No I don't consider '2+1' to be anything really. It's an incomplete mathematical statement I guess.

We have named various entities 1,2 and 3. Each are members of various sets/fields/rings/groups and we have learnt various properties about them. Considering them as members of the 'usual' fields/groups we might consider (which have an associated defined operation called '+') it turns out that 1+2 usually equals 3 (although not always as in the group consisting of the set {0,1,2} with the operation of addition modulo 3.)

We don't have to do anything in particular, nor do we have to even specifically define a number to know some of its properties. We know that any even perfect number ends in ....28 or ....6. That is (probably) a fact about some numbers so incomprehensibly big that we'll never explicitly reference them. Nonetheless, we've learnt something about them. Our knowledge is about a set of numbers, just as knowledge of infinity is best understood as a kind of 'global knowledge' about a set, not as the endpoint of some construction we have undertaken.

Someone tell me what they think intuition is.

In so far as intuition comes out of the blue then scientist's intuition= religious revelation. yes, no?

The main difference between the traditional Thomistic version of the CA and the KCA version is how they treat time. All cosmological arguments rely on some premise about a first cause. In KCA, this first cause is understood temporally--as at the beginning in a temporal sense of a causal chain of events (specifically, the cause of the beginning of the universe).

In the traditional Thomistic cosmological argument, the first cause is not just a temporal cause (and so this version doesn't imply that the universe had a beginning), but rather a "sustaining cause." Here the first cause is understood as a necessary cause, an unmoved mover, something that acts without being acted on, but necessarily so. The main point about this kind of first cause is that it is what sustains the universe (including time as part of the universe) in existence. As such, it is not itself in time or at the beginning of a time, and so might have created/caused a universe with a beginning point of time (a beginning to the temporal causal chain) or universe without a beginning point.

No. In philosophy "intuition" usually refers to an idea based on rational insight. This is distinguished from the sense it also has in ordinary language of getting an idea/solution from out of the blue or as a kind of inspiration (this ordinary language sense is how scientists quoted by Notready are using the term).

This ordinary language sense of "intuition" is not like religious revelation because no responsible scientist regards this second kind of intuition as being in any way a reason to believe an idea, whereas religious people do think that revelation provides a reason to believe an idea. We can see this by contrasting two sentences: "I have a real intuition that p, but p is actually false," and "I have a real revelation from God that p, but p is actually false." The first sentence is very ordinary and common, the second is accepted by almost no one.

I'm actually more familiar with Craig's discussion of the KCA in his academic work--that was what I was referring to.

This is a misuse of the excluded middle. Assuming that "An actual infinite is impossible" is meaningful, then it is true or false. That doesn't mean that I must have an intuition that it is true or false. I might have no intuitions about this at all. Or, I might regard (as I in fact do), my intuitions about actual or potential infinities as basically worthless.

This is also incorrect. We can recognize that in some areas of life going with our intuitions is useful whereas in others it is not. So, our intuition can be useful in understanding the behavior of other people. That doesn't mean it is useful in developing ideas about the beginning of the universe or the properties of the infinite.

Also, since the properties of the infinite are extremely complex and difficult to understand, I doubt that people who don't have at least a Ph.D's level of understanding are able to even understand, let alone evaluate the intuitions about infinity that Craig is relying on here. This is why it is completely ridiculous from an intellectual point of view for Craig to present this argument in a popular debate. The percentage of the audience that can even understand his argument is minuscule.

Yeah, in my opinion this is basically a dodge on Craig's part. The problem with thinking about the infinite is that there is basically no view (including Craig's) that doesn't require us to accept some very implausible conclusions. Craig wants us to reject certain parts of the modern understanding of infinity in math, even though that understanding has been extremely fruitful. I don't think this is crazy--when biting bullets there is some level of choice on which ones you choose to eat--but Craig doesn't really give us any reason to prefer his way other than pointing out the weirdness of actual infinites. And since he doesn't really address the reasons to prefer the Cantorian understanding of the infinite, this isn't really a very fair-minded appraisal (incidentally, this is a big part of why Craig's appealing to intuitions in his popular presentation on this point is so worthless--almost no one in his audience understands modern set theory and so don't understand what he is asking them to give up in accepting an intuition about the impossibility of actual infinites).

I didn't make a statement. I asked a question. What Craig's argument does is force you to say you have no intuition about the character of Hilbert's Hotel. If you don't, time to move on I guess.

We haven't precisely defined intuition. Your "acceptable use" (to judge other people) is one form. But most people base enormous decisions at least in part on intuition. We have to. We don't have the time, data or intelligence to make 100% certain decisions. And I think you are enormously condescending to people who have a college level education, which is the main audience for Craig's popular work. Nor does he intend for his debates to be full explication of all the ideas he mentions. If someone is interested, most of Craig's popular work is a starting point, not a final resting place.

I challenge you to provide any evidence from Craig that he even makes a hint of a suggestion that anyone should reject any established mathematical theory, anywhere. From Reasonable Faith, p. 117:

Seems a negative for a claim rather than a positive because most of our intuitions about the real world have proven to be "not even wrong".

What an intuitive concept.

This misses the point of what I'm saying. My criticism here is that the whole exercise of identifying our intuitions about the character of Hilbert's Hotel is pointless. I'm not claiming that we don't have them. What I'm saying is that our intuitions about the character of Hilbert's Hotel have almost no evidentiary value because our intuitions about the infinite have been shown to not be very accurate. In other words, I can tell Craig that I have the exact same intuition that he has, and then say that I reject his conclusion because I think my intuitions about the infinite doesn't tell me anything useful about the properties of the infinite or its relation to reality.

So? I'm not interested in what people have time to do. I'm interested in whether the KCA is a successful. It is no kind of defence to say: I don't have sufficient time to correctly decide whether the intuitions that underlie the KCA are correct. If you don't have this time, then don't use this argument.

I find this sort of intellectual populism tiring. Here's what I said: "Also, since the properties of the infinite are extremely complex and difficult to understand, I doubt that people who don't have at least a Ph.D's level of understanding are able to even understand, let alone evaluate the intuitions about infinity that Craig is relying on here."

Do you think that the properties of the infinite are not extremely complex and difficult to understand? Or do you think that most people have closely studied the properties of the infinite? Because otherwise my claim stands, and if your complaint is with God, not me.

Again, I'm talking about the academic discussion of the KCA, not his popular debates.

Most obviously, if you accept Craig's interpretation of the transfinite ordinals and cardinals, then you will probably have to reject classical mathematics in favor of intuitionism or some other non-classical variant. If you accept classical mathematics, then there is no reason to suppose that actual infinites couldn't exist and that if they did they would not in fact have the properties that Craig finds so counter-intuitive. Obviously under classical mathematics it is not conceptually impossible--so why couldn't they be physically (i.e. temporally) possible?

How have our intuitions about the actual infinite been shown to not be accurate since no one even knows if an AI is even possible?


I said that what Craig does at the popular level is try to reach people with a certain message - it's insane to say that he has to give a full, academic explanation of a concept or he's somehow being dishonest. If the KCA is valid then there's nothing wrong with presenting a truncated version. Otherwise, you might as well not try to communicate anything unless you not only understand all linguistic philosophy but have solved all the problems of the universe. That Craig doesn't present a 4 volume discourse on the properties of the infinite isn't an argument that what he does present isn't valid.


I have no idea if this is correct or not. I don't lie awake at night trying to figure out if I'm an intuitionist or a classicist re mathematics. From what I can see, there is an ongoing controversy in higher math concerning these topics. If they don't know, how would I? I also reject the notion that I need a Ph.D. in math to understand the theistic arguments.

Special relativity and quantum mechanics have shown us that some of our basic intuitions about the nature of time, space, and causation are false. Modern set theory and its application to the infinite have shown us that some of our intuitions about the nature of the infinite are false. Thus, we have good reason to believe that our intuition about a question to which we don't otherwise have a clear answer (whether an actual infinite is possible) is not particularly reliable as some of our intuitions about related ideas have turned out to be false.
You keep appealing to the fact that Craig isn't just a popularizer--that he's written serious philosophical essays on this subject. I'm addressing those serious essays. I'm not interested in his popular debates except as sociology. My criticism of his appeal to intuition is when he does this in his serious philosophical work, not in his popular debates, so I'm not talking about the "truncated" version.
I find your attitude here baffling. If you don't care to understand the kalam argument, then why do you bother defending it? It seems to me that you are dangerously close to sophistry here.

And your rejection of the need for advanced mathematical knowledge is just wishful thinking. Modern-day philosophy is a technical field that often requires detailed knowledge of math, econ, history, physics, etc. The kalam argument relies on a premise that utilizes advanced mathematical concepts. You can't simply decide by fiat that it is unnecessary to understand these concepts as a condition for understanding or evaluating the argument.

This is my perspective on the issue:

1. There are mathematicians who deny the existence and perhaps the possibility of an actual infinite, not only in nature but in theoretical math.
2. An actual infinite collection of things, whether objects or past events, has never made sense to me, even before I heard of Craig or even before I was a Christian - since I can remember, in fact.
3. No one has shown that it even makes sense to talk about an actual infinite number of things, or even defined what that could be.
4. So the KCA makes sense to me. As Craig said, maybe I'm wrong. If the KCA seems flawed to you because you think there can be an actual infinite number of things, so be it.

An actual infinite number of things is a clearly delineated set (I) of real objects such that, no matter what positive integer x you chose, you could always find a proper subset (S) with x objects in it and such that the set I/S would be able to placed in one-to-one correspondence with I.

Speak English.

Obviously this is not a reason to think that the actual infinite is impossible.

That's fine--it is a difficult idea. However, have you studied math deeply enough to decide whether this not making sense is because you simply don't understand the math or because it actually doesn't make sense? If it is the former, then this is not a reason to think that the actual infinite is impossible.

As noted by bunny, people have certainly defined the infinite. Is your claim here that none of these definitions are successful? I'll also note that the distinction between an actual and potential infinite was originally developed by Aristotle as an attempt to solve paradoxes that no longer exist in modern mathematics.

The KCA makes sense to me as well. I just don't see how it functions as a reason to believe that there is a First Cause for you.

Let me explain this in more detail. Here's a brief summary of what it seems to be is going on. One of the premises of the KCA is that an actual infinite is impossible. One reason we might accept this premise is because of expert testimony by mathematicians, physicists, or philosophers. However, there is no consensus among experts that this is so (if anything, I suspect the opposite view is the majority position). So we can't support this premise on the basis of authority.

We might also support this premise by direct mathematical or philosophical argumentation--e.g. showing a contradiction in the idea of an actual infinite or as an implication of other mathematical or philosophical ideas. However, you are not doing this because this kind of argumentation requires relatively deep background in the respective fields. So this also isn't a support for your accepting this premise.

Finally, you might accept the premise because of its intuitive appeal. However, as I've already pointed out, the purported intuitive support for this premise also provides us with little reason to accept the premise because of the unreliability of our (unschooled) intuitions about the infinite.

In summary, unless there is something else that I've missed, it seems to me that you have no reason to accept this premise. Now, you can still choose to accept the premise even in the absence of a reason to do so. However, in that case any argument predicated on this premise doesn't provide you with a reason to accept the conclusion. Thus, the KCA doesn't provide you with a reason to believe that there is a First Cause, or a God. Thus, it is a failure as an argument.

Define an actual, physically existing collection of real objects that is infinite.

From Craig's formal argument:

2.The universe began to exist.
2.1 Argument based on the impossibility of an actual infinite:
2.11 An actual infinite cannot exist.
2.12 An infinite temporal regress of events is an actual infinite.
2.13 Therefore, an infinite temporal regress of events cannot exist.
2.2 Argument based on the impossibility of the formation of an actual infinite by successive addition:
2.21 A collection formed by successive addition cannot be actually infinite.
2.22 The temporal series of past events is a collection formed by successive addition.
2.23 Therefore, the temporal series of past events cannot be actually infinite.
2.3 Confirmation based on the expansion of the universe.
2.4 Confirmation based on the thermodynamic properties of the universe.

(2.1) is one of four supporting arguments for the second premise of the KCA; it's not one of the KCA's premises. In other words, even if (2.13) is false, if (2.3) is true, then the second premise is supported. In other, other words, even if an actual infinite is possible that is not to say that the universe is an actual infinite and the universe did not begin to exist.

We define an infinite series in terms of sets. A finite set is easy enough to understand. For instance, the set of numbers between 1 and 5 would be {2,3,4}. One property of this set is that it has three elements. We call this property (the number of elements it has) its cardinality. This set has a cardinality of 3.

We can use this notion of cardinality to distinguish infinite from finite sets. It is a feature of all finite sets that any proper subset has a different cardinality from the set as a whole, i.e. the subset of numbers between 1 and 4 {2,3} has a cardinality of 2 (there are only two elements in this set). However, infinite sets have the feature that a proper subset has the same cardinality as the set as a whole. For instance, the set of all positive integers {1,2,3,...} is an infinite set. However, the subset of all positive odd integers {1,3,5,...} has the same cardinality (same number of elements) as the set of all positive integers. So we can use this as a definition for an infinite set--any set with a proper subset with the same cardinality as the set is an infinite set.*

Now, let's suppose that there are an infinite amount of stars. This would mean that if you took every single star and numbered it, then the amount stars with odd numbers would be as large as the amount of all the stars. If reality has this feature, then there are an infinite amount of stars. If it doesn't, then there are not an infinite amount of stars.

*Bunny, please correct me if I'm wrong here.

Which part didn't you understand?

An 'actual infinite' is a collection of real objects.

That collection has the following properties:

1 There is a rule which tells us whether any particular object is in the collection or not
2 The collection can be subdivided
3 No matter how large a subdivision you exclude from the original set, what remains can be paired off with the original set

I looked up Gamow's account of Hilbert which is what Craig uses, almost verbatim. At the beginning of the paragraph Gamow says "A part can be equal to the whole".

To me, if you literally accept this, you are saying the same thing as "A = non-A". Maybe that's true. I reject it.

By the way, in case by this you mean for me to provide an example of an actual infinite. I'm not claiming that one exists, I don't know if one does or not. What I am saying is that I don't think an actual infinite is impossible.

What I am defining above (though off the top of my head, so no doubt the definition isn't perfect. I'm trying to speak english after all) is what an actual infinite would be like - it doesn't matter if there aren't any (like a black, female president of the US during the twentieth century is well defined even though non-existent).

What properties do you think the infinite has? If you reject this then how can you apply the reasoning inherent in Hilbert's Hotel? The 'paradox' arises from accepting this feature of infinite sets.

The relevant premise is (2). Craig presents basically two arguments for this premise: (a) his arguments based on the properties of the infinite and (b) scientific arguments related to the Big Bang. I've been focusing on his arguments related to the properties of the infinite because NotReady said that he views the philosophical/logical arguments to be much stronger than the scientific ones. However, as you say, if the scientific arguments lead to the conclusion that the universe began to exist, then even if the logical/philosophical arguments are unsuccessful, (2) might still be justified and the argument as a whole be successful.

However, note that I've not been arguing that the universe did not begin to exist or that the actual infinite is possible. Rather, I'm pointing out that the reason to think these claims are false endorsed by NotReady is not actually such a reason.

If I understand it you are saying the same as OrP. So you have an infinite set, subdivide out an infinite set, and have remaining an infinite set equal to the original, undivided infinite set.

If that's the case then I don't see how that cures the problem of an actual infinite.

BTW, see this from Wiki:

They are using "actual" to mean something other than physical - is that the problem we're having? Because Hilbert said the infinite is not to be found in nature, which is what I'm talking about.

I haven't had time to read the "actual infinity" conversation carefully, but I do want to say that it might be worth distinguishing "actual infinity" with "a verification of actual infinity."

If an actual infinity exists in the universe, it seems unlikely that scientific methods would be sufficient to determine that this is the case. Essentially, we would have to know of a way to account for an infinite amount of information in a finite amount of time. This cannot be done with turing-like devices, and I'm not sure if even theoretically possible with the invocation of quantum computational methods.

In other words, I can imagine that the universe has infinitely many stars, but I cannot imagine that we would ever "count" infinitely many stars.