On The Cardinality of Infinite Sets And Omniscience

[_brade]

This brings back fond memories of that epic thread we had on infinite sets over at MAD.

[_LDSToronto]

An interesting argument, and I think there some fallacy at its core. It is tempting to use Cantor and Godel to disprove all sorts of things, but there is some rashness in extending such theories beyond their field. Cantor and Godel deal specifically in the mathematical realm. Take Godel, for example. His theorem states that every consistent theory has one at least one statement that can not be proven within that theory. However, Godel only works within the strict confines of arithmetic, so to extend the theory beyond analogy is incorrect.

The same can be said of Cantor, Heisenberg, Einstein, and other celebrated theorists: the scope of their theories are encompassed by the field upon which they operate. To broaden their reach to the humanities is tempting, but often very difficult and fruitless.

I could be way off base - heaven knows that a few of you are over-the-top skilled in these matters.

H.

[_Milesius]

The philosopher-logician Gary Mar (student of Alonzo Church) quashes this argument here.

Abstract. Recent attacks on God’s omniscience employ a metaphysical application of
Cantor’s theorem. Two variations of this atheistic “Cantorian” argument can be
distinguished. A quantificational form of the argument can be demonstrated to be
invalid employing a defensive strategy championed by Plantinga. Turning the tables on
an argument used to dismiss the failure of Cantor’s theorem within mathematical
systems such as Quine’s New Foundations, it can be shown that a set-theoretical form
of the argument is question-begging. Such atheistic “Cantorian” arguments are not only
philosophically untenable, but also historically uninformed since the resources for
answering them are contained within Cantor’s own writings about the infinite and its
relation to theology.

Also, see this paper by Selmer Bringsjord.

Excerpt:

If the power set axiom is the cause of our troubles, then, the Christian
may insist, start instead with "beefed-up" ZF-Power (cf. [9]) (or, for
that matter, VNGB, KPU,...). Such a move vitiates, with one blow,
the cottage industry that Grim has helped develop out of the
power set axiom.

KeithB is only a tick more respectable here than JAK trying to refute Gödel, the non-physicist who thinks he has upended the theory of relativity or the people whom a physical chemist once told me bombard him with schematics for perpetual motion machines.

[_The Nehor]

Let us then consider the set of all possible subsets of the universe. We could construct these by, for example, considering the universe with one more or one less electron, a slightly different gravitational constant, etc. We could then define the super set of universes from this original universe. In order to be omnipotent, God would have to also know the properties of these universes as well. However, from this new set of universes and subuniverses, we could additionally define a new set of universes, which is the set of all possible subsets of the new set. And so on. Hence, would could define an infinite cascade of universes and subsets of increasing cardinality, so that there would be no set which an omnipotent God could know to contain the knowledge of all those universes and subsets. Hence, omnipotence is impossible.

Here is where I see the flaw. You are extending mathematics into another sphere of knowledge. Numbers and universes are not interchangeable as sets and subsets.

You also assume that God has a set of knowledge. I don't think God's knowledge can be defined by declaring what He knows. It doesn't work that way.

[Guest]

The philosopher-logician Gary Mar (student of Alonzo Church) quashes this argument here.

Abstract. Recent attacks on God’s omniscience employ a metaphysical application of
Cantor’s theorem. Two variations of this atheistic “Cantorian” argument can be
distinguished. A quantificational form of the argument can be demonstrated to be
invalid employing a defensive strategy championed by Plantinga. Turning the tables on
an argument used to dismiss the failure of Cantor’s theorem within mathematical
systems such as Quine’s New Foundations, it can be shown that a set-theoretical form
of the argument is question-begging. Such atheistic “Cantorian” arguments are not only
philosophically untenable, but also historically uninformed since the resources for
answering them are contained within Cantor’s own writings about the infinite and its
relation to theology.

Also, see this paper by Selmer Bringsjord.

Excerpt:

If the power set axiom is the cause of our troubles, then, the Christian
may insist, start instead with "beefed-up" ZF-Power (cf. [9]) (or, for
that matter, VNGB, KPU,...). Such a move vitiates, with one blow,
the cottage industry that Grim has helped develop out of the
power set axiom.

KeithB is only a tick more respectable here than JAK trying to refute Gödel, the non-physicist who thinks he has upended the theory of relativity or the people whom a physical chemist once told me bombard him with schematics for perpetual motion machines.

Okay, so I read the paper that you sent to me, along with a few other papers. Here are my thoughts on this matter.

Basically, (Bringsjord 1989) makes two arguments against my OP. First, he challenges the use of the Zermelo–Fraenkel set theory formulation with the axiom of choice (AC). Second, he makes an argument that saying something is possible for God to know and that God knows something are equivalent. Let me address the second argument first because it is, in my opinion, the much weaker argument.

If you look at (6) on page 189 of (Bringsjord 1989), you get the crux of the second argument. To write this statement and (D) on page 188 in plain English, you have:

(D) S is defined as being omniscient if, for the set of all propositions P, saying that S knows P is equivalent to saying that it is possible for P to be known by S.

(6) It is not possible for God to know anything that can be derived from a human perspective (i.e. what it is like to be finite).

To me, this is just a silly argument. I would argue that God should be able to know the set of all knowledge, including things known to human beings, hypothetical situations, etc. To say anything else is almost a tautology where the author is saying, "Well, God is omniscient because he knows everything that it's possible from him to know." I could say the same thing about the mentally handicapped person living in a rest home, but this doesn't make him omniscient.

The first challenge is more interesting. I guess that I should have been more specific and said that, if the Zermelo–Fraenkel set theorems hold and we insist upon the axiom of choice (AC) (http://en.wikipedia.org/wiki/Axiom_of_choice), then my thesis holds, a point upon which both (Bringsjord 1989) and (Grim 1988) agree. I just assumed that one could assume these two facts, since this is the most common formulation of set theory and the one that I learned in school. However, I failed to mention that there is another formulation of set theory using the axiom of determinacy (AD) (http://en.wikipedia.org/wiki/Axiom_of_determinacy) instead. The "other' formulations of set theory alluded to in (Bringsjord 1989) are made basically using the ZF+AD formulation of set theory.

If you want to see a brief (and non-rigorous argument) why this negates the power-set argument from the OP, skip to the end of the article. The basic idea here is that, in this formulation (ZF+AD), the cardinality of the power set is equal to the cardinality of the original set, so there is no cascading hierarchy of infinities to deal with, each with a larger cardinality than the last infinity (and thus the set of all knowledge remains bounded in a set and there is no need for a proper class formulation of the set of all knowledge.

Besides certain intuitive problems with this formulation of set theory (like the disappearance of definite meaning in the '=' sign and problems that arise with the inductive process), I am still not sure that this is any more convincing of a case for God's omniscience than the ZF+AC formulation of set theory. One of the major problems is that this means numbers (and by extension the universe, if the properties of numbers are being projected onto the universe) have some "internal structure" that is irreducible to a single point. Indeed, Physics and the formulation of quantum mechanics suggest that this is a plausible phenomenon in the universe (for example the uncertainty principle in quantum mechanics), but it still bespeaks to a situation where God is an observer looking in on a phenomenon too complex for him to understand (because a number can't be known to an arbitrary accuracy.

For example, consider the question to God: "How many strings of 2's longer than 100 digits long are contained in the number pi?" Under the ZF+AD formulation of set theory, God shouldn't be able to answer this question, since a number can be known to an arbitrary accuracy (and thus it's impossible for even God to "look" at the number and "see" how many strings like this exist in that number), meaning that each section of the real number line has an infinite "internal complexity" that remains hidden from God.

So, in conclusion, it looks like God is more or less in trouble either way. Either he has a situation where he can determine numbers "exactly" (ZF+AC) and thus has a hierarchy of infinite sets to deal with or he can't determine each number "exactly" and has the infinite internal complexity of each number span to deal with in trying to be omnipotent.

It's tough to be God :(


*******************

Without involving a technical explanation of the two different formulations of set theory, one intuitive way to understand the difference is to look at the effect of a Dedekind cut (http://en.wikipedia.org/wiki/Dedekind_cut) on the real number line. By the formulation of the real numbers, the cut must occur on at least one point (i.e. there is no "gap" in the real numbers between which to cut -- the real numbers are infinitely dense).

The question becomes then a matter of whether a cut can occur at one point or must be on an interval with a non-zero Lebesgue Measure (http://en.wikipedia.org/wiki/Lebesgue_measure). In the ZF+AC formulation, it can occur on a single point. In the ZF+AD formulation, it can't (and must occur on an interval of size epsilon where epsilon is arbitrarily small but nonzero).

ZF+AD then leads to a situation where the real number line can be split into, at most, countably many bins using a series of Dedekind cuts. To see this, imagine cutting a section of the real number line using cuts of size epsilon in the interval [a, b]. This would be equivalent to a Cartesian product of intervals and these could be enumerated epsilon_1 -> epsilon_n.

This leads to the elimination of a power set with a different cardinality because, intuitively, the terms from the Cantor diagonal argument in the OP cannot be specified to an arbitrary accuracy, meaning that a seating between the power set and the real numbers is possible, which means that the power set will also have a finite cardinality. (Again, I am skipping rigorous proofs of these statements for the sake of clarity and so that I don't spend 5 hours writing this post).

However, as noted in (Grim 1988), there are some things that are disturbing about ZF+AD. One of them is that the = sign becomes less meaningful, since, at a certain level, the values on the real number line kind of "smoosh" together and it's impossible to tell whether two numbers are exactly equivalent (only to some arbitrarily high accuracy). Also, according to (Grim 1988), ZF+AD invalidates the idea of mathematical induction (although I am still awaiting the book -- on hold from the library -- to see exactly how that works).

*************************

Sources:

*Patrick Grim, 'Logic and the Limits of Truth', Notis 22 (1988) 341-67.

*Grim on Logic and Omniscience
Selmer Bringsjord
Analysis
Vol. 49, No. 4 (Oct., 1989), pp. 186-189
(article consists of 4 pages)
Published by: Oxford University Press on behalf of The Analysis Committee
Stable URL: http://www.jstor.org/stable/3328556


Also, for a primer on the mathematics involved:

http://digarchive.library.vcu.edu/dspac ... sis_AD.pdf


I am going to check some of my logical arguments on here. I think that some of my conclusions need work (although the arguments seem intuitively correct. However, I am sure that I am going to get ripped by the mathematicians on this thread if I'm not careful, lol.

[_MrStakhanovite]

Bump.

Tarski, get in this.

[_keithb]

Here is where I see the flaw. You are extending mathematics into another sphere of knowledge. Numbers and universes are not interchangeable as sets and subsets.

The whole premise of Physics is that mathematics IS transferable to knowledge in the real world. Check any elementary book on Physics for examples.

Also, even if the concepts of mathematics weren't transferable, it still comprises a system in which there is an emergent set of knowledge. If God is omnipotent, he should know and understand math, IMHO.

You also assume that God has a set of knowledge. I don't think God's knowledge can be defined by declaring what He knows. It doesn't work that way.

Um, huh? What do you mean that God's knowledge can't be defined by defining (declaring) what God knows? This statement doesn't really make sense, no offense.

Of course I assume that God has a set of knowledge. How else would you interpret omnipotence? A set is (in naïve set theory) just a collection of mathematical objects that is an object in it's own right, including statements of truth, etc.

[_keithb]

Also, sorry to the scientist and mathematicians in the group (Tarski) because I was not as careful with definitions, proofs, etc. as I could have been.

Your summary of Cantor's proof contains some errors. I would encourage interested readers to a Google search and maybe read the Wiki articles on transfinite numbers.


In general, these problems come from self-referential techniques whether it be Cantor, Godel, or Turing. These difficulties are why Mathematicians use categories instead of sets in some places. I don't see why we can't do the same for omniscience.

You may be interested in learning about Oracle machines and hypercomputers.

It probably does contain some errors, I admit. It's not my field of expertise, and I wasn't being very rigorous at all. I apologize again for this.

[_MrStakhanovite]

Keith,

Don't bother with Nehor, he's waaaaay out of his element here. Anything that can be systematically defined can be analyzed with math.

[_ludwigm]

Anything that can be systematically defined can be analyzed with math.

For example one simple system can be analyzed with this simple math:

14,000,001 - 14,000,002 - 14,000,003 - 14,000,004 - 14,000,005 - 14,000,006 - ...


What system am I talking about?