Infinite regress does not occur
Various cosmological and teleological arguments are based on a sequence of events or explanations that are a terminated by a final cause, which is assumed to be God. However, it is conceivable that there is no final cause and an infinite sequence of causes extends into the past: this is known as an infinite regress. Apologists argue that infinite regress does not occur, which would otherwise make the first cause unnecessary.
Potential and actual infinities
A potential infinity is a process that can continue indefinitely or a set that can expand indefinitely and without limit. While the process may continue without limit, the number of steps that have actually occurred in the process is finite. The existence of potential infinities is generally accepted. When a potential infinite value actually occurs, it takes some arbitrarily large but finite value at the present moment and may reach any other arbitrarily large but finite value in future.
An actual infinite is a actual set with an uncountably large number of items. A set containing all whole numbers (natural numbers) arguably is an actual infinite which exists as a mathematical concept. The controversy of infinite regress often involves the question of the existence of actual infinities in physical reality. The mathematics of actual infinities, pioneered by Georg Cantor, is notoriously tricky and sometimes counter intuitive.
Hilbert's hotel is an alleged paradox which rules out the possibility of actual infinities existing in nature. Imagine a hotel with an infinite number of rooms and is full of guests. If another guest arrives, the manager wants to create a room vacancy. He moves the guest in room 1 to room 2, the guest in room 2 to room 3 and so on. This makes room 1 vacant to accommodate the extra guest. However, after the addition of an extra guest, the hotel still has an infinite number of guests and is apparently full. Mathematically, ∞ + 1 = ∞.
Similarly, an infinite number of new arrivals can be accommodated by moving the guest from room 1 to room 2, room 2 to room 4, room 3 to room 6. This makes all the odd numbered rooms available. The hotel is still occupied by an infinite number of guests. Seemingly ∞ + ∞ = ∞.
We can also consider a single guest checking out or all the odd numbered rooms checking out. The manager decides to consolidate the remaining guests to fill the rooms but this leaves an infinite number of guests remaining and the hotel is still apparently full. ∞ - 1 = ∞ and ∞ - ∞ = ∞. Stranger still, all guests with room number above 3 could check out (which is an infinite number), leaving a finite number of guests. ∞ - ∞ = 3. These apparent contradictions arguably implies that the existence of such a hotel is absurd and therefore impossible.
William Lane Craig uses the example of coins to make the same point:
"In each case, I took away an identical number of coins from an identical number of coins, and came up with contradictory results. In fact, you can subtract infinity from infinity and get any answer from zero to infinity. [...] This, and many other examples, suggest that infinity is just an idea in your mind, not something that exists in reality."
If Hilbert's hotel is impossible then it is argued that an infinite regress cannot occur. 
"Hilbert’s Hotel is absurd.... if an actual infinite number of things were possible, then such a hotel must be possible. Hence, it logically follows that if such a hotel is impossible, then so is the existence of an actual infinite."
An actually infinitely old universe
William Lane Craig argues that the past is a potentially infinite collection of successively added moments. If the age of the universe is an actual infinite, the amount of past moments that can exist is never enough, which is absurd. Therefore the age of the universe is not an actual infinite and an infinite regress cannot occur.
"One cannot form an actually infinite collection of things by successively adding one member after another. Since one can always add one more before arriving at infinity, it is impossible to reach actual infinity."
- — William Lane Craig 
- "Moreover, you can ’t add anything to something that is infinite, but tomorrow we will add another day to our timeline. So our timeline is undeniably finite. "
Actual infinities do not occur in nature
Although mathematically interesting, actually existing instances of actual infinities are not seen anywhere in nature.
"The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought....The role that remains for the infinite to play is solely that of an idea."
- — David Hilbert 
"Infinite totalities do not exist in any sense of the word (i.e., either really or ideally). More precisely, any mention, or purported mention, of infinite totalities is, literally, meaningless."
- — A. Robinson
- "[...]as we discussed in the last chapter, an infinite number of finite things—whether we’re talking about days, books, bangs, or universes—is an actual impossibility. There can’t be an unlimited number of limited universes."
However, since we can't directly observe an actually infinite set, how can we know they can't exist?
Progression of conscience
- "There cannot be an eternity of the past of cause and effects because mankind would have approximated into that alleged past eternity and not still be sinning to the extent we still do along the exponential progression of conscience we are clearly on. "
Even if there is such a thing as a "progression of conscience", there is no reason why it could not increase indefinitely or be followed by a "decline in conscience", followed by another progression, in a cyclic fashion.
The real implication of Hilbert's hotel is the mathematical concept of infinity used in Hilbert's hotel is flawed. The paradox can be resolved by the theory of transfinite cardinal numbers, proposed by Georg Cantor. There are arguably different types of infinities, so the subtraction of two infinite sets can result in a finite number, infinite number or zero, depending on their "cardinality". William Lane Craig claims this unspecified result as supporting his view:
- "Dr Millican says that infinity minus infinity is not defined in transfinite arithmetic – there’s more than one solution to the equation. And that is precisely the problem when you try to translate this into the real world. "
Subtraction of "infinity" is meaningless since infinity is the cardinality of a set and not a number that can be used arithmetically . In the case Hilbert's Hotel, if all guests check out except for three, there will obviously be three guests remaining and there is no paradox. Craig's flawed concept of "infinity" does not exist in transfinite arithmetic. Therefore there is no paradox in Hilbert's hotel.
Not all mathematicians, such as the intuitionism and finitism movements, or philosophers accept the concept of transfinite cardinal numbers. Some accept the mathematical concept but reject the actual existence of actual infinities.
Are God's attributes actual infinities?
- Main Article: Infinity
If actual infinities do not occur, God cannot have actually infinite attributes either.
Actual infinities are not observable
Apologists point out that actual infinities are not observed in nature. However, an actually infinite quantity of anything is not directly observable, so we would not expect direct evidence. Such observation would require infinite time and infinite resources. Therefore, actual infinities may exist in nature but without our knowledge. Just as the possibility of God cannot be ruled out with absolute certainty based on a finite number of observations, the possibility of the existence of actual infinities cannot be rule out. Therefore infinite regress is possible, as far as can be known. It is unreasonable to demand direct evidence of the physical existence of actual infinities.
An actually infinitely old universe
The argument is begging the question because it assumes the past is a potentially infinite set, which directly implies an (arbitrarily large but) finite age of the universe! On the other hand, it is possible that the age of the universe is an actual infinity and the amount of past moments is an exactly equal actual infinite - in which case there is no absurdity. It is conceivable that moments are successively added to an actually infinite set of past moments.
The past might exist in a different form than supposed by Craig, such as the theory of block time, which also avoids the absurdity. This apologetic argument also ignores the possibility of cyclic time.
Supposed absurdity of the physical existence of actual infinity
Modern physics have wacky notions, both mathematically and physical existence, that are very counter intuitive. This includes things being in two places at once (two slit experiment), virtual particles, cosmic inflation, black holes, quantum field theory, etc. If everyone accepts an argument's conclusion as absurd, an argument is a valid reductio ad absurdum. However, skeptics and scientists should not be so hasty in dismissing possibilities in an area so removed from human experience. Apologists' arguments are therefore an argument from personal incredulity. 
Proof by logic
- Main Article: Proof by logic
As much as people try, abstract mathematics cannot rule out the existence of a physical entity.
Apologetics that rely on the impossibility of infinite regress
- First cause argument
- Cosmological argument
- Leibniz cosmological argument
- Argument from contingency
- Kalam 
- ↑ 1.0 1.1 James, Infinity minus infinity, Reasonably Faithless, Mar 25, 2013
- ↑ 2.0 2.1 2.2 William Lane Craig, The Existence of God and the Beginning of the Universe
- ↑ William Lane Craig, Reasonable Faith: Christian Truth and Apologetics, Crossway Books, 2008
- ↑ 4.0 4.1 I Don't Have Enough Faith to be an Atheist
- ↑ David Hilbert, On the infinite, translated from Mathematische Annalen (Berlin), vol. 95 (1926), pp. 161-90
- ↑ [http://biblocality.com/forums/showthread.php?3476-4-Step-Proof-for-God-amp-Minimal-Facts-Approach&p=4195#post4195
- ↑ 
- ↑ 
- Common Sense Atheism, Hilbert’s Hotel
- Hilbert's paradox of the Grand Hotel, wikipedia
- Hilbert's Program, Stanford Encyclopaedia of Philosophy, Jul 31, 2003
- Rafael E. Núñez, Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals, Journal of Pragmatics 37 (2005) 1717–1741