Are Some Infinities Bigger than Other Infinities? (Part II)

I believe the idea that there are different sizes of infinite sets is based on a flawed reasoning. This also applies to other downstream problems that are based on this assumption, like the continuum hypotheses.

The question that raises all this is this: “why can’t we find a bijection between the reals and the naturals?” And the standard answer is “because they are not equal in size”. Because for two sets to have a bijection, they must be equal in size. But I think claiming there is a size difference between two sets which have by definition the same infinite size is absurd. There are others that share my view — the finitists (not to be confused with strict finitists). These are people who do believe that there is such a thing as an infinite set, but that all infinite sets are the same size. One finitist argument that explained the lack of one-to-one correspondence between the reals and naturals is given below:

If a set S has a well-ordering [i], and if S is also the same size as the set of natural numbers [ii], then this implicitly generates a one-to-one correspondence between S and the natural numbers [iii]. The “first element” of S corresponds to 1, the “first element” of the remainder corresponds to 2, the “first element” after that corresponds to 3, and so on for all of the rest. Therefore, the conclusion of the diagonal argument can be recast as follows: “There is no possible counting scheme that, even with infinite time, can list the reals in such a way that a new item that is guaranteed to not be on the list can’t be generated. Hence, there exist some infinite sets for which one of the following must be true: either such a set has no well-ordering, or it is larger than the set of natural numbers.” And it is only a matter of preference to go with either one of the explanations. So, it is perfectly valid to explain the lack of one-to-one correspondence between the reals and the natural numbers, by saying that this is due to the fact that the reals don’t have a well-ordering, not because they are larger in cardinality.

(Source)

The issue I have with this explanation is that the well-ordering property is neither a sufficient condition (two sets can have [i] but not have [iii], for e.g., {1,2,3} & {1,2,3,4}) nor a necessary condition (two sets automatically have [iii] if they have [ii], regardless of whether they have [i] or not). And if it is not a necessary condition, i.e., its existence doesn’t guarantee [iii] nor does its nonexistence explain the lack of [iii], and you don’t believe in [ii], then you are still left with the burden of explaining the lack of one-to-one correspondence between the reals and the naturals even though they are equal in size. I will now try to take on this burden.

I believe the core mistake in all this is confusing epistemic limitations for ontological truths. First, let me outline the most popular argument for the ∣R∣>∣N∣ theorem, Cantor’s Diagonalization Argument (CDA), as clearly as I can.

Imagine an exhaustive enumeration of the reals between [1, 2]
1.0110….
1.0111….
…ad infinitum

Imagine we take the diagonal of this list and toggle the values (toggle the 1st value for the 1st row, the 2nd value for the second row, …). If you look at this newly constructed value, you’ll find that it is different from every item in the list at least in one decimal point. So, by constructing this new number, we have shown that a complete enumeration of the reals isn’t possible. While we can find a one-to-one correspondence between any two countably infinite sets, we can’t find such correspondence for the reals because we have just showed that they are uncountable.

I believe there are several mistakes in this argument, but I will get back to this later.

You will find most explanations talking about how there exists no bijection function or one-to-one correspondence between the naturals and the reals and how this implies that the latter is a bigger set than the former. They use different proofs, mainly CDA, to show that the set of reals is an uncountably infinite set. But the reals are a superset of the naturals, so it’s better to just focus on the part of the reals that gives the set its uncountable quality. The rationals and the algebraic irrationals are countable, and what gives the reals their uncountable property is the remaining set — the transcendentals. So, I’ll just focus on the comparison between the set of naturals and transcendentals.

For a bijection to exist between the naturals and the transcendentals, they must be equal in size. I contend that we can’t show this correspondence between the two sets not because one is larger than the other, but because we don’t know the size of the set of transcendentals. I define knowing the size of a set as:
A) knowing all its items
OR
B) knowing the algebra that relates all its items (even if you can’t list all the items)

So, do we know the size of the transcendentals? I don’t think so.

Imagine a set S, whose size you don’t know. Does a bijection exist between S and the naturals? well, you don’t know. I think the same is true for the transcendentals.

But doesn’t CDA show that a finite listing of the transcendentals isn’t possible, hence proving there are infinitely many of them? No, it doesn’t. Because any finite listing of the transcendentals will have a finite diagonal (even if can’t be a perfect diagonal since the list will be infinitely long in one dimension but finitely long in the other). So, by systematically altering the diagonal and creating a new number, you haven’t produced a new item that the list failed to include, because the set of transcendentals doesn’t include a number with a finite representation.

But what about if you define a sequence like π, π+1, π+2, …? since all items in this sequence are transcendental, haven’t you just shown that there are infinitely many of them? But the set of integers (and all other algebraic/non-transcendental numbers) is countable, so if you try to define a sequence by an arithmetic combination of some transcendental and an algebraic sequence, you are only creating a countable sequence, not an uncountable one.

This concludes my argument. But I will just say a bit more on why I don’t think CDA is particularly illuminating. I will do this by showing that the methods applied on the set of reals can be applied on the set of naturals and the same conclusions can be reached.

Argument 1.1: You might protest by saying that we can’t talk about the “largest” item in the imagined exhaustive list of naturals because such a list has no such number and because considering such a number will make the list finite. However, we can’t talk about the diagonal of the list of reals as an item either because at any snapshot in time we are talking about a finitely long diagonal of a finite list which tells us nothing about the numbers yet to be listed. But instead, if we are talking about an idealized diagonal which is an infinitely long diagonal of an exhaustively listed set (with no more items left unconsidered), then we must also be allowed to consider such an idealized last or largest item for the exhaustive list of the naturals as well.