Why Proof by Contradiction is Unreliable and there is no such thing as an Infinite Set Pravin K. Johri Proof by contradiction, relying on the Law of Excluded Middle, is used universally in mathematics even though it is considered unreliable in other sciences. The most notable application is in the Cantor Diagonal Argument to infer that the unit interval is uncountable. Cantor’s theorem uses it to conclude that there are infinitely many cardinalities of infinite sets. The very idea of “Infinite infinities” should boggle everyone’s mind. Excerpts from the Wikipedia page “Georg Cantor” Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied. Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such as algebra, analysis and topology) in a single theory, and provides a standard set of axioms to prove or disprove them. Proof by contradiction was severely criticized by Kronecker, who disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. If proof by contradiction is indeed unreliable, it would negate Cantor’s infinite set theory and invalidate many areas in mathematics. This profound conclusion requires background on the genesis of such a method, and this is provided in Appendices A and B. 1 The Law of Excluded Middle (LoEM) In logic, the Law of Excluded Middle (or the principle of excluded middle) is the third of the three classic laws of thought. It states that For any proposition, either the proposition is true or its negation is true. Excerpts from the Wikipedia page “Brouwer-Hilbert controversy” The earliest known formulation is Aristotle’s principle of non-contradiction - that of two contradictory propositions (i.e. where one proposition is the negation of the other) one must be true, and the other false. It just seems “logical” that an object of discourse either has a stated property (e.g. “This truck is yellow”) or it does not have that property (“This truck is not yellow”) but not both simultaneously. The LoEM implies two absolutes, the proposition and its negation, and no gray area or middle (ground) between the two. The likelihood of a middle is excluded in this line of reasoning. Even Quinn [6] acknowledges that the LoEM may not be justified. Excerpts from Quinn [6] Excluded-middle arguments are unreliable in many areas of knowledge, but absolutely essential in mathematics. Indeed we might define mathematics as the domain in which excluded middle arguments are valid. Instead of debating whether or not it is true, we should investigate the constraints it imposes on our subject. Given the doubts as to the efficacy of excluded-middle arguments, should they be used in a pure science? Perhaps there is no clear answer. If that is the case, it is unlikely one can come up with meaningful constraints when such logic applies and when it doesn’t. Can one analyze whether or not the LoEM is true? The best way to show it is untrustworthy is to demonstrate that it leads to different conclusions than the ones adopted in mathematics. 2 Complement of a Set It turns out finite set theory has a basic concept which is quite similar. In a Venn diagram, a rectangle represents the Universe U universe U which contains all the entities under consideration. Sets are represented by circles and can only be subsets of U. The complement Sc of a set S is S given by the part of the rectangle outside of S's circle. If an object in not in S it must be in its complement, and vice versa. Hence, showing x S is sufficient to establish that x Sc, and vice versa. It does rely, however, on an implicit assumption that x was in the universe to begin with. In the previous example, the universe can be all trucks and the set S can be all yellow trucks. Then Sc should be made up of all trucks that are not yellow. But is it? Proof by Contradiction A proposition can be established by contradicting its negation. For example, if the proposition this truck is not yellow is negated then one can establish that this truck is yellow. The postulate this truck is not yellow seems to be simple, specific and straightforward, and so is its negation. They both assert a single property of an object in a clear concise manner. Is the same true for more complicated premises like the ones in the Cantor Diagonal Argument (CDA) and in Cantor’s theorem? The CDA negates the proposition that the unit interval is countable, which on the surface is also a succinct description. However, below the surface, there are many axioms, definitions, methods, etc. that have allowed one to reach the point when such a claim can be made. Should any or all of these be included in the proposition? Are any implicit assumptions negated by the proof by contradiction argument along with the main proposition? 3 Failure of Proof by Contradiction It turns out that even the simplest of propositions need not be absolutely precise. What if the truck is red and yellow? Now, the previously incompatible statements this truck is yellow and this truck is not yellow are both neither entirely true nor wholly false! Proof by contradiction failed in the simplest of examples as soon as a second dimension of multiple colors is introduced. Note that the original proposition and its prescribed negation didn’t say anything about single or multiple colors. As stated, it should have applied in both cases. Potential Fixes? Perhaps, the proposition needs to include a condition that the truck is of a single color: IF the truck is of a single color, THEN either it is yellow or it is not yellow. But now the result only applies to a part of the Universe (trucks of a single color) and not the entire Universe (all trucks) and it is no longer true in general. Perhaps, the original proposition has be enhanced to include a second claim that the truck is of a single color in order to rule out the red and yellow truck: Proposition The truck is of a single color AND it is yellow 4 Now the proposition is a conjunction of two assertions, and there is more than one negation of such a compound postulate with multiple claims. Either claim, the truck is of a single color or the truck is yellow, can be negated resulting in three possible negations. Negation 1 The truck is of a single color and it is not yellow Negation 2 The truck is not of a single color and it is yellow Negation 3 The truck is not of a single color and it is not yellow Some negations, like negation 2, need not be consistent. If the truck is not of a single color, then just specifying that it is yellow is not a sufficient description. However, negations 1 and 3 are valid. Which negation does the proof by contradiction argument establish? Negation in Logic A conjunction (AND) of logical propositions p q is true only if both propositions are true. A disjunction (OR) of logical propositions p q is true if at least one proposition is true. Negation is an operation that takes a logical proposition p to another proposition "not p", written ¬p, which is true when p is false and false when p is true. The negation of a conjunction of two propositions p and q is defined as ¬(p q) ¬p ¬q The right hand side evaluates to true if: (1) ¬p is true and ¬q is false; or (2) ¬p is false and ¬q is true; or (3) ¬p is true and ¬q is true. In general, a proposition with n simultaneous claims has 2n – 1 negations. 5 The Cantor Diagonal Argument (CDA) How many logical claims are there in more complicated propositions like the one in the CDA? The CDA makes an explicit assumption that the unit interval is countable, creates a contradiction, and concludes that the unit interval must, therefore, be uncountable. It is normal to expect the explicit assumption caused the violation. However, some mathematicians have asserted that this need not be the case - that the CDA is a contradiction of the concept of actual infinity itself and is denying an implicit assumption instead of the explicit supposition. The CDA is based on many prior assumptions: the concept of actual infinity, the axiom of infinity, the notion of one-to-one correspondence between infinite sets, the Field axioms, etc. The contradiction could easily be indicating that one or more of these concepts is flawed. A proof-by-contradiction argument, if used improperly, can arrive at a wrong conclusion. In Appendix C the CDA itself is used to show that the CDA is ill-defined and does not actually examine numbers to infinite digits. Lots of underlying assumptions are, in fact, wrong! The Two Notions of Infinity and the Axiom of Infinity Aristotle’s abstract notion of a potential infinity is “something” without a bound and larger than any known number. Excerpts from the Wikipedia page “Brouwer-Hilbert controversy” Cantor extended the intuitive notion of "the infinite" – one foot placed after the other in a never-ending march toward the horizon – to the notion of "a completed infinite" – the arrival "all the way, way out there" in one fell swoop, and he symbolized this [as] … ℵ0 (aleph-null).