Kenny

No, when I say that the elements cannot be put in a list, I mean any sort of list whatsoever, even an infinitely long one. Another way to say it is that there is no way to index the elements of such a totality in such a way that all of them are accounted for. One can index the set of all integers by assigning the value of each integer to its corresponding element. Even though the set of real numbers is not countable, the same sort of thing applies to them. As long was we define an appropriate indexing system it is possible to assign each element a location on the list (even if our particular indexing system is arbitrary). Such is not true, however, for the totality of all sets or the totality of all propositions.

Once again, I would like to remind everyone that I am not an expert in set theory, so it is possible that I am off base somewhere here. I am open to the possibility of correction.

God Bless,
Kenny

Kenny

P.S. -- Correction

It occurs to me that that this paragraph of my above post is poorly worded and thus leaves me open to attack. I should not have used the phrase “members of that totality” as the notion of “membership” involves the notion of grouping, which I have denied can be legitimately applied to certain types of totalities. What I should have said is something like: “there are more elements which can be said to constitute that totality than there can be members of any list.” Where the denial of the diagonal comes in, with respect to the notion of the totality of propositions, is the denial that the sub-totality of all true propositions is itself something which can be thought of as a “group” such that it can be regarded as an "subset" of the totality of all propositions. I affirm that it is meaningful to speak of all propositions and to speak of all true propositions, but I deny that it is appropriate to think of “all propositions” or “all true propositions” as “groups” to which notions such as “membership” apply.

[ February 17, 2002: Message edited by: Kenny ]
<further rewording performed>

[ February 17, 2002: Message edited by: Kenny ]</p>

jpbrooks

Please forgive my persistence on this matter, but my understanding in this area needs to be more complete if I am to follow your reasoning.

(BTW, I was almost totally confused throughout the duration of my course in basic set theory, and I'm still astounded over the fact that I was actually able to get a passing grade in it! Set theory is only now beginning to make some sense to me because I have had time away from class lectures to ponder some of the vast number of "insights" that were crammed into my head during my class sessions.)

What you seem to be suggesting is that there is a more general branch of mathematics (with its own set of axioms and definitions) that deals with "totalities", of which set theory is only a sub-category.
But I confess that this whole matter is extremely counter-intuitive to me.
So can you show how the procedure for determining when a "totality" possesses elements that cannot be "indexed" works? If not, how can you know that the "totalities" in question have elements that cannot be "indexed"?

[ February 17, 2002: Message edited by: jpbrooks ]</p>

jpbrooks

Kenny,

I have been thinking about the points that you have made above about "totalities", "indexing", etc., and perhaps what you are saying is that 'totalities" that cannot be "indexed" all share the same properties as the "set of all sets" (assuming, for the moment, that such a thing can exist). In other words, the "set of all sets" represents a whole category of "entities", that don't have a finite cardinality.
I realize that this is just the wild speculation of a novice in set theory, but if this is what you are suggesting, I follow you. I'm just not certain how the "totality" of all true propositions can be such a "set of all sets".

On second thought, I do see it!
Hm. This is quite interesting!

[ February 18, 2002: Message edited by: jpbrooks ]</p>

Kenny

I think you are approaching the basic idea, but I don’t think that’s the best way to word it. The “set of all sets” has no properties because it is incoherent. But, I do think that it is meaningful to think of “all sets” (i.e. the totality of sets) and that there are other types of totalities for which similar paradoxes arise for the set of all sets (such as the set of all propositions). For such totalities, there are more elements which can be said to constitute that totality then there are members of any sort of list of those elements.

There is no set of all sets, only a totality of all sets. Were there a set of all sets, this set would have cardinality and thus it would be possible to index all the elements belonging to that set. The totality of all sets is such that it is impossible to index all the elements which constitute it; therefore the totality of all sets has no cardinality (i.e. it is not a set). The same is true for the totality of all propositions.

God Bless,
Kenny

jpbrooks

So, isn't this the same thing as saying that these "totalities" have an infinite cardinality? If they had a finite cardinality, then they would be sets and their elements could, in principle, be placed into one-to-one correspondence with some "power set" of the (real) numbers.

I have to leave, but I'll be back later.

[ February 20, 2002: Message edited by: jpbrooks ]</p>

bd-from-kg

jpbrooks:

We're talking about sets far, far bigger than you seem to have any conception of. Just for starters, let I0 be the set of integers. then P(I0) (the power set - i.e. the set of all subsets) of I0 is bigger than I0. Call this I0(1). Then for each n, let I0(n) = P(I0(n-1)). Each of these sets is larger than the preceding one. Now let I1 be the union of all of the I0(n). I1 is larger than any of the I0(n). And I1(1) = P(I1) is lager than I1, etc. We can go on like this, defining I2, I3, etc, and eventually take the union of I0, I1, I2, ..., which is again larger than any of these. And we're just getting started.

Are you starting to get the picture? We're not talking about infinite sets as opposed to finite ones, and we certainly aren't saying that if it isn't finite it isn't a set.

If you're really interested in this stuff, try Rudy Rucker's <a href="http://www.amazon.com/exec/obidos/ASIN/0691001723/qid=1014254199/sr=1-6/ref=sr_1_6/104-7587674-0201533" target="_blank">Infinity and the Mind</a>. It's fun (as far as anything about this sort of thing can be) and (besides giving a pretty good introduction to infinite sets) discusses some interesting relationships between set theory and philosophy and theology.

Kenny:

Sorry for the delay in responding. I'll try to get a response posted tomorrow.

jpbrooks

bd-from-kg,

thanks for taking the time to respond.

Precisely!
My point is that for any given set (let's call it S), no matter how large S gets, it will always be possible to find some set in the sequence {I0, I1, I2, ...} whose elements can be placed into one-to-one correspondence with those of S. That is what I meant when I said that any set (infinite or not) has a "finite cardinality". I used that term for lack of a better one. I'm not yet familiar enough with the terminology of set theory to come up with the appropriate term to
describe every concept in set theory that I have in mind.

For each of the "totalities" that are being discussed in this thread, there is no set in the sequence {I0, I1, I2, ...} whose elements can be placed into one-to-one correspondence with the elements of the "totality". Thus the "totality" cannot be a set.

I understand this. See my point immediately above.

Thanks.
Yes, I have indeed read parts of that book (out of a copy that I borrowed from the library a couple of years ago), and found it very interesting, though at the time, a liitle "over my head". It's probably time for me to get my own copy.

bd-from-kg

Kenny,

Here at last is a reply at least to the first of you back-to-back posts. I’m working on a reply to the second and should have it ready some time tomorrow at the latest.

It appears that you have two main points in your discussion of Grim’s argument. the first is that the proposition that God is omniscient” can be reasonably understood in a way that does not involve quantification over all propositions, and the second is that I still haven’t resolved the questions raised by Plantinga designed to cast doubt on the notion that quantification over all propositions is always illegitimate. But I have to add a third point to discuss your use of the term “totality”, which is highly nonstandard. Of course you can use any word to mean whatever you like, but if you want to communicate successfully it’s best to stick with standard meanings.

Point 1: Redefining omniscience

But do you want statements like “P is true, if and only if, P is known by God” or “God is omniscient” to be tautologies? Remember, any tautology is empty of content. So the price of making “P is true, if and only if, P is known by God” a tautology is that it no longer says anything about God; it only says something about what you mean by “true”. Thus, if God believed that the Japanese attacked Pearl Harbor on Dec. 7, 1841, then according to this definition that would be “true” even though there was no Pearl Harbor in 1841, much less a Japanese attack on it.

The problem can perhaps be made clearer by looking at what happens if I try the same maneuver, but replace “God” with “bd-from-kg”. I say that “P is true” means “P is known by bd-from-kg”. Is there a problem with this? I suspect that you’d probably object that the fact that bd-from-kg believes P doesn’t mean that P really corresponds to reality – or in other words, that some of the things bd-from-kg believes do not correspond to reality. Well, let’s test this proposition. Does bd-from-kg believe it? No. So you’re wrong: everything that bd-from-kg believes does correspond to reality.

By this point the basic problem should be clear: this definition severs all connection (in the conceptual sense) between “truth” and correspondence to reality. It might still happen to be true that any proposition that is true corresponds to reality, but this is not true by definition.

Thus to “save” the proposition that God is omniscient from involving quantification over all propositions you have twisted the meaning of “true” beyond all recognition.

In fact, this raises another problem with this move. If you say that “P is true” means “P is known by God”, you are saying that anyone who does not believe in God cannot in principle know what “P is true” means. Moreover, you can’t really disagree with such a person, because when he says “P is true” and you say “P is false” the proposition that he is asserting is completely different from the one you are denying.

There’s one other problem (that I can think of). On this showing, (if we assume that “P” and “P is true” are logically equivalent) the proposition “God exists” means “God knows that God exists”. So if God doesn’t exist it is meaningless to assert that He does. Similarly, if God does exist it is self-contradictory to assert that He doesn’t. It looks as though you’re trying to slip the (patently fallacious) original version of the Ontological Argument through the back door.

All in all, this seems to me to be a completely unsatisfactory way to avoid having the proposition the God is omniscient involve quantification over all propositions.

Point 2: The unavoidability of quantifying over all propositions

Yes, and the latter statement implicitly involves quantification over all propositions. But note that not only does A imply B; it is equivalent to B. As I pointed out earlier, statements like “All propositions are true or false” and “Some propositions are true” are legitimate even though they involve quantification over all propositions because the propositions they express can be expressed in other ways that do not involve this kind of quantification. The same is true in your example. B involves quantification over all propositions, but it can be rephrased in the form of A, which doesn’t. So it expresses a meaningful proposition.

Point 3: What is a “totality”?

Well, certainly not if one defines “totality” the way you do. But on the other hand, it certainly is true if you define it the way I (and practically all logicians) do, because nothing counts as a “totality” unless it can be so viewed (or conceived). Intuitively, a “totality” means a “completed whole”, which in turns means that one can conceive (without self-contradiction) of its being “completed” in some sense – i.e., of constituting an entity, even if one that only exists in the imagination. (I know, I’m using neo-Platonic terminology shamelessly. But it’s almost impossible to discuss such things at an intuitive level without doing so. When we’re actually doing set theory [as opposed to philosophizing about it] we’re all neo-Platonists.) A “class”, on the other hand (as the term is commonly used by logicians – or at least as it was commonly used when I was in grad school many years ago) doesn’t necessarily have this property. Thus, there is a “class” of all sets, but not a “totality” of all sets. However, it’s not clear to me that “class” in this sense is really a useful concept. A class is defined by a property; anything with a given property is by definition a member of the corresponding class. Thus, if X is a set, X is a member of the “class” of sets. But this means that the concept of a “class” is basically equivalent to the concept of a “property”.

I’m not following you here: the reason Grim says this is precisely that “all propositions” do not constitute a totality. Probably not important though.

[ February 22, 2002: Message edited by: bd-from-kg ]</p>

jpbrooks

Hello again, bd.

It is not my intention to interfere with your exchange with Kenny. But your first point reminded me of a question about logical rules for which I have never received an adequate answer from anyone.

You wrote:

I think your point about grounding the concept of omniscience on a tautology is cogent, but I would like to pose the following question:

Isn't there a similar "problem" with logical implication itself?
That is, in the truth table for logical implication:

a | b | a-&gt;b
----------------------------
T | T | T
T | F | F
F | T | T
F | F | T

if the antecedent statement of the implication is false, then the whole implication is true no matter what the truth value of the consequent is.

So my point is, if this "problem" in the truth table for logical implication is analogous to the one in your quoted paragraph above, and is already accepted as an aspect of our system of logic, why must the "problem" that you cited be unacceptable?

But even if I am wrong here and it is unacceptable, the fact that there was no Pearl Harbor in 1841 would seem to argue against the claim that God believed that an attack occurred there on that date.

[ February 23, 2002: Message edited by: jpbrooks ]</p>