Cantor – Maverick Philosopher

World + God = God: A Mathematical Analogy

The Big Henry offers the following comment on my post, World + God = God?

"World + God = God" is (mathematically) analogous to "number + infinity = infinity", where "number" is finite. If God embodies all existence, then God is "existential infinity", and, therefore, no amount of existence can be added to or subtracted from God's totality.

The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction. Similarly, if God is presumed to be the embodiment of all existence, He does not comply with the rules of arithmetic addition or subtraction.

To supply an example that supports Big Henry's point, 8 + $\aleph_0$ = $\aleph_0$ . $\aleph_0$ (aleph-nought, aleph-zero, aleph-null) is the first transfinite cardinal. A cardinal number answers the How many? question. Thus the cardinal number of the set {Manny, Moe, Jack} is 3, and the cardinal number of {1, 3, 5, 7} is 4. Cardinality is a measure of a set's size. What about the infinite set of natural numbers {0, 1, 2, 3, 4 . . . n, n + 1, . . .}? How many? $\aleph_0$ . And as was known long before Georg Cantor, it is possible to have two infinite sets, call them E and N such that E is a proper subset of N, but both E and N have the same size or cardinality. Thus the evens are a proper subset of the naturals, but there are just as many of the former as there are of the latter, namely, $\aleph_0$ . How can this be? Well, EACH element of the evens can be put into 1-1 correspondence with an element of the naturals.

So far the analogy holds. But I think Big Henry has overlooked the transfinite ordinals. The first transfinite ordinal, denoted

, is the order type of the set of nonnegative integers. (See here.) You could think of

as the successor of the natural numbers. It is the first number following the entire infinite sequence of natural numbers. (Dauben, 97) The successor of

+ 1. These two numbers are therefore different. Here the analogy breaks down. God + Socrates = God.

+ 1 is not equal to

Moreover, it is not true to say that "The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction." This ignores the rules of transfinite cardinal arithmetic and those of transfinite ordinal arithmetic. Big Henry seems to be operating with a pre-Cantorian notion of infinity. Since Cantor we have an exact mathematics of infinity.

In any case, I rather doubt that mathematical infinity provides a good analogy for the divine infinity. God is not a set!

Infinity and Mathematics Education

A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie);
on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical. If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory. Cantor sought to achieve an exact mathematics of the actually infinite. But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's Elements, David Hilbert's Foundations of Geometry, Richard Dedekind's Continuity and Irrational Numbers, Cantor's Contributions to the Founding of the Theory of Transfinite Numbers, etc. Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc. Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of construction the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to supplant a textbook-driven approach, but that the latter ought to be supplemented by the foregoing. I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No!

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite. The countably infinite has nothing to do with the potentially infinite. I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity. In so doing they took a lot of the excitement and wonder out of it. So what did you learn? You learned how to solve problems and pass tests. But how much actual understanding did you come away with?

The Axiom of Infinity as Easy Way Out?

I posed the question, Can one prove that there are infinite sets? Researching this question, I consulted the text I studied when I took a course in set theory in a mathematics department quite a few years ago. The text is Karl Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel Dekker, 1978). On pp. 53-54 we read:

It is useful to formulate Theorem 2.4 a little differently. We call a set A inductive if (a) 0 is an element of A; (b) if x is an element of A, then S(x) is an element of A. [The successor of a set x is the set S(x) = x U {x}.]

In this terminology, Theorem 2. 4 is asserting that the set of natural numbers is inductive. There is only one difficulty with this reformulation: We have not yet proved that the set of all natural numbers exists. There is a good reason for it: It cannot be done, axioms adopted so far do not imply existence of infinite sets. Yet the possibility of collecting infinitely many objects into a single entity is the essence of set theory and the main reason for its usefulness in many branches of abstract mathematics. We, therefore, extend our axiomatic system by adding to it the following axiom.

The Axiom of Infinity. An inductive set exists.

Intuitively, the set of all natural numbers is such a set.

Therefore, if we turn to the mathematicians for help in answering our question, we get the following. There are infinite (inductive) sets because we simply posit their existence! Thus their existence is not proven, but simply assumed. Philosophically, this leaves something to be desired. For it is not self-evident that there should be any infinite sets. If there are infinite sets, then they are actually, not potentially, infinite. (The notion of a potentially infinite mathematical set is senseless.) And it is not self-evident that there are actual infinities.

I will be told that there is no necessity that an axiom be self-evident. True: axiomhood does not require self-evidence. But if an axiom is an arbitrary posit, then I am free to reject it. Being a cantankerous philosopher, however, I demand a bit more from a decent axiom. I suppose what I am hankering after is a compelling reason to accept the Axiom of Infinity.

A comparison with complex (imaginary) numbers occurs to me. They are strange animals. But however strange they are, there is a sort of argument for them in the fact that they 'work,' i.e. they find application in alternating current theory the implementation of which is in devices all around us. But can a similar argument be made for the denizens of Cantor's Paradise? I don't know, but I have my doubts. Nature is finite and so not countably infinite let alone uncountably infinite. But caveat lector: I am not a philosopher of mathematics; I merely play one in the blogosphere. What you read here are jottings in an online notebook. So read critically.

A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

A commenter in the 'Nothing' thread spoke of possible worlds as sets. What follows is a reposting from 1 March 2009 which opposes that notion.

…………….

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions. If this is right, then the actual world, which is of course one of the possible worlds, is the maximally consistent set of true propositions. But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.