Is there any logical reason that our world or rather existence of any sort should have a beginning? In other words does the assumption of 'eternal' existence create any contradictions?
The Kalam argument, which is an Islamic version of the cosmological argument maintains that the universe cannot have existed for an infinite period of time. This is because if there was an infinite period of time before us we never would've reached the present day. This is a view shared (or may have even been his own original work) with William Craig too.
Thomas Aquinas wrote about five ways to God, which may contain something useful on this topic. In the second way, the way of the 'first cause', he claims that if there would be no first cause, there wouldn't be any cause:
Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false.
Aquinas' five ways are heavily criticised and can certainly not be considered solid logic proofs; nor were they meant that way. Especially the sentence "Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause" is very, very debatable.
I am not aware of any proof for or against eternal existence.
Logic?... Our world? How could something belong to us if we did not exist?
It seems that everything existing before humans would be outside of our world. Taken to the extreme, was the initial instance after the big bang our world? After humans become extinct and the Sun implodes/explodes, is that also our world? Taken further, is today our world? If I died today, would tomorrow be my world? The idea that this is our world seems quite self centered and reminds me of how Galileo was imprisoned for life for positing that the Earth was not the center of the universe.
My thinking leads me to believe that our world has existed only as long as we have, not any longer. That's just my 2 cents (my own answer).
Physicists continue to spend a great deal of effort in understanding the origin of the Universe. You might get some neat answers if you ask a community of physicists.
Further to Gabs' answer, a passage from Wikipedia states:
On actual infinities, Craig asserts the metaphysical impossibility of an actually infinite series of past events by citing David Hilbert's famous Hilbert's Hotel thought experiment and Laurence Sterne's story of Tristam Shandy. Michael Martin objects:
"Craig's a priori arguments are unsound or show at most that actual infinities have odd properties. This latter fact is well known, however, and shows nothing about whether it is logically impossible to have actual infinities in the real world. ... Craig fails to show that there is anything logically inconsistent about an actual infinity existing in reality."
I tend to agree. My unemphatic opinion is that nonexistence (the 'existence' of nothing) is precluded by existence (of something), and that the case does not change - whether across cyclical Big Bangs or other forms of existence. The most obvious logical contradiction appears to be the proposition that something could come from nothing.
The most weighty concerns regarding logical problems with eternity come from considering causality: if there was no beginning, how do you get a causal chain started? It seems like there might be room for an inconsistency to creep in with some uncaused events.
However, there are at least four ways out of such problems.
Uncaused events are okay. There are some of those; they just have to be stable over time. We know from observation that there aren't many on our time- and length-scales, but that doesn't mean that there are literally none.
Changes diminish to zero as one travels infinitely far back (or finitely far back--a periodic structure is okay as long as there's no information transfer from one period to the next).
Although we experience causality forward, that is our bias; what is physically constraining is consistency, which can run in either direction (including infinitely far into the past).
We are finite. Reality isn't. The infinite computation is just not a problem; it's equivalent to infinite initial state, and that's okay with reality too.
To my knowledge, the other problems are all merely kind of unexpected, not deeply troubling like lack-of-causality could be. But since we have a menu of ways out, there is no known logical barrier to an infinite past (of sorts).
Because you mentioned logic, it is worth looking at causality and how it could affect logic.
Causality is one common axiom of time. Without it, there is virtually no reason to assume an eternal world is inconsistent. All of the arguments I am aware of for a non-eternal world stem from an application of the rules of causality.
With causality, a "first cause," and a finite universe, it is trivial to fit all of existence into mathematical sets. Sets and logic tend to go hand in hand. With causality, and a perfectly cyclical infinite universe, it is also easy to fit all of existence into mathematical sets, but it requires a little creativity to work the loops out.
However, there are many causal universes which do not fit into these two categories. Many of them cannot be described with sets, due to their self-referential topology. They must be described in classes. While logic can be applied to classes, most of our intuition of how logic should behave and all of our First Order Logic proof assume the universe can fit into a set. This can lead to all sorts of surprising contradictions if you simply try to treat classes like sets (see Godel, Russel, et.al.)
What's logically possible depends on your axioms.
If your axioms about the Universe are satisfied by, say, ordinary Minkowski space, then your axioms cannot imply that the Universe had a beginning, because Minkowski space had no beginning.
If your axioms about the Universe are sufficiently restrictive that they are satisfied only by our own Universe and no other, then they semantically imply that the Universe had a beginning, because our Universe had a beginning. Whether they logically (i.e. syntactically) imply that the Universe had a beginning is, I suppose, an open question.
I agree with @Rex Kerr when he states that the consideration of causality may actually be the issue that creates a logical problem for us. (Please correct me if I'm misstating you.)
Specifically I mention:
- Although we experience causality forward, that is our bias;
I see no problem considering a non-cause-and-effect paradigm operating outside of our known, physical reality. While perhaps this would be non-ascertainable with our current knowledge and experience, I leave the door open for any kind of possibility that doesn't need to fit within the confines of our current understanding of cause, effect or - more to the point - time itself.
My understanding of humans is that we find the concept of non-time quite challenging (I know I do). While time governs the definitions of eternity, infinity and finitude, I see no dilemma in calling time a construct and thus, so too the very concept of these terms.
To answer your question more directly, I would say, "Yes, anything is possible."
The dilemma of extracting something from nothing is quite perplexing. It is like dividing by zero. It simply does not compute. Such questions enter into a metaphysical domain where one has to reexamine what they actually deem as "material", or "real".
In my experience, such experiences are a manifestation of both. That is, both infinite and finite. We our creatures of "now", that is, consciously all we have is the moment.
Ironically we don't even know what the moment is ( see Jiddu Krishnamurti ). Normally, when we reflect on the ontology of ourselves and the nature of the cosmos, our attention is attracted to things we do know about and our familiar with. Questions that deal with our spatial awareness of time, causality, and ultimate demise. Of course these our important questions but how we perceive their relationship to what we do not know is even more important. The beauty of this is in not knowing; allowing room for surprises.
The Universe we live in, had a beginning, it is The Creator that is eternal. If you don't like "Creator", then it is the entity, substance, energy, etc. that is eternal. The Universe was created from this "eternal substance," about 13.7 billion years ago.
EDIT: I can think of one way that the world could be thought of as eternally existing. Our world is part of God, God is eternal (always existed), therefore our world is eternal (always existed).
An analogy might clarify this. The ocean (liquid water) is God, a small iceberg is the world, and even though the iceberg has a different form, it was created from the same water. So, if you want there is a starting point for the iceberg form, but the water molecules always existed.