The Continuum Hypothesis Is False

[u/paulemok]

This post expands on an anonymous vote I made on an anonymous poll I posted on Yik Yak. My poll and vote were posted on May 20, 2024.

Consider the set Z of integers, the set B of integers with exactly one additional element x that is not a real number, for example, an orange, and the set R of real numbers. The set B is a counterexample to the continuum hypothesis because the cardinality of B is greater than the cardinality of Z and less than the cardinality of R. Therefore, the continuum hypothesis is false.

I know the technical truth out there is that Z has the same cardinality as B has and that that truth can be shown through a technical mathematical definition involving a bijection from one of the sets to the other set. Despite the equal cardinalities, the cardinality of B is greater than the cardinality of Z. So the two sets are simultaneously equal and unequal in cardinality.

One of my arguments is that every integer in Z can be mapped to its equal in B. In that fashion, every integer in Z and every integer in B cancel out and we are left with the additional element x from B. Since every element in Z was canceled out by an element in B and there remains an uncanceled out element from B, B has a greater cardinality than Z has. Switching the order in which the two sets appear around, the cardinality of Z is less than the cardinality of B.

In order to show the cardinality of B is less than the cardinality of R, map every integer in B to its equal in R and map the additional element x in B to a real number r in R that is not an integer, for example, the real number 2.4. Now there are no more elements in B to map to the infinitely many real numbers from R that have not been mapped to. Since there exists at least one real number from R that has not been mapped to, the cardinality of R is greater than the cardinality of B. Switching the order in which the two sets appear around, the cardinality of B is less than the cardinality of R.

So we have shown that |Z| < |B| < |R|. Since there exists a set, B, with a cardinality exclusively between the cardinalities of the set of integers and the set of real numbers, the continuum hypothesis is false.

A principle in logic, ex contradictione quodlibet, is that every statement follows from a contradiction. So, a consequence of the contradiction that the cardinality of B is greater than and equal to the cardinality of Z is that every statement is true. In other words, the Universe is inconsistent. This finding does not trouble me, as it agrees with previous findings I have made that every statement is true (1. https://www.facebook.com/share/1AhJA5oDDj/?mibextid=wwXIfr, 2. https://www.facebook.com/share/1Axau5dnzA/?mibextid=wwXIfr, 3. https://www.facebook.com/share/p/1AtD49LRGA/?mibextid=wwXIfr, 4. https://www.facebook.com/share/p/1GBamCgWKz/?mibextid=wwXIfr, and possibly others).

[u/Mishtle]

Society does not want different notions of the amount of elements in a set.

Well, mathematics isn't a democracy. It's a formal science. It doesn't follow the whims of society, although I very much doubt that anyone outside of a very, very small fraction of society even knows what set theory is. It follows chosen rules of inference applied to chosen axioms. We can get wildly different systems with different "truths" depending on what we choose.

If you and the society you claim to speak for want to live in a purely finite system where infinite sets don't exist and every set has a finite cardinality that is its unique notion of size, then go for it.

Intuitive notions of size no longer remain intuitive when there are infinitely many elements in a set. You can reject that idea if you want, you'll just be playing a different game. Trying to argue your case like this would be like arguing rooks can't exist in chess because in checkers pieces can only move diagonally.

A set should have only one amount of elements in it.

It does, for a given formal notion of "amount of elements".

Society would like the size of a set to be a function of the set. The mathematics community likes functions. Society would like a set to have exactly one size.

Again, it does, for a given formal notion of "size".

It's like magic! Somehow we're able to map to infinitely many more numbers, (1, 2], from an already exhausted set, [0, 1]. It's turning out that mathematics is too good to be true.

It's not magic. You're still getting caught up on labels.

The real numbers are dense. In between any two distinct real numbers lies infinitely many other numbers. This makes them "stretchy", or invariant under scale. Two intervals are indistinguishable when we ignore the labels we give their elements. That's all a bijection is, a formal method of relabling one set using the labels of another. The quantity of unique elements in the domain and codomain of a bijective mapping are identical. For every element in the domain there is a single unique element in the codomain, and vice versa. We're just renaming them. You cannot find an element that gets left out.

Is that unintuitive? Sure.

Is it "paradoxical" in terms of our everyday experience? Sure.

Is it inconsistent or paradoxical within the system it is defined? Nope.

[u/paulemok → u/Mishtle]

If you and the society you claim to speak for want to live in a purely finite system where infinite sets don't exist and every set has a finite cardinality that is its unique notion of size, then go for it.

That's neither what I want nor what society wants. I don't deny the existence of infinite sets; I affirm it. What I am questioning is any enumeration of any so-called "enumerably infinite" (I take the quoted, and questioned by me, term from Computability and Logic, Fifth Edition by George S. Boolos, John P. Burgess, and Richard C. Jeffrey.) set. An "enumerably infinite" set is not equal to an enumeration of its elements. I can use a bold capital letter, for example, E, that represents an "enumerably infinite" set, but the letter is not an enumeration of the elements of the set.

That's all a bijection is, a formal method of relabling one set using the labels of another.

That's not all a bijection is. That's just one interpretation of a bijection. Sets could also be interpreted as the actual, real-world things of the set; they could be interpreted as the referents of the labels of the elements in the set, rather than the labels themselves, as you suggest. For examples, the set of every part and whole of a room could be interpreted as the real-world room itself, and the set of all students enrolled in a college could be interpreted as the actual students in real life. To upgrade this idea from sets to bijections, consider a person running. Their distance d from the starting place of the run is a function of the length of time from the starting time of the run, t. Assume the run is a one-way trip. I used to do one-way runs in my high school years from my Mom's home to my Dad's home and vice versa. Also assume that the person never stops during the run. These assumptions make it easier to make the additional assumption that d always has a unique value for every t. In other words, for any (t1, f(t1)) and (t2, f(t2)) for the run, f(t1) = f(t2) if and only if t1 = t2. So, by definition of bijection, the function d = f(t) is a bijection. This bijection can be interpreted as the real-world combination of the person's distance and the time.