# WL Craig's Kalam Cosmological Argument



## sotzo

I've seen several critiques of Craig's argument from the atheist camp. I have heard that there are Reformed Christians who may disagree with Craig's conclusion or premises. Two questions:

1. How is the Craig's argument any different than the traditional cosmological argument?

2. Does anyone know of any critiques of Craig's Kalam argument from a Reformed Christian perspective? (BTW, not looking for a presupp critique of the Kalam argument as it relates to whether evidential vs. presupp apologetic approaches...i'm really interested in the actual validity of the argument itself.)

Here is the argument:
Premise 1: Everything that begins to exist has a cause.
Premise 2: The universe began to exist.
Conclusion 1: Therefore, the universe must have a cause.


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## turmeric

What's Kalam?


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## sotzo

I've never been able to find what "Kalam" means - of course, nobody has ever accused me of being a Google genius either. 

Seriously though, that is part of my question...finding out what Kalam means and how it is different than the traditional cosmo. arguments.


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## Beoga

turmeric said:


> What's Kalam?



Kalam I believe was a Muslim apologist who was the first to formulate the Cosmological argument for the existence of god (well, in his case Allah). Don't completely quote me on that though.


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## BrianLanier

sotzo said:


> I've seen several critiques of Craig's argument from the atheist camp. I have heard that there are Reformed Christians who may disagree with Craig's conclusion or premises. Two questions:
> 
> 1. How is the Craig's argument any different than the traditional cosmological argument?
> 
> 2. Does anyone know of any critiques of Craig's Kalam argument from a Reformed Christian perspective? (BTW, not looking for a presupp critique of the Kalam argument as it relates to whether evidential vs. presupp apologetic approaches...i'm really interested in the actual validity of the argument itself.)
> 
> Here is the argument:
> Premise 1: Everything that begins to exist has a cause.
> Premise 2: The universe began to exist.
> Conclusion 1: Therefore, the universe must have a cause.



The argument is certainly vaild. I assume you are inquiring into its *soundness*. I think the argument has a prima facie persuasiveness to it, and can be used to increase the warrant of the reasonableness of Theism. However, used as a pre-dogmatic formulation of a demonstative argument for the existence of God, it will have its share of problems. So if one were to say the belief in God is properly basic, it could function as piece of natural theology. I don't think Christians should have a problem with the argument per se, just if it is used as a foundation for belief in God (pre-dogmatic).


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## VictorBravo

turmeric said:


> What's Kalam?



I first heard of the word "kalam" in the context of early Islamic philosophy/theology. کلام

The word literal means "to speak", but it was used to describe a dialectic approach: Propositions and counterargument. Why it is used for this particular argument I don't know.


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## VictorBravo

BrianLanier said:


> sotzo said:
> 
> 
> 
> I've seen several critiques of Craig's argument from the atheist camp. I have heard that there are Reformed Christians who may disagree with Craig's conclusion or premises. Two questions:
> 
> 1. How is the Craig's argument any different than the traditional cosmological argument?
> 
> 2. Does anyone know of any critiques of Craig's Kalam argument from a Reformed Christian perspective? (BTW, not looking for a presupp critique of the Kalam argument as it relates to whether evidential vs. presupp apologetic approaches...i'm really interested in the actual validity of the argument itself.)
> 
> Here is the argument:
> Premise 1: Everything that begins to exist has a cause.
> Premise 2: The universe began to exist.
> Conclusion 1: Therefore, the universe must have a cause.
> 
> 
> 
> 
> The arguement is certainly vaild. I assume you are inquiring into its *soundness*. I think the argument has a prima facie persuasiveness to it, and can be used to increase the warrant of the reasonableness of Theism. However, used as a pre-dogmatic formulation of a demonstative argument for the existence of God, it will have its share of problems. So if one were to say the belief in God is properly basic, it could function as piece of natural theology. I don't think Christians should have a problem with the argument per se, just if it is used as a foundation for belief in God (pre-dogmatic).
Click to expand...


I'd add that it appears to depend upon an empirical observation: the universe had a beginning. That has to be established first.


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## sotzo

> I'd add that it appears to depend upon an empirical observation: the universe had a beginning. That has to be established first.



Craig's response to this point is that we know the universe had a beginning because if it did not, the universe would be infinite and it is impossible to traverse an actual infinite. He then makes the point that since you cannot traverse an actual infinite we would never be at this place in time having our discussion (if the universe indeed had no beginning) since there would be an infinite series of events to get to this point...in other words, there would be no time zero from which to come to this point in time.


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## VictorBravo

sotzo said:


> I'd add that it appears to depend upon an empirical observation: the universe had a beginning. That has to be established first.
> 
> 
> 
> 
> Craig's response to this point is that we know the universe had a beginning because if it did not, the universe would be infinite and it is impossible to traverse an actual infinite. He then makes the point that since you cannot traverse an actual infinite we would never be at this place in time having our discussion (if the universe indeed had no beginning) since there would be an infinite series of events to get to this point...in other words, there would be no time zero from which to come to this point in time.
Click to expand...


Wow, that seems like a bigger and more difficult argument than the first one. I see a hint of Thomas Acquinas in that one, but I'd have to go back to my books to be sure.


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## Davidius

sotzo said:


> I'd add that it appears to depend upon an empirical observation: the universe had a beginning. That has to be established first.
> 
> 
> 
> 
> Craig's response to this point is that we know the universe had a beginning because if it did not, the universe would be infinite and it is impossible to traverse an actual infinite. He then makes the point that since you cannot traverse an actual infinite we would never be at this place in time having our discussion (if the universe indeed had no beginning) since there would be an infinite series of events to get to this point...in other words, there would be no time zero from which to come to this point in time.
Click to expand...


If you were Zeno you could just say that we don't really traverse space anyway. 

Also, it seems like one could object that, even if a cause were conceded, the cause need not be the Trinity.


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## tellville

CarolinaCalvinist said:


> Also, it seems like one could object that, even if a cause were conceded, the cause need not be the Trinity.



No, and he admits that. That is why he then argues for the resurrection. He attempts to show that there needs to be a timeless, immaterial, personal cause that starts the universe. But why would such a cause start the universe? He then points to the resurrection. Jesus being resurrected by God is the only explanation of the facts that we have, and if this is the case then Jesus has God's "stamp of approval" (because only God could raise Jesus from the dead). Well, Jesus gave his stamp of approval to the Jewish religion as it was fulfilled in him. Thus, Christianity has God's stamp of approval. Therefore, whatever Christianity says about why we are here and who God is is true. Given that he is an evidentialist, he will then say that while this is not an infallible proof for Christianity or God, it is on the whole the most probable explanatory model we have given the evidence. 

As for Kalam, it is either the name of the Muslim philosopher or the name the Muslim philosopher who thought up the argument gave it.


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## cih1355

I do not know if there is a traditional cosmological argument. I just know that there are different versions of the argument.

One could say that the argument does not necessarily prove the existence of the Christian God, but Craig would respond that the argument only intends to prove that God exists and that another argument is needed to prove that the God who exists is the Christian God.


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## Civbert

CarolinaCalvinist said:


> sotzo said:
> 
> 
> 
> 
> 
> 
> I'd add that it appears to depend upon an empirical observation: the universe had a beginning. That has to be established first.
> 
> 
> 
> 
> Craig's response to this point is that we know the universe had a beginning because if it did not, the universe would be infinite and it is impossible to traverse an actual infinite. He then makes the point that since you cannot traverse an actual infinite we would never be at this place in time having our discussion (if the universe indeed had no beginning) since there would be an infinite series of events to get to this point...in other words, there would be no time zero from which to come to this point in time.
> 
> Click to expand...
> 
> 
> If you were Zeno you could just say that we don't really traverse space anyway.
Click to expand...


Although you were joking, that's actually a good point. Zeno shows that we traverse the infinite every time we move from A to B. So it is not a given that we can not traverse an actual infinite.


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## BrianLanier

Civbert said:


> CarolinaCalvinist said:
> 
> 
> 
> 
> 
> sotzo said:
> 
> 
> 
> Craig's response to this point is that we know the universe had a beginning because if it did not, the universe would be infinite and it is impossible to traverse an actual infinite. He then makes the point that since you cannot traverse an actual infinite we would never be at this place in time having our discussion (if the universe indeed had no beginning) since there would be an infinite series of events to get to this point...in other words, there would be no time zero from which to come to this point in time.
> 
> 
> 
> 
> If you were Zeno you could just say that we don't really traverse space anyway.
> 
> Click to expand...
> 
> 
> Although you were joking, that's actually a good point. Zeno shows that we traverse the infinite every time we move from A to B. So it is not a given that we can not traverse an actual infinite.
Click to expand...


Well, there are arguments on both sides. I tend to think that Craig has the better argument (i.e., the impossibility of an actual infinite). Again, I think most (if not all) Christians would agree with that the universive has not *always* existed. Perhaps not so for the unbeliever, but then again he has an metaphysical stake in denying the argument (not to say that believers don't as well!). So again, this argument may have value as a piece of natural theolgy, so long as the Christian does not use it as providing a foundation to the Christian faith. (Then there is the question as to how Craig uses the argument himself and I'm not sure that is clear either.)


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## Brian Bosse

Hello Gentlemen,



> Premise 1: Everything that begins to exist has a cause.
> Premise 2: The universe began to exist.
> Conclusion: Therefore, the universe must have a cause.



The argument is valid. An actual infinity is incoherent and oxymoronic. All me to make a distinction between a finite interval of time that is subdivided into an infinite number of smaller intervals, and an infinite interval of time divided into an inifite number of intervals. Zeno's arguments play on the former, whereas the latter is the claim being made by those who say that the universe is uncreated. Consider an infinite string of digits (for instance, the number one-seventh) that is put into a one-to-one coorespondence with the set of natural numbers. 



Code:


N:                   1 2 3 4 5 6 7 ... n ...
One-Seventh (1/7) : .1 4 2 8 5 7 1 ... 5 ...


Mathematicians will note that for any given 'n' in the set of natural numbers corresponding to a particular digit in the sequence of one-seventh, there are an infinite number of digits that remain. If for some given 'n' this is not the case, then the sequence is not infinite and has an ending. We will call this the non-actuality theorem (NAT). 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...



Code:


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 infinte, 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.*

Brian


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## sotzo

Brian

Nice proof...you've explained precisely why Zeno's paradox doesn't apply here. The famous tortoise racing Achilles example (where Achilles never catches up) that is used to illustrate the paradox would be apples to apples with Craig if there were no starting line for the race. As it stands, they have a T(0) and, therefore, the paradox resolves.


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## tewilder

The problem with Criag is that he assumes ZFC set theory. Pick a different set theory that allows proper classes and you can build a coherent transfinite arithmetic and the paradoxes go away.


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## Brian Bosse

Hello T.E.,

I am no expert, but Mathematical Logic and Set Theory are a hobbie of mine. I am not sure what you mean by "there being a coherent transfinite arithmetic and the paradoxes go away." Can you explain? What paradoxes go away? How do "Proper Classes" solve any of these paradoxes?

Thanks,

Brian


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## tewilder

There is a guy named Philip Ehrlich who has worked out a transfinite mathematics. Craig argues on the basic of the old paradoxes of cardinality where, for example, the cardinality of the set of natural numbers is Aleph, and Aleph + 1 = Aleph and Aleph + 2 = Aleph, and so on, because it can't get any bigger, until you jump up to the next order of cardinality, say that of the real numbers.

So Craig says if you have an actual infinite where for example you add a moment to an already elapsed infinite number of moments you wouldn't have any more moments because the total of them is still Aleph. 

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. 

Ehrlich has also done interesting studies of thermodynamics. For example, he has a book _Negative, infinite, and hotter than infinite temperatures_. 
http://www.springerlink.com/content/j45862lt261g3q34/
He said that although thermodynamics does not require an actual infinite, the math for it stands ready in case some application for it does turn up.

Craig knows nothing of this stuff, and his book does not take it into account.

Actuallly, it is a much longer story than that, but there is the main point. 

A book to read along side of Craig would be _Infinity: An Essay in Metaphysics_ by Jose Benardete. What Craig does is turn over the supposed paradoxes of the infinite until you mind boggles and you agree with his intuition that the actual infinite is impossible. But Benardete's mind does not boggle as easily as Craig's, and he does not report the same intuition.


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## Brian Bosse

Hello T.E.,



> 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.



I am unfamiliar with Hilbert-Akerman set theory. Can you point me to the axioms that make up this formal system? 



> What Craig does is turn over the supposed paradoxes of the infinite until you mind boggles and you agree with his intuition that the actual infinite is impossible. But Benardete's mind does not boggle as easily as Craig's, and he does not report the same intuition.



The concept of an actual (completed) infinity is incoherent to me. I can grasp potential infinity just fine. The touble with an actual infinite is that it must contain a potential infinity. The natural numbers are such an example. There is no 'n' such that 'n+1' does not exist. Yet, those who hold to an actual infinity speak of the exhaustion of such a set. In what sense can such a set be exhausted? It cannot be in the sense that there is a last 'n'. I say all of this just to say that I am suspect of anyone whose mind is not "boggled as easily Craig's." 

Brian


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## tewilder

Hilbert and Ackerman published a book on set theory in 1928. It was later translated into English. I don't have the list of axioms. But what I do know is that for a transfinite arithmetic you need a set theory, such as the Hilbert Ackerman, that allows proper classes, that is classes that are not capable of being members of classes. 

"The concept of an actual (completed) infinity is incoherent to me."

I don't see why God, for example, couldn't make infinite things, even if we could never get through them.


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## Brian Bosse

Hello Tewilder,



> I don't see why God, for example, couldn't make infinite things, even if we could never get through them.



Can God square a circle? Can God list all of the natural numbers? No. Just as 'square' and 'circle' are not compatible, so is 'list all' and 'natural numbers'. The natural numbers are precisely defined. There is no ambiguity. Part of that definition is that for all 'n' there exists a successor to 'n'. To speak of this as being completed (actual) is the same type of talk as squaring a circle. 

Brian


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## ChristianTrader

Brian Bosse said:


> The concept of an actual (completed) infinity is incoherent to me. I can grasp potential infinity just fine. The touble with an actual infinite is that it must contain a potential infinity. The natural numbers are such an example. There is no 'n' such that 'n+1' does not exist. Yet, those who hold to an actual infinity speak of the exhaustion of such a set. In what sense can such a set be exhausted? It cannot be in the sense that there is a last 'n'. I say all of this just to say that I am suspect of anyone whose mind is not "boggled as easily Craig's."
> 
> Brian



Does not any attempt to deny actual infinity have serious theological consequences? Either God is finite or actual infinity. No one wants any part of saying that God is finite. If one was to attempt the potentially infinite angle, then one would have to move over into completed infinity unless one wants to deny that God has exhaustive self knowledge.

CT


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## Brian Bosse

Hello C.T.,



> Does not any attempt to deny actual infinity have serious theological consequences? Either God is finite or actual infinity.



One needs to be very careful and precise in these types of discussions. The mathematical concepts of the 'finite' and 'infinite' are not the same when one speaks of creation being finite or when we speak of the everlasting God. The Cantorian actual infinity is not the same thing as saying God has no beginning or other similar sayings.



> If one was to attempt the potentially infinite angle, then one would have to move over into completed infinity unless one wants to deny that God has exhaustive self knowledge.



Again, you need to be precise about what you are saying here. The idea of actual versus potential infinity is an idea that is proposed at our level of creation. So, this is the level the argument takes place in. 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. Things predicated at this level simply do not translate to our level of existence. So, if God is everlasting in the sense of existing always or before creation, because this necesarily is indpendent of creation, we cannot say that in our creation there is an everlasting past. The Bible speaks analogously to us. We cannot comprehend it in any other way. I have no problem saying that at the second level of existence there is no actual infinite. Regarding the other level, I cannot speak of. No created being can speak of this level except that which is revealed to us by God. Even then, God can only use second level references in analogy to describe the first level of existence. The bottom line is that this level of existence is ultimately unaccessible to us. So, when you speak of God's self-knowledge, you are necessarily speaking of something that is beyond us. It is not part of this level of creation. In other words, it is not what we are concerned with when we are speaking of concepts like actual or potential infinity, which are second level concepts. I believe any theological problems that you might see arising from this can be dealt with by making fine distinctions. 

Sincerely,

Brian


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## CubsIn07

Although the KCA is sound in and of itself from what I have seen, you will notice that when Bahnsen debated Stein, Bahnsen said that he does not use the KCA apart from already presupposing the Christian worldview because good philosophers can get around the argument. I think Bahnsen would say that you are not allowed to use contrary premises in separate proofs. The problem is that you will have a hard time arguing the eternality of God, an uncaused cause, and the KCA to make one larger argument for Christianity. Yes an actual infinite seems impossible and the only way to get around this and maintain traditional Christianity it to postulate God's timelessness, but that is a difficult proposition. My point is that if you say that everything that begins must to have a cause, how do you know that God didn't begin to exist?


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## Brian Bosse

Hello Cubsin7,



> The problem is that you will have a hard time arguing the eternality of God, an uncaused cause, and the KCA to make one larger argument for Christianity.



Yes, Bahnsen would say that _in the Christian worldview_ God is uncreated. As such, He is not subject the KCA. This means the Christian worldview can account for the universe being here. God created it; whereas, those worldviews not positing an uncreated creator fail. However, one needs to posit more than this because it is incoherent to speak of an uncreated creator in time. He needs to be eternal, and as such the Christian must embrace a God _sans_ time even with the difficulties it presents. Any other alternate that I am unaware of just is not tenable. 



> My point is that if you say that everything that begins must to have a cause, how do you know that God didn't begin to exist?



The only way to "prove" this is to assume it. Scripture is clear that God is the uncreated one. Therefore, the pragmatic analysis of worldviews (presuppositional apologetics) will show the sufficiency of the Christian worldview. However, KCA is not an argument for God _per se_. It only is an argument for the necessity of a creator for the universe _whatever that is_. In many ways it works the same as ID. 

Sincerely,

Brian


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## tewilder

What Craig is arguing against is a composed infinite, and infinite made up of an infinite number of parts.

To make his argument is uses mathematical examples that presuppose a certain type of set theory, although he does not tells us (because he did not know himself) that other types of set theory the do not yield this result are possible. 

To make his argument he also argues that time is made up of parts, so an actual infinite time of a universe existing is a composed infinite of an infinite number of time parts, or events, or some such.

Very quickly the argument gets complex, and not all the assumptions are out in the open.

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

For the argument to not apply against the infinity of God, Craig has to argue that God is not composed, either in his being or in his duration. The sort of God that, say, the Federal Vision believes in that is essentially relational and goes through changes, emotional states, etc. would not be possible if Craig's argument is valid, but only the God of the philosophers, who is a purely simple, unchanging being. 

Of course, the Federal Vision people say that their God is Van Til's God. So if they are right, then Craig has made a dividing line between two basically different theologies.


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## Brian Bosse

Hello T.E.,

The issues are interesting. 



> To make his argument is uses mathematical examples that presuppose a certain type of set theory, although he does not tells us (because he did not know himself) that other types of set theory the do not yield this result are possible.



This may be the case. His appeals to things like the infinite hotel, etc... do point to non-intuitive issues that arise within Set Theory. However, I do not yet see how any set theory gets around these issues. For instance, all set theories will argue that the cardinality of the nuatural numbers equals the cardinality of the even numbers. It seems you are saying that there is a set theory where this is not the case. This is news to me, but I am no expert. 

With all of that said, I proposed an argument that does not presuppose a particular set theory. Rather, it denies the actual infinite in terms of the real world. It does not deny it in strict theoretical constructs. Most set theorists would not find this problematic. They will admit that the transfinite may not have a real world application and remain content with their construct. I love to study formal systems just for the sake of the logic behind formal systems (set theories are in this catagory). 



> For the argument to not apply against the infinity of God, Craig has to argue that God is not composed, either in his being or in his duration.



I disagree. The argument as stated only argues against an actual real world interval that is infinite as opposed to an interval that is finite divided into infinite parts. Also, when you speak of the infinity of God, I suspect you are equivocating on the term 'infinity'. I do not think the Bible uses the term in such a technical sense as mathematicians have defined it. A discussion regarding this would be interesting. 

Sincerely,

Brian


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## tewilder

Brian Bosse said:


> Hello T.E.,
> 
> 
> This may be the case. His appeals to things like the infinite hotel, etc... do point to non-intuitive issues that arise within Set Theory. However, I do not yet see how any set theory gets around these issues. For instance, all set theories will argue that the cardinality of the nuatural numbers equals the cardinality of the even numbers. It seems you are saying that there is a set theory where this is not the case. This is news to me, but I am no expert.



Sure. But transfinite arithmetic is done with ordinal numbers, not cardinal numbers. 

What's more, there is a reason for that. Cardinality is defined as relative ordering. Two sets have the same cardinality if the members can be mapped one to one onto each other. This is the same as saying that each set can order the other (just so long as at least one can generate an order). For finite sets, this is the case when there are _just as many_ members in each set. But if set A is mapped onto set B one to one, and set A still has members left over when B runs out, then A has a greater cardinality, and it is also the case that it has more members, is bigger than B.

But what about infinite sets? The natural numbers vs. the real numbers. You cannot order the real numbers by the natural numbers, as Cantor's diagonal proof shows. 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_?

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.

This is why in the transfinite realm, cardinal numbers fail. Cardinality, in my view, is not a fully arithmetic concept, and in the transfinite realm this becomes apparent. 



Brian Bosse said:


> With all of that said, I proposed an argument that does not presuppose a particular set theory. Rather, it denies the actual infinite in terms of the real world. It does not deny it in strict theoretical constructs. Most set theorists would not find this problematic. They will admit that the transfinite may not have a real world application and remain content with their construct. I love to study formal systems just for the sake of the logic behind formal systems (set theories are in this catagory).
> 
> Brian



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.


Brian Bosse said:


> For the argument to not apply against the infinity of God, Craig has to argue that God is not composed, either in his being or in his duration.
> 
> 
> 
> 
> I disagree. The argument as stated only argues against an actual real world interval that is infinite as opposed to an interval that is finite divided into infinite parts. Also, when you speak of the infinity of God, I suspect you are equivocating on the term 'infinity'. I do not think the Bible uses the term in such a technical sense as mathematicians have defined it. A discussion regarding this would be interesting.
> 
> Sincerely,
> 
> Brian
Click to expand...


Well, we aren't talking about what the Bible says about God. We are talking about what philosophical theology says about God, and has said since ancient times. God is simple, unchanging etc. 

There are plenty of people who reject this sort of traditional metaphysics, but as far as I know only process theology people have an developed answer about what they would put in its place. 

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. 

Of course this gets Craig (if his argument is valid) only to the God of the philosophers. He then has to call on supplementary arguments as to why the God that his argument proves to exist would be the God of the Bible. Craig is well aware of that.


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## tewilder

Brian Bosse said:


> Hello T.E.,
> 
> 
> I disagree. The argument as stated only argues against an actual real world interval that is infinite as opposed to an interval that is finite divided into infinite parts. Also, when you speak of the infinity of God, I suspect you are equivocating on the term 'infinity'. I do not think the Bible uses the term in such a technical sense as mathematicians have defined it. A discussion regarding this would be interesting.
> 
> Sincerely,
> 
> Brian



Then there is this guy:

http://www.springerlink.com/content/g482wh68q1t27p66/

who argues that God could create an actual infinite past, and yet Craig's argument would still be valid!


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## 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


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## 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)?


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## 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.


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## 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.


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## 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. 



T.W. said:


> 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.


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## tewilder

sotzo said:


> 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.


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## tewilder

Brian Bosse said:


> 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


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## 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.


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## Brian Bosse

Hello Gentlemen,



T.E. said:


> 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


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## 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?


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## 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


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## sotzo

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


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## 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


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## tewilder

Brian Bosse said:


> 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.


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## 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


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## tewilder

Brian Bosse said:


> 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
Click to expand...


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.


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## Civbert

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



Brian Bosse said:


> 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...
> 
> 
> Code:
> 
> 
> 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.


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## Brian Bosse

Hello T.E. and Anthony,



T.E. said:


> 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.



T.E. said:


> 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. 



Anthony said:


> 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.



Anthony said:


> 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.



Anthony said:


> 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. 



Anthony said:


> 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.



Anthony said:


> 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


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## 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.


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## sotzo

tewilder said:


> "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?


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## tewilder

sotzo said:


> tewilder said:
> 
> 
> 
> "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.
Click to expand...


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?



sotzo said:


> 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.


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## 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


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## 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


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## 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.


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## tewilder

sotzo said:


> 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.
Click to expand...


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.


----------

