}. 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

“¬P⇔Q_{1}⇔Q_{2}⇔Q_{3}.” It’s

actually “¬P. Q_{1}. ¬P∧Q_{1}∧…↔Q_{3}.” (I left the dots, because

there are still missing steps; ¬P∧Q_{1} isn’t enough to get

get to Q_{2}.)

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 Q_{3} provides a contradiction, that it

invalidates Q_{2}; which invalidates Q_{1}. 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 “Q_{i} ∧ Axiom ⇔ Q_{i+1}.” The negation of

that doesn’t imply the negation of the axiom; it implies (¬Q_{i})

∨ ¬Axiom).

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.