The Different Versions of Infinity and Cantorian Sets

by Max Andrews

When we think of infinity we usually think of the usual two categorical distinctions:  a potential infinite and an actual infinite.  A potential infinite suggests that infinity is only an idea or a concept but doesn’t actually exist in the Platonic sense or in the physical sense. In any set, one may always be added.  An actual infinite is the notion that there exists such a set, Platonic or physical, which is infinite.  A potential infinity may be symbolized by a lemniscate:  ∞.  An actual infinite can be depicted by the aleph-null or aleph-nought:  ℵ0 (The Hebrew letter aleph with a subscript zero).

First, let’s have a brief refresher on set theory. A set is any collection of things or numbers that belong to a well-defined category.  In a set notation, this would be written as {2, 3, 5, 7, 11} being the first five prime numbers, which is a finite set of things.  Let’s simply signify this set as S.  There is a proper subset (SS) of S.  There are members in S that are not in SS, but no member of SS that is not in S.  The set of first three primes in a proper SS of the above S is {2, 3, 5}.  A dense set is a set where there is always room for one more in between another two elements.  Where there is an infinite set is with a set of cardinality, or natural numbers, it’s simply called a power set or an infinite set.  A series is an ordered set of numbers.  A finite series has a finite fixed number of terms.  An infinite series has an infinite number of terms.  A series with m terms, or the sum of the firs m terms of an infinite series, can be written as Sm or ∑a.

Georg Cantor developed the idea of what real numbers “are” by use of notions that do not directly refer to geometry.  In 1874-75 Cantor developed a theory of different sizes of infinities (with the infinitude of natural numbers as the smallest infinity).  The first notion of these sets is a 1:1 correspondence where two sets have the same cardinality (same number of elements).  (Remember, a cardinal number is the ‘number’ of elements in some set or a simply being mathematical entities in Plato’s world.)  This way, no elements of either set fail to take part in the correspondence.  In infinite sets there is a novel feature introduced by Galileo Galilei in 1638 that suggests an infinite set has the same cardinality as some of its proper subsets (recall proper refers to other than the whole set).  Consider the case of the set ℕ of natural numbers:

ℕ = {0, 1, 2, 3, 4, 5, …}

Remove 0 and there’s the new set ℕ-0 and there’s still the same cardinality as ℕ because we can set up the 1:1 correspondence in which the element r in ℕ is made to correspond with the element r+1 in ℕ-0.  Also, consider the cardinality of set ℤ of all the integers is again of the same cardinality.

ℤ = {0, 1, -1, 2, -1, 3, -3, 4, -4, …}

This could be simplified to ℕ if we pair off the elements.  Again, the cardinality of ℕ and ℤ are the same.

Moving on to the axiom of choice.  The axion of choice states that if we have a set A, all of whose members are non-empty sets, then there exists a set B which contains exactly one element from each of the sets belonging to A.  Roger Penrose admits this is quite obvious but advises caution with this.  The trouble, according to Penrose, it that this axiom is a pure ‘existence’ assertion without any hint of a rule whereby the set B might be specified.  Consider the Banach-Tarski theorem, which says that the ordinary unit sphere in Euclidean three-space can be cut into five pieces with the propert that, simply by Euclidean motions (i.e. rotations), these pieces can be reassembled to make two complete unit spheres.

Continuing on past that detour… Any natural number is ≤ any infinite cardinal number (umm… usually smaller!).  Let’s suppose that b ≤ a, with a being infinite then the cardinality of the union A B is simply the greater of the two, namely, a, and the cardinality of the product A×B is also a.  Okay, admittedly, we aren’t near infinities yet; so, let’s keep going.

So far we can see that the number of rational numbers is the same as the number of natural numbers.  Enter Cantor stage right.  Let’s now use ℵ0 for the cardinality of the natural numbers ℕ which is the same as the cardinality of the integers in ℤ.  The infinite number in ℵ0 is the smallest of the infinite cardinals.  Now, what is the cardinality of ρ of the rational numbers? Choosing the lowest terms for each rational, the 1:1 correspondence between the set of rationals with a subset of the set ℕ×ℕ.  Thus, ρ is ≤ the cardinality of ℕ×ℕ. But the cardinality of ℕ×ℕ is equal to the cardinality of ℕ, namely ℵ0. Thus, ρ≤ℵ0.  But the integers are contained in the rationals, so ℵ0≤ρ.  Hence, ρ=ℵ0.

After Cantor clarified this aspect of set theory he went on to demonstrate that there are infinities larger than ℵ0 and the cardinality of a set of real numbers is such an infinity (to be discussed in a future post).

(The majority of information and content is taken from Roger Penrose’s The Road to Reality).

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7 Responses to “The Different Versions of Infinity and Cantorian Sets”

  1. Hi Max – Do you think this information has any implications for the effectiveness of Dr. Craig’s argument using Hilbert’s hotel?

  2. Taking college math and a masters course in quantitative business decisions doesn’t qualify me to enter this discussion; however, I am intrigued to hear what it is you believe. I don’t see that you’ve made any belief statements on this blog. Do you belive there is an actual chronological infinite set of numbers that exists? I personally think there are only potential chronological infinites. Wouldn’t it be more correct to say that “an actual infinite set that can be defined, discussed, and plugged into a useful math equations is symbolised by the aleph-null, but that its solution is alway translated in finite terms?

    • Mike. When I was researching this I was relearning a lot of it haha. I learned a bit of set theory least year in my metaphysics course when studying kalam. Hold tight and I’ll answer that on my next post tonight or tomorrow. I didn’t get to Cantor’s larger infinities. I’m makin you wait. Bwhahaha!

  3. I can’t conceive the idea of different sizes of infinities. To know that a set has a certain size (e.g., the first five prime numbers), I must know how many elements it has. Therefore, when I know that a set S has a certain size, this implies that S is finite – otherwise I couldn’t know its size! But how can it be that a set has a size and is infinite at the same time? This thesis is not logically inconsistent with the axioms of set theory? (I’m really curious about it).
    Nice post – txs

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