I know this has been done approximately a million times already, but I wanted to do it myself, just as an intellectual exercise.
The classical Kalam cosmological argument for the existence of God consists of six statements:
1. Everything that has a beginning of its existence has a cause of its existence.
This first premise has two major flaws:
1) It assumes that things can begin to exist. On what grounds is this assumption made?
Sure, common-day objects such as tables and chairs "begin to exist" in the sense that the arrangement of matter that people agree are "tables" and "chairs" begin to "exist" when someone arranges the matter in those precise ways. However, that's not what the premise is arguing. The matter itself didn't "begin to exist". It was just rearranged. The argument refers to first there being nothing, and then something beginning to exist.
This is an unfounded premise which cannot just be assumed without any justification, and currently there is no justification to make it according to known physics. According to current knowledge energy (which is what matter consists of) cannot be created nor destroyed. The energy that exists in this universe is, as far as we know, permanent: It doesn't change. Nothing of it can be destroyed, nor can it be created. The only thing that can happen is for the energy to change from one state to another, but the total amount is always preserved.
Comparing concepts like "the chair began to exist when the carpenter created it" and "the universe began to exist", and considering them equivalent, would be a fallacy of equivocation. It's certainly not the same thing. (In the first case nothing actually came into existence. A more accurate assessment would be that matter was transformed from one shape to another.)
Did the energy in this universe "begin to exist" at some point, or has it always existed (the definition of "always" not being trivial, as we will see later)? This is currently an unknown. There is no justification to say either way. Thus, especially in this context, there is no justification to say that it did begin to exist. Hence the premise is unfounded.
2) Even if we make the assumption that energy "began to exist", the second assumption made by the premise is also unfounded: That it must have had a cause for it beginning to exist.
Again, there is no justification for this claim according to modern knowledge of physics. For example, according to current knowledge, in quantum mechanics so-called virtual particles begin to exist (in a sense) without any cause for their existence. Their appearance is spontaneous and stochastic. (Spontaneous means that there is no external cause for the phenomenon happening, and stochastic means that it's not possible to predict when or where it will happen, in other words, the phenomenon is completely non-deterministic and uncaused.)
It is currently unknown whether the energy in this universe has always existed or if it appeared spontaneously (inside a "metaverse", possibly paired with a "negative universe" with an equal amount of "negative energy"). There is no justification to make a claim one way or the other. It is certainly not completely out of question that it was a phenomenon without a deterministic cause.
2. The universe has a beginning of its existence.
Unfounded assumption. We don't know if the universe began to exist or if it has "always" existed in some form. (The Big Bang theory does not state that the universe began to exist. It only states that at some point all the universe was compressed into an infinitesimally small point, from which it expanded. It does not make claims on where that point came from or what happened "before".)
It's possible that this universe did begin to exist, but that the phenomenon was spontaneous and uncaused. (If there is a "metaverse" inside which our universe resides, this kind of phenomenon might be happening all the time, eg. with positive-negative pairs of "universes" popping up spontaneously. This is one possibility among many.)
3. Thus the universe has a cause of its existence.
The conclusion is invalid because the two premises are invalid. (It might be true, but there's no justification for making the claim.)
4. This first uncaused cause must transcend physical reality.
There is an odd leap in this argument. The meaning of "uncaused cause" has not been defined previously, but is being used here as if it had. Where did this concept come from, and what does it mean?
As said, there doesn't necessarily have to be a cause for the existence of the universe. It might have "always" existed (and hence has no well-defined beginning when it "began to exist"), or its existence might have been completely spontaneous, without cause. (As said, this concept is not completely out of the realm of possibility as shown by quantum physics. This doesn't mean this is what happened, but it does mean that it's not impossible.)
Even if there was some kind of cause, this cause might indeed be due to some kind of property of whatever "metaverse" our universe might reside in, and hence this "cause" would indeed "transcend physical reality" in the sense that it's not a phenomenon from inside this universe, but from the "metaverse". However, if this is the true explanation, it would still be a natural one, even if it happens on a level that transcends this universe of ours. (It would simply mean that our universe is not the only thing that exists, but that there's more.)
Perhaps there is no "metaverse" and our Univserse is indeed the only thing that exists, and the explanation for its existence is something else completely. However, without further knowledge no assertions can be made about this, only hypotheses. Hypotheses cannot be taken as premises of logical arguments because they have not been proven to be true (and this is the major error the whole argument is making).
5. This uncaused cause that transcends physical reality is the description of God.
Even if we granted all the previous arguments (even though every single one of them was completely unfounded), this fifth argument is a complete null statement. It says nothing.
Putting a label on an unknown is useless and says nothing. You can substitute the word "God" above with a myriad of other words, and the argument would still be equally valid (in other words: not valid at all.)
For example, a completely valid alternative for the term "God" above would be: "the properties of the metaverse in which our universe resides" (or "the metaverse" for short.) You can make the substitution, and the fifth argument is equally valid. We can call this unknown "God", "the metaverse" or something else completely, and it would be still equally valid.
Well, in principle at least. In practice using the term "God" is actually less rational because that word carries a lot of assumed baggage that many other terms don't. For example "the metaverse" does not carry much extra "baggage" with it because nobody makes assumptions on its properties. However, the term "God" does because there is an enormous amount of assumptions attached to that word. (Just as one example, most people assume one property of "God" to be that it's sentient, which in this context is, of course, completely unfounded.)
6. Therefore God exists.
Substitute the word "God" with "the metaverse" in this sixth argument, and you will see why it does not hold. You can say "therefore the metaverse exists", and the argument would still be the same, but not any more (or less) valid. However, nobody would start believing that there is a metaverse outside of this universe simply because of this argument. Why not? Because the whole argument is flawed.
The argument is, basically: "We define 'God' as whatever it was that caused the universe to exist, hence God exists." You can substitute the word "God" with anything else you want, and the statement would still be equally valid. Just because you put a label on an unknown and "define" that label to have the (assumed) properties of that unknown doesn't make either the label nor attaching the properties to it valid.
Again, substitute "God" with "the metaverse", and "define" the metaverse as being the cause of our universe, and you have an equally (if not even more) valid argument.
Also remember that in the fifth argument above I granted all the previous arguments in order to make a point. Of course granting them is not valid because the previous arguments were all unfounded and thus nothing can be deduced from them.
Not only is this applying a label to an unknown (in a manner that is completely unjustified), but it's asserting that this unknown, whatever it might be, actually "exists". That is, in fact, yet another unjustified claim.
There's a shortened version of the argument that has only three statements. I will likewise present shortened refutations (because they are basically the same as above).
1. Whatever begins to exist has a cause.
How do you know?
("It makes sense" is not a valid answer. Just study a bit of quantum mechanics, or just general relativity, to see that not everything in the universe "makes sense", as we traditionally understand that concept.)
2. The universe began to exist.
How do you know?
(Remember: Modern science makes no such claim. The Big Bang theory makes no such claim. It is an unknown.)
3. Therefore, the universe has a cause.
You can't make a conclusion from unfounded premises. If the premises happened to be false (which they might well be in this case), the conclusion will be false as well.
What you can say is: "If whatever begins to exist has a cause and if the universe began to exist, then the universe has a cause", and that would be a logically true statement. However, the conditionals are not necessarily true (in which case the conclusion would be false).
This shortened version is accompanied by sub-set arguments:
1. An actual infinite cannot exist.
Again: How do you know?
What is the velocity of a photon (in vacuum), from the photon's own point of view (rather than an external observer)? If you don't know the answer, you'd be surprised.
(Of course one could argue whether this constitutes true "infinite velocity" or simply "zero distance traveled in zero time, hence technically speaking infinite velocity", but nevertheless the concept of "infinite" is not in a sense impossible in physics. There are many, many other examples of this in physics. Just a couple more: How long does it take for an object to cross the event horizon of a black hole, from the point of view of an external observer? How long would it take for a spaceship to travel from Earth to the edge of the Universe?)
If there exists a metaverse, could it perhaps be "infinite" in size (by whatever meaning of the term that would make sense there)? Could "infinite time" exist in this metaverse (perhaps in the sense that "time" as we know it does not actually exist there at all)? Impossible to know, but not impossible to be so. We can't assume anything.
2. An infinite temporal regress of events is an actual infinite.
Not necessarily. Non-linear time could converge towards a finite point, even though there is no beginning. (We know from general relativity that time even inside this universe is certainly not linear throughout the universe.)
And even if there was an infinite temporal regress of events, so what? On what basis can one argue that it's impossible?
3. Therefore, an infinite temporal regress of events cannot exist.
The conclusion cannot be made because it makes an unfounded assumption.
There is a second sub-set of arguments that talks about "successive addition" and how it cannot be infinite. The argument makes no sense whatsoever because it doesn't explain what it means by "addition" or by "successive addition" in relation to the universe.
In fact, that reminds me of the "unmoved mover" argument, originally by Aristotle, which goes like:
1. There exists movement in the world.
Ok, although it's an interesting question if the total amount of momentum in the universe is zero (it may very well be). Not that it invalidates the premise per se. It would just mean that all the movement might be "self-contained" in the sense that it does not need an original source external to the universe.
2. Things that move were set into motion by something else.
In most cases yes, but not necessarily always. For example the radioactive decay of an atom certainly causes movement (at least of the individual particles being emitted), but it's spontaneous and non-deterministic, in other words, not produced by any external cause, nor predictable.
The total amount of momentum is preserved, but there is certainly movement of the individual parts produced by the process (which is why the total momentum of the entire universe is an interesting question).
(Of course neither Aristotle nor Thomas Aquinas could have known this, so it would be unfair to demand this knowledge from them. However, it does invalidate their argument.)
3. If everything that moves were caused to move by something else, there would be an infinite chain of causes. This can't happen.
The first sentence is correct because of the conditional. The second sentence is unfounded. How do you know that an infinite chain of causes cannot happen?
4. Thus, there must have been something that caused the first movement.
Not necessarily. It could have been spontaneous, with no external cause. (As already stated several times, spontaneous phenomena do happen even inside this universe, not to talk about it possibly happening at a much larger scale in a hypothetical metaverse.)
5. From 3, this first cause cannot itself have been moved.
Argument 3 was not shown to be correct, hence nothing can be deduced from it. Likewise the assumption of there being a "first cause" is unfounded all in itself, because the "cause" could have been spontaneous (and hence not a cause at all).
There are three possibilities: There was an infinite regress of causes, the first movement was not caused but spontaneous, or the first movement was caused by an unknown deterministic cause, which itself did not participate in this chain of events. Why is this third option considered the only possibility (when one could in fact argue that it's the less likely of the three, at least according to the Occam's razor principle)?
6. From 4, there must be an unmoved mover.
Not necessarily. The first movement could have been spontaneous, if a "first movement" even existed in the first place.
Even if there was a "first movement", if the sum total of all momentum in the universe is zero, there is actually no need for an external source for the movement if movement (between different parts) can be generated entirely inside the universe itself (which is certainly possible, as we saw eg. with the radioactive decay example).
One can make the argument that the expansion of the universe has no "counterpart" that balances out this "movement" making the overall zero. However, we do not know if there is possibly a metaverse where this is happening in one way or another with pairs (or larger groups) of universes. (It is possible, for example, that what we perceive as "expansion" is actually some kind of movement in the metaverse, and that a "twin universe" to ours is moving to the opposite direction, cancelling out the total. If you study a bit of physics, this is not actually completely far-fetched of an idea.)
Another different argument is the so-called transcendental argument for the existence of God. The version by Matt Slick is summarized as (emphasis mine):
Logical absolutes exist. Logical absolutes are conceptual by nature, are not dependent on space, time, physical properties, or human nature. They are not the product of the physical universe (space, time, matter), because if the physical universe were to disappear, logical absolutes would still be true. Logical Absolutes are not the product of human minds, because human minds are different, not absolute. But, since logical absolutes are always true everywhere, and not dependent upon human minds, it must be an absolute transcendent mind that is authoring them. This mind is called God.
The argument is much sounder than the cosmological and other similar arguments in that it doesn't build up on one false premise after another and coming to a logical conclusion based on them. Instead, it kind of does the opposite: It builds up on sound premises but then makes a huge leap and pulls up a conclusion from nowhere which does certainly not follow from the premises (in other words, a non-sequitur).
There are three major problems with the argument. They are:
The non sequitur comes from the sudden assumption that logical absolutes must have been "authored" by some mind, created, rather than existing on their own right. This is completely unfounded.
Logical absolutes don't need to be "created" or "authored". That's because they are simply properties of existence. (For example "something is what it is, and isn't what it is not" is a logical absolute, and a property of existence, something that describes the nature of existence.)
This is completely comparable to any other property of any other thing. For example, "roundness" is a property of certain objects: Some objects are "round" when their shape conforms to certain specifications. The concept of "roundness" would exist even if there were no objects nor universe. However, it's still just a property of something, a concept that describes a feature. The property "round" doesn't need to be "created" or "authored" in order to "exist". It's not like there would have to be some superior "mind" coming up with the concept of roundness before any round objects become possible. Objects can be round completely regardless of whether there is any mind of any kind describing them as such and coming up with the concept. This property is completely independent of anything.
Likewise logical absolutes describe the properties of existence, and they likewise are independent of anything and do not need any kind of "mind" to have "created" them. It's not like existence would become impossible if there was no superior mind coming up with its properties.
The premises state that logical absolutes are not dependent on any mind (because they would be true even if there were no minds coming up with them), yet in the conclusion it is stated that they must have been authored by a mind (and thus they are dependent on a mind). This is a contradiction. The argument is claiming that logical absolutes both are and are not dependent on a mind for their existence at the same time.
If nothing can exist without the properties of existence (in other words, logical absolutes) being first created, and if God exists, then something other than God would had to have created the properties of existence before God could exist.
This dilemma is, in fact, similar to the classic "but who created God?" The circular logic can be summarized as (using similar argumentation style as all the previous):
The answer to the contradiction is simple: Logical absolutes do not need to be "authored" by anything, but are intrinsic properties of existence. Hence logical absolutes are not proof of anything.