WL Craig's Kalam Cosmological Argument

[Brian Bosse]

Hello T.E.,

I love this topic!

But what about infinite sets? The natural numbers vs. the real numbers.

If this interests you, we should probably start another thread. But I will make some comments here.

You cannot order the real numbers by the natural numbers, as Cantor's diagonal proof shows.

To make this proof go through, one must assume an actual infinity.

But does that mean, in the case of infinite sets, that one has more members or is bigger? Or in the case of infinite sets is a difference of cardinality merely a matter of relative orderability and does not imply the informal concept of more?

Mathematicians are partly responsible for the confusion between the number of elements in a set and the set's cardinality. Even though they have carefully defined 'cardinality', when speaking informally they often speak like "Some infinities are larger than other infinities" without qualifications. You seem to make the same point when you say...

If you look at the standard texts: Kleene, Church, etc. the all assert without argument or even raising the question of equivalency that greater cardinality means "more", but I maintain that this is a unjustified metaphysical interpretation of set theory. Cantor just assumes this also, which I suppose is where the problem started.

I would agree with you that when strictly speaking greater cardinality does not correspond to more elements. Of course, this only matters when sets are infinite. Otherwise cardinality corresponds precisely to number of elements. It also hints at the problems inherent in trying to have an actual infinity encompass a potential infinity.

In this form it is question begging. You don't believe and actual infinite can exist in the real world, because you think that in the real would it is impossible.

It is not quite question begging. I have already provided an argument for why there cannot be an actual infinity earlier in this thread. I have a reason that is more than just pure conjecture or some crass assumption on my part.

We are talking about what philosophical theology says about God, and has said since ancient times. God is simple, unchanging etc.

I am open to the idea that philosophical conceptions of God may be incorrect. Our philosophy of God, so to speak, must be Biblically informed. However, I am fairly confident that one can overcome any objection rooted in actual infinity. In other words, I am not sure I agree with you that traditional formulations do assume an actual infinite. Maybe they do? I have been wrong before.

Craig has a section in at least one of this books (I remember reading two or three) where he talks about the simplicity of God and how that exempts God from the argument, and he has a discussion trying to show that time must be thought of as consisting of moments or events and is therefore composed.

If am not mistaken, Craig's view of God and time is not traditional. Me, on the other hand, hold to the more traditional view. Interesting stuff...much of which is beyond me.

Sincerely,

Brian

 

[sotzo]

Thanks to everyone thus far for posting. Let's see if we can put some summary statements together based on what you've said...this will help me with my original questions:

1. The Kalam argument is helpful as far as natural theology goes, but its conclusion requires only an intelligence and not specifically our God of the Bible.

2. The set theory Craig presupposes may not account fully for transfinite events, the latter being the issue at hand in the Kalam argument. This is the gap in the armor of the argument.

Do all agree to this summary of the dialogue thus far?

Brian, you said:

"Being that God created this level of existence there exists another level not accessible to us as creatures that necessarily is a level where God exists. We cannot comprehend this level."

When you say "a level not accessible to us" do you mean a part of reality that is perceptible but not explainable (i.e., Heisenberg's uncertainty principle)?

 

[tewilder]

Every part of the argument has been attacked by somebody. You can find a lot of these articles online.

-------------

Maybe I should explain how I got into this. I was in seminary with Craig. So when I saw he had books coming out I bought and read them. I actually wrote out a long paper going after his proof, on the grounds that he was just setting up his intuitions about infinity against other peoples intuitions, who had equal or better reasons to be taken seriously. Craig replied that this was a weak argument because it was an "appeal to authority". That his confidence in his own intutions was equally an appeal to authority did not register with him.

Shortly after I ended up on a search committee hiring two faculty for temporary lecturer jobs. We sorted through a huge box of applications and resumes and Philip Ehrlich ended up on the short list. I voted for him as not only was his resume impressive but I was interested in his topic. He was hired and I took his course on infinity.

Also I started reading some other stuff, read Cantor's book, checked on how the standard texts defined terms, etc. and came to the conclusions I have been stating in the previous posts.

 

[sotzo]

tewilder:

thanks for the explanation. You said:

"Ehrlich showed that if you don't used restrictive set theories such the Zermelo-Fraenkel set theory but instead something like Hilbert-Akerman you can construct a transfinite ordinal arithmetic where Aleph does not equal Aleph + 1 which does not equal Aleph + 2 and you can do normal operations of addition and subtraction. "

Doesn't Godel's theorem render Hilbert-Ackermann sets meaningless? My VERY limited understanding of this area is that Hilbert was not attempting a new theory of sets, but a way to justify them self-referentially. I may be completely misunderstanding however.

 

[Brian Bosse]

Hello Sotzo and TW,

1. The Kalam argument is helpful as far as natural theology goes, but its conclusion requires only an intelligence and not specifically our God of the Bible.

I agree.

2. The set theory Craig presupposes may not account fully for transfinite events, the latter being the issue at hand in the Kalam argument. This is the gap in the armor of the argument.

I disagree. The mathematics of the transfinite has nothing to do with reality. It is simply a formal construct. A formal game played by mathematicians if you will. On the other hand, I presented an argument as to why the universe cannot have already existed for an infinite period of time. Let me be real clear here: appealing to the mathematics of the transfinite is nothing but a smokescreen. The mathematics of the transfinite does not substantiate an actual infinity in the real world in any way.

When you say "a level not accessible to us" do you mean a part of reality that is perceptible but not explainable (i.e., Heisenberg's uncertainty principle)?

No. I am speaking about a part of reality that is independent of this creation. Prior to creation God existed. This state of existence is independent of creation since it is prior to creation. One can think of this priority as logical priority rather than temporal priority.

I actually wrote out a long paper going after his proof, on the grounds that he was just setting up his intuitions about infinity against other peoples intuitions, who had equal or better reasons to be taken seriously.

I would enjoy reading your paper. I tend to disagree that Cantor and others had different intuitions about the infinite. When you hear Cantor and others talk about the ideas of the transfinite, they always qualify these things as either sounding strange to the common person or going against intuition. In fact, they let their imaginations run when developing the tansfinite rather than relying on their intuition.

Again, the transfinite has no correspondence to reality. It is simply part of a metamathematical game played by logicians and mathematicians. I, personally, enjoy the game, but I do not confuse the game with the real world.

Sincerely,

Brian

Godel's Theorem does not play a part in this discussion other than his argument can be thought of as a type of diagonal argument. Cantor used a diagonal argument to prove that the cardinality of the natural numbers was smaller than that of the real numbers. This was the birth of the transfinite. Diagonal arguments are fascinating, and a favorite topic of mine.

 

[tewilder]

tewilder:

thanks for the explanation. You said:

"Ehrlich showed that if you don't used restrictive set theories such the Zermelo-Fraenkel set theory but instead something like Hilbert-Akerman you can construct a transfinite ordinal arithmetic where Aleph does not equal Aleph + 1 which does not equal Aleph + 2 and you can do normal operations of addition and subtraction. "

Doesn't Godel's theorem render Hilbert-Ackermann sets meaningless? My VERY limited understanding of this area is that Hilbert was not attempting a new theory of sets, but a way to justify them self-referentially. I may be completely misunderstanding however.

Ehrlich's view was that there was no one true set theory. You chose the one that suited what you needed to accomplish. He thought that Godel's Platonism was laughable. What makes the Hilbert-Ackerman type of set theory different is that it allows proper classes. These are classes that are not capable of being members of classes so they are not sets. A set theory that only allows classes that are sets is more restrictive.

Ehrlich had done a lot of work on set theory. He developed extremely large and inclusive classes. He even came up with extreme cases where you have added in so much that you couldn't add more without taking something out. Here we are talking about infinities that would make the Aleph0 (natural number cardinality), Aleph1 (real number cardinality) etc series so small but comparison it would be a "spit in a bucket" as Erlich put it.

I don't know the details, but he published these results in various math journals. At the time I took his class all his articles were still awaiting publication, although they had be accepted in journals.

Godel's result is that you can't have a complete and consistent axiomatic system for arithmetic. There is always more that can't be proved.

I think that Hilbert and Ackerman did work on the problem that Russell found of "the set of all sets that are not members of themselves". (I.e. is that set a member of itself?) There are a variety of solutions to this. Russell's ramified theory of types is one, H-A had a different one, I think, but I have not read about it.

 

[tewilder]

Hello Sotzo and TW,

I would enjoy reading your paper. I tend to disagree that Cantor and others had different intuitions about the infinite. When you hear Cantor and others talk about the ideas of the transfinite, they always qualify these things as either sounding strange to the common person or going against intuition. In fact, they let their imaginations run when developing the tansfinite rather than relying on their intuition.

Brian

I threw the paper away long ago. I was not thinking of Cantor as a case with intuitions opposed to Craig, but to people like Bernadote.

Here is another argument for actual infinity:

http://stripe.colorado.edu/~morristo/kalam-not.html

 

[sotzo]

So Brian, in your view an actual infinite is mathematically possible, but actually impossible?

And TE Wilder (does the TE refer to your office as Teaching Elder or are those your initials?), in your view and actual infinate is both mathematically and actually possible?

Do I understand you both correctly? I'm not playing moderator here...I am just trying to approach Craig's argument, alternating between both of your positions.

 

[Brian Bosse]

Hello Gentlemen,

Here is another argument for actual infinity:...

I quickly read the article. My analysis, because the skim, will not be sharp. The first argument dealt with a finite space divided into and infinite number of subspaces, which is not analagous to the situation where we are speaking of an infinite space. The second argument ignores the necessary situation that if the uiniverse has been around for an eternity, i.e. it had no temporal beginning, then there exists a time U(1) in the past that is infinitely away from today. This is just simply ignored. The argument I put forth dealt with this. Three is simply an argument against Craig's conception of God relative to time. Even I do not agree with Craig here. But this is not an argument for an actual infinity. The fourth argument simply states that the empirical evidence is not conclusive that the universe had a beginning. The fifth argument questions the whole nature of causation. Again, this is not a good argument for the actual infinite. I could go on, but in short the article does not impress me in the least. It does not make any real case for actual infinity.

So Brian, in your view an actual infinite is mathematically possible, but actually impossible.

I am not sure what it means to say that an actual infinite is mathematically possible. Set theorists assume that infinite sets are actual. Cantor's diagonal argument would be impossible without this assumpition. In fact, there is a whole school of mathematics that deny this assumption. I am not so militant against the assumption. However, I do not allow those making the assumption to ignore the potentiality of the infinite. Nor do I sit by when those who are informed or those not so informed speak as if the transfinite says something about the real world.

I have presented an informal argument against the actuality of infinity in terms of the universe. Maybe T.E. would like to explore my argument? I think it is a good argument.

Sincerely,

Brian

 

[sotzo]

Brian:

I've lost you here and, again, it may be my nascent familiarity with set theorythat is to blame.

You said:

"Set theorists assume that infinite sets are actual. Cantor's diagonal argument would be impossible without this assumpition. In fact, there is a whole school of mathematics that deny this assumption. I am not so militant against the assumption. However, I do not allow those making the assumption to ignore the potentiality of the infinite."

But wouldn't protagonists of set theory (those who assume that infinte sets are actual) be the same folks who do not ignore the potentiality of the infinte based on their assumptions?

But then you say:

"Nor do I sit by when those who are informed or those not so informed speak as if the transfinite says something about the real world."

This seems to be contrary to your desire to not allow some to ignore the potentiality of the infinte.

I guess this is the crux of the matter...either the transfinite is an attribute of the real world or it is not. If it is, then Craig's argument fails because there has never been a point in time when the world began to exist. If it is not, then Craig's argument may be sound. No?

 

[Brian Bosse]

Hello Joel,

Yes, I can see how my last post was confusing. Forgive me.

1. Cantor's diagonal argument requires the assumption that all sets (including infinite sets) are actual.
2. There is a school of mathematics that deny this. In fact, from Aristotle onward there have been people who denied actual infinity.
3. I am not so against the concept of taking a set, such as the set of natural numbers, and treating it is a complete whole.
4. However, I am completely against those who think transfinite mathematics cooresponds to the real world just because they have developed some formal system.

I guess this is the crux of the matter...either the transfinite is an attribute of the real world or it is not. If it is, then Craig's argument fails because there has never been a point in time when the world began to exist.

I have argued against the universe being eternal independently of the transfinite. In other words, my argument does not have anything to do with the transfinite. Those who appeal to the transfinite in an effort to say the an eternal universe could be possible may as well have appealed to the Telly Tubbies. Neither speak to the issue.

Again, sorry for the confusion. The KLM is sound, but it does not prove much on its own.

Brian

 

[sotzo]

Thanks Brian...I see your point and it helps me understand Craig's argument better, especially where it is deficient.

 

[Brian Bosse]

Hello Sotzo,

You are welcome. I am not sure, though, that T.E. agrees. I think he still holds that one is not able to dispell the possibility that the universe represents an actual infinity in terms of its age, and as such KLM is not proved sound. In this regard, he has not commented on the argument I provided.

Sincerely,

Brian

 

[tewilder]

Hello Sotzo,

You are welcome. I am not sure, though, that T.E. agrees. I think he still holds that one is not able to dispell the possibility that the universe represents an actual infinity in terms of its age, and as such KLM is not proved sound. In this regard, he has not commented on the argument I provided.

Sincerely,

Brian

I don't see why God can't create an actual infinite, and in addition to the space time continuum we are in, perhaps in another work he did.

 

[Brian Bosse]

Hello T.E.,

I don't see why God can't create an actual infinite, and in addition to the space time continuum we are in, perhaps in another work he did.

Earlier, I asked if God could create a square circle. Do you care to answer that question? Also, to be very precise here, the debate is not whether or not God can create an actual infinite, but whether the universe has been around for an infinite amount of time. Again, I have provided an argument for why it has not.

Sincerely,

Brian

 

[tewilder]

Hello T.E.,

I don't see why God can't create an actual infinite, and in addition to the space time continuum we are in, perhaps in another work he did.

Earlier, I asked if God could create a square circle. Do you care to answer that question? Also, to be very precise here, the debate is not whether or not God can create an actual infinite, but whether the universe has been around for an infinite amount of time. Again, I have provided an argument for why it has not.

Sincerely,

Brian

Your argument assumes a cardinality of Aleph0 and that every moment corresponds to a natural number. Therefore every moment is only a finite time away.

But suppose we don't limit the cadinality this way. Then take moment Aleph+6. That moment is a specific moment, and it is infinitely remote.

 

[Civbert]

Brian,
I think the following was your argument against an "actual" infinity:

Now, let's apply this to the universe. If the universe is infinite, then there exists a point u(1) that is an infinite time span away from today T(0). Given an infinite set of intervals from u(1) corresponding to the set of natural numbers we would get something like this...

N:                1    2    3  ...   n ... 
Time Interval:  u(1) u(2) u(3) ... u(n) ...

If we are here today, T(0), then there exists a time from u(1) such that it is a finite time to T(0). (If there is no interval of time from u(1) such that it is a finite distance of time to T(0), then we cannot be here. There is no interval of time to get us here.) Since we are here, then if the universe is infinite, there exists a time from u(1) such that it is a finite time to T(0). However, by the NAT there does not exist any 'n' such that u(n) is a finite interval of time from T(0). Therefore, the universe is not infinite.

This argument establishes the truth of premise 2. Premise 1 seems uncontroversial, and as such the argument can be declared sound. Q.E.D.

I think Premise 1 is controversial. "If we are here today, T(0), then there exists a time from u(1) such that it is a finite time to T(0)." This already presumes a finite universe. I think to make this claim, you must assume that there is alway a finite distance between u(1) any u(n). But if the distance from u(1) to u(n) is infinite, then the time to T(0) is also infinite.

One could have easily said: For every point with a distance of x from here, there is a point that has distance of x+1. This seems to be an uncontroversial statement.

It applies to the material universe. One can not define a point beyond which there is no further point, because there is always the point that is 1 measure beyond it.

I also have a problem with the meaning of "reality" being assumed.

I disagree. The mathematics of the transfinite has nothing to do with reality. It is simply a formal construct. A formal game played by mathematicians if you will. On the other hand, I presented an argument as to why the universe cannot have already existed for an infinite period of time. Let me be real clear here: appealing to the mathematics of the transfinite is nothing but a smokescreen. The mathematics of the transfinite does not substantiate an actual infinity in the real world in any way.

It appears that you are using the term "reality" to mean very specifically, the material world. In other words, you seem to imply that the set of natural numbers are"unreal" because you can not conceive of them have actual correspondence to the material universe. This seem to be a metaphysical commitment that leads to a whole slew of problems when speaking about the existence of God and spiritual things.

 

[Brian Bosse]

Hello T.E. and Anthony,

Your argument assumes a cardinality of Aleph0 and that every moment corresponds to a natural number. Therefore every moment is only a finite time away.

If the universe really is infinitely old, then there exists some point in the past, say u(1), that is an infinite interval from today. Here is the point, there is no successive amount of time from u(1) that is still not an infinite amount of time from today. If there ever is, then u(1) was not an infinite amount of time from today. Therefore, there cannot be a successive interval of time from u(1) to today. However, there must be if we are here. Since we are here, then u(1) is not in the infinite past.

But suppose we don't limit the cadinality this way. Then take moment Aleph+6. That moment is a specific moment, and it is infinitely remote.

Call this infinitely remote point u(1). There exist no successive interval of time from u(1) to today. But we are here. Therefore, there must be such an interval. Therefore, u(1) is not infinitely remote.

I think Premise 1 is controversial. "If we are here today, T(0), then there exists a time from u(1) such that it is a finite time to T(0)." This already presumes a finite universe. I think to make this claim, you must assume that there is alway a finite distance between u(1) any u(n). But if the distance from u(1) to u(n) is infinite, then the time to T(0) is also infinite.

Don’t you have a Scripturalism thread you should be attending to?

The key point is that we are here today. Mathematicians point out that if in fact you do have an infinite sequence such as a succession of time (I used the sequence of digits in the expansion of 1/7 as an example), then no matter what point you pick in the sequence there will always be an infinite number of successions to follow. If ever this not the case, then the sequence was not infinite. So, if there really was a point in the infinite past that the universe existed, say u(1), then there is no succession of sequences, say ‘n’, such that ‘n’ is not an infinite distance from today. But since we are here today, then there must be a succession of sequences, say ‘n’, such that ‘n’ is a finite distance from today. Again, if this is not the case, then we cannot be here today.

One could have easily said: For every point with a distance of x from here, there is a point that has distance of x+1. This seems to be an uncontroversial statement. It applies to the material universe. One can not define a point beyond which there is no further point, because there is always the point that is 1 measure beyond it.

I am not sure what the pertinence of this is. If the universe is infinitely old, then there exists a point from today, say u(1), that is an infinite distance from today. The problem is getting from u(1) to today. There is no amount of time from u(1) such that it still isn’t an infinite distance from today. If there is such a time, then u(1) was not an infinite distance away. Therfore, if u(1) really was an infinite distance away, then we can’t be here. However, we are here. Therefore, u(1) is not an infinite distance away.

I also have a problem with the meaning of "reality" being assumed. It appears that you are using the term "reality" to mean very specifically, the material world.

Well, that is what we are talking about. We are talking about the universe itself - material existence. I am saying that it is impossible for the material universe to be eternal. I have provided an argument for why it is not. This has nothing to do with God being eternal.

In other words, you seem to imply that the set of natural numbers are"unreal" because you can not conceive of them have actual correspondence to the material universe.

I can conceive of there being a correspondence. I just cannot conceive of the natural numbers being exhausted in terms of the age of the universe. This is what those who appeal to an actual infinity in terms of the age of the universe do. They say that the age of the universe is at least a surjection with the natural numbers. If this is actually the case, then we cannot be here today. Any point from the infinite past is still an infinite distance from today. Think of it this way: how long is eternity? Let’s say T(aleph0) represents some point in the infinite future. We are here today at time T(1). In a billion years we will be at T(10^9). What is the distance from this point to T(aleph0)? It is still an infinite distance away. There will never be a time T(x) such that the distance is finite. Yet, for those who say the universe is eternal, this must be the case for us to be here today.

This seem to be a metaphysical commitment that leads to a whole slew of problems when speaking about the existence of God and spiritual things.

I disagree. It is incoherent to speak of a completed infinite. By definition, the infinite is potential (given any ‘x’, there exists an ‘x+1’). Speaking of an actual infinity is like speaking of a round square. If you think you need to be able to posit an actual infinity for God, then please state where. I will be happy to try and provide a formulation or explanation that is both orthodox and avoids the incoherence of an actual infinity.

Sincerely,

Brian

 

[tewilder]

"There exist no successive interval of time from u(1) to today."

Why not? It doesn't seem any crazier than, say, the Axiom of Choice.

 

[sotzo]

"There exist no successive interval of time from u(1) to today."

Why not? It doesn't seem any crazier than, say, the Axiom of Choice.

I don't see the parallel between the AC and the issue of whether matter has existed eternally.

Also, what is your counterargument to the view that there is a Tzero, and that if there was no Tzero, there would be no Tnow?

 

[tewilder]

"There exist no successive interval of time from u(1) to today."

Why not? It doesn't seem any crazier than, say, the Axiom of Choice.

I don't see the parallel between the AC and the issue of whether matter has existed eternally.

Well, try it. Produce a pair of real numbers where one is the real number that is next larger than the other.

If you can't, why should analogous demands be expected of infinite sequences?

Also, what is your counterargument to the view that there is a Tzero, and that if there was no Tzero, there would be no Tnow?

It's just an assertion.

 

[tewilder]

You might want to take a look at some of Ehrlich's work.

Here is a view of an anthology:

http://projecteuclid.org/DPubS/Repo...ew=body&id=pdf_1&handle=euclid.rml/1081878082

It is has an article by Conway on the surreal numbers, and Erhlich's work on infinity is based on Conway's earlier studies. Some essays into into the matter of infinite real numbers, infinitesimals, and their reciprocals. As an alternative to the surreal numbers (of which the reals are a part) there is a paper by H. Jerome Keisler on the hyperreals which also have infinite and infinitesimal numbers.

For some sense of what the course I took from Ehrlich was like see this:

http://www.pitt.edu/~pittcntr/Being_here/last_donut/donut_2006/Apr_11_2006.htm

 

[tewilder]

See also:

Abstract of:
The absolute arithmetic continuum and the unification of all numbers great and small.
http://www.univ-nancy2.fr/poincare/colloques/symp02/abstracts/ehrlich.pdf

 

[sotzo]

Well, try it. Produce a pair of real numbers where one is the real number that is next larger than the other.

If you can't, why should analogous demands be expected of infinite sequences?

I don't think the issue is infinte sequences per se, but infinte sequences involving time. I subscribe to a realist view of time (Newtonian time). In such a framework, of course, time is bound up with the existence of matter. If matter is eternal, then time is not eternal..if time is not eternal then tzero occurred at some point in the past.

 

[tewilder]

Well, try it. Produce a pair of real numbers where one is the real number that is next larger than the other.

If you can't, why should analogous demands be expected of infinite sequences?

I don't think the issue is infinte sequences per se, but infinte sequences involving time. I subscribe to a realist view of time (Newtonian time). In such a framework, of course, time is bound up with the existence of matter. If matter is eternal, then time is not eternal..if time is not eternal then tzero occurred at some point in the past.

That is merely the time in this creation. It does not limit what God could have created, or may have created in another act of creation.