On The Cardinality of Infinite Sets And Omniscience

[_keithb]

Introduction:

I have been thinking a lot lately about what it really means to be omniscient in the universe. Consider the following paradox:

If God is said to be omnipotent, then there exists a set T containing all of the truths which God knows. This set T may be infinite (and is, according to classical Mormon theology). Indeed, I am not arguing anything as to the nature of this set, or the size thereof, only that the set itself exists.

Now, let us take this set of knowledge, however large, and see if, by construction, we can define a set larger than the original set. I claim that such a set exists and can be constructed by taking the super set (the set of all possible subsets) of the original set T.

To see this, I first need to define a few terms so that the layperson will understand what I am talking about. First, let us define the cardinality of an infinite set. This goes into the problem of how to count up a set of things which is infinite. Usually, when one is counting something, one can simply count up the elements in a set, however large, and affix a number corresponding to the number of elements in that set.

However, when counting up elements in an infinite set, one has to take a different approach. One possible way to do this is by what is called a "seating" approach. A simple analogy is if one wants to compare the number of seats in a football stadium to the number of people in attendance. The simplest way to do this would be to have everyone in the stadium try to "sit down" in a seat. If there are seats left over, then there are more seats than there are fans. Similarly, if there are still people left standing, then there are more fans than there are seats. By logic, the number of seats in the stadium and the number of fans in attendance are the same if and only if, after everyone has been seated, there are no seats left empty and no fans left standing.

Consequently, the "seating" process can be thought of as a function mapping the number of fans onto the number of seats and the number of seats onto the number of fans. Hence, the numbers are the same if the functions are surjective and injective, meaning that, let s an element in the set S containing the set of all seats and f be an element in the set F containing the set of all fans. The cardinality of the two sets is said to be equivalent if there exists functions g and h such that g(s) -> F and h(f) -> S for all s in S and f in F.

This process can be extended to sets with arbitrary properties, even an infinite number of elements. For example, consider the set of the integers (i.e. . . . -3, -2, -1, 0, 1, 2, 3, . . . ) and the real numbers (all numbers on the real number line). Let i define an element of the integers I and let r define an element of the real numbers R. We can say that the cardinality of the two sets is again equivalent if there again exists functions g and h such that g(i) -> R and h(r) -> I for all i in I and r in R. Indeed, one of these functions must always exist since the cardinality of one set will always be larger, meaning that elements from the smaller set can always be mapped onto elements from the larger set.

Cantor, in his papers (Cantor 1874), (Cantor 1891), showed that while the function g(i) -> R exists (by construction), the function h(r) -> I does not exist. As a brief outline and adaptation of the proof from his paper, consider a mapping of the whole numbers (0, 1, 2, 3 . . . ) onto the real numbers using the following method:

Lets "match" the real numbers in the interval [0 1] to the whole numbers in the following way. We will pick a number in the interval [0 1] at random, assign the number 1 to it, pick a second random number, assign the number 2 to it, and so on until infinity. The matching would look something like this.

1 0.1494812023023478193 . . .
2 0.2430832753200899237. . .
3 0.3422193285721058239. . .
4 0.4326568223408235020. . .

etc.

Now, if we can construct one or more numbers that are not contained on that list, we will have proven the cardinality of the real numbers to be greater than that of the counting number, since the counting numbers are certainly all matched by such a system. Such a number can be constructed by taking a number that differs from the first number in the first decimal place, the second number in the second decimal place, the third number in the third decimal place, and so on. To show this:

1 0.1494812023023478193 . . .
2 0.3430832753200899237. . .
3 0.3422193285721058239. . .
4 0.7326568223408235020. . .

Now pick 0.2537 . . . as a number not matched on the list above. Hence, we have shown that the cardinality of the real numbers is greater than the cardinality of the counting numbers. Also necessary (but not shown) is a proof that the cardinality of the counting numbers is equal to that of the integers. This can be shown by considering a pairing between the two of the following description:

0 0
1 1
2 -1
3 2
4 -2
5 3
6 -3

and so on.

As it turns out, the real numbers form the super set (set of all possible subsets) of the integers. To see this more easily, we can construct these subsets systematically by rewriting the real numbers in binary notation and then exhausting all possible combinations of those binary numbers. I will not do this here, but it is an elementary exercise (and a more detailed explanation can be obtained in several places on the internet which talk about infinite set cardinality).

Also, as it turns out, the idea of creating a super set of a set is not constricted to the construction of the real numbers from the integers. Indeed, we can define the real numbers as another set and, through a constructive process similar to that of shown above, show that the super set of the reals also comprises a set with cardinality greater than that of the original real numbers. Again, the details of this proof are left to the reader.

Relationship to the omniscience of God:

Consider again the set F containing the set of all knowledge known to God. That such a set exists is, at least to my understanding, the condition for the existence of an omnipotent God. If such a set F exists, then we can define a set with a larger cardinality than the set F by constructing the super set of the original set F. Thus, the existence of this original set F is a logical contradiction, and hence impossible. Hence, the existence of an omnipotent God is impossible.

As a practical model of this, consider a universe, over which God presides, and the set of all facts about this universe given by the set U. If the universe is indeed infinite or certain quantities in the universe are non-discreet (such as time), then the set of knowledge about this universe would also be infinite. If on the other hand, the universe was finite and all quantities in the universe were discrete, then the set U would indeed also be finite (though large). However, this set could be expanded to include an infinite amount of knowledge in several ways, including an abstraction to include a logical system such as mathematics, which does contain an infinite set of knowledge.

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.

Conclusion: I have shown that omnipotence is impossible due to the mathematical logic behind the cardinality of infinite sets. If indeed we define a God that is intelligent, but not omnipotent, I would argue that this fails to satisfy the definition of God; but this is an idea better left for another post.

Sorry to be long winded here. 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. Finally, I tried to make the argument accessible to the layperson in mathematics as well. I am additionally sorry if the argument was difficult to follow.

Also, I am sure that someone out there has made an argument similar to this one before. I wasn't trying to steal their thunder; I just felt too lazy to do a proper literature survey.

Thanks, and tell me what you think.

Cantor, Georg (1874), "Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik 77: 258–262 .

Cantor, Georg (1891), "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der Deutschen Mathematiker-Vereinigung 1: 75–78 .

For more information, also check out http://en.wikipedia.org/wiki/Cantor%27s ... lity_proof

[_Gadianton]

Also, I am sure that someone out there has made an argument similar to this one before

You might be, actually, not sure you want the honor though.

I just felt too lazy to do a proper literature survey.

But this is important, you don't want to be one of these guys who invests a lot of time into understanding some deep, theoretical idea, and then go about wielding it dogmatically to a controversial hobby-horse application with no interest in generally accepted opinion on how to interpret the results.

There are plenty of theists who have used the paradoxical results of mathematical logic to argue that, contrary to your thesis, science (or scientism) fails because there are truths existent that can't be proven, defined, listed, etc. Not that I agree, but the suggestion is no less valid than yours.

[_keithb]

Also, I am sure that someone out there has made an argument similar to this one before

You might be, actually, not sure you want the honor though.

I just felt too lazy to do a proper literature survey.

But this is important, you don't want to be one of these guys who invests a lot of time into understanding some deep, theoretical idea, and then go about wielding it dogmatically to a controversial hobby-horse application with no interest in generally accepted opinion on how to interpret the results.

There are plenty of theists who have used the paradoxical results of mathematical logic to argue that, contrary to your thesis, science (or scientism) fails because there are truths existent that can't be proven, defined, listed, etc. Not that I agree, but the suggestion is no less valid than yours.

However, I would argue that, if there are truths that exist that can't be observed in a scientific sense, then they are not particularly useful and/or affect humanity in any meaningful way.

As an example of this, consider dark matter and dark energy. Indeed, about 95% of the known universe is made of these two things. However, these things are only of practical interest on length and time scales that are galactic in nature, which is the reason why they have eluded detection until recently. Similarly, there could be a giant spaghetti monster out there that is invisible and comprises an even larger portion of the universe -- the world may never know.

However, my blurb above isn't about that. It's about the fact that it is impossible for God to be omniscient -- to know everything -- because the term "everything" is ill defined. Give me a set that defines your "everything", no matter how large, and I will use it to make a set of everything bigger than the first.

[_Gadianton]

Keith, would you be concerned at all about your theory to learn the following straight from wiki on Cantor:

[quote="wiki]The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.[51] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[52] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God,[53] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world[/quote]

It's surprising to me that your observation escaped Cantor himself. I'm not saying your suggestion is terrible, it's intuitive and apparently caused controversy for Cantor by fellow believers. But is it really the knock-down argument you think it is?

My point was that the same results that interest you, from logic, interests Christians (or at least believers) for the opposite reasons.

For instance, Godel's similar insight regarding "undecidable propositions" was apparently in part driven by his belief that truths could be known that couldn't be proven. Even Bertrand Russel became depressed I believe I read once for ten years, after discovering his paradox; the "gotchas" in math definitely have implications for philosophy, potentially good and bad for both God and philosophy of science. But your articulate way of drawing out a labored explanation of the diagonal argument, which is well understood and accepted, and then offering it up to a fairly questionable end that you sort of tack on and admit you "haven't really looked into" might come across to some as a little odd, or arrogant, when then offered in total as a "proof" - with footnotes and everything.

I mean, I think you gave a good explanation and all, there are other skeptics I will not mention who screw up that part, but it's always to your advantage to poke around and see what others have done with or have thought about an idea like this that raises a pretty obvious point.

[_keithb]

Keith, would you be concerned at all about your theory to learn the following straight from wiki on Cantor:

[

It doesn't bother me in the slightest. It doesn't matter if the man thinks his discovery came from God, Jesus, a werewolf, or the talking pirate who lived on his shoulder. What does matter are his ideas and whether his proofs are correct or not.

The one thing that I think I will do is to give a bit more credit to previous authors who have had the same idea. After doing a bit of research, it appears that there have been others who have done work on this idea, and I need to do a brief survey and include this in a future post.

Also, on the idea of truths that can be "known and not proven", I would concede this fact. Even in mathematics, there are certain propositions which are provably undecidable (using axiomatic set theory), like the one of whether there exist infinite sets with a cardinality between aleph_0 and aleph_1. Even in the universe, there could be phenomena which occur on a time, length, or energy scale which make them undetectable by science. However, again, I would question whether these are indeed useful truths.

I guess one of the problems is that I didn't clearly define the scope of my post. My intention was not to attempt to disprove the existence of God or something silly like that. My intention was to move the idea of omniscience from the realm of "mysteries of Godliness" into the realm of "falsifiable through logical contradiction". I think that there is still just a bit of wiggle room up at the top of this, where we say the knowledge of God comprises a proper class instead of a set, but I think I can resolve this by saying that, because God is a well defined being in space and time (especially in Mormon theology), then the set of knowledge which God knows should also be well defined. If it is indeed well defined, then it would comprise a set, hence the Cantor's paradox.

Thanks for your replies, by the way.

[_MrStakhanovite]

http://plato.stanford.edu/entries/omniscience/#OmnCar

[_moksha]

If an infinite set were to fall down in a forest and no one heard it, that would really put the kibosh on omnipresence.

[_keithb]

http://plato.stanford.edu/entries/omniscience/#OmnCar

I am going to have to check out the articles, etc. quoted in that a bit more carefully. Thanks for the link.

[_brade]

http://plato.stanford.edu/entries/omniscience/#OmnCar

Patrick Grim is making a mess ;)

[Analytics]

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.