Here, I use to exist as generally as possible; if it is an object, it exists; if it is conceivable, it exists; if it is anything, it exists; even the properties and relations themselves exist. Existence as a property, applies to absolutely everything, even paradoxical and impossible things. Those impossible things do not have an instantiation (by definition of impossible), but under this most general definition of existence, they do exist. All of existence is the collection of everything that exists; thus, it's everything.
*To have an instantiation means to be actualized in the relevant manner. An instantiated physical object has a physical extension, an instantiated property is applicable to an object that is instantiated, or to a property that is applicable to an object that is instantiated, or to a property of a property that (...), etc. A relation is instantiated if it relates two objects/properties/relations that are instantiated. Sometimes, state is used identically to property. Other times, it refers to both the the object(s), and the relation(s)/propert(y/ies). The latter is instantiated if everything within is instantiated.
Here's a pretty logical explanation for why all of existence must be infinite in space and time:
An object O cannot exist outside of all of existence, because if it exists, it is a part of all of existence, by definition of all and existence. This is simply an a priori truth, as it follows from the definition. Thus, all of existence is spatially infinite.
This applies both to space and time. Thus, there is no outside of existence, nor is there before or after. I've heard this being called the Closure Principle, and it is completely logical as far as I see it. The next part though, I'm a bit more iffy about.
First, a definition. D is the duration of the physical extension of all of existence; that includes completely empty space too. Now, here's two propositions:
P = D possesses a duration before, and a duration after.
Q = D is of finite length
P if Q, since if it is finite, it is made of n units of time. One of those units must be the first, and one of those must be the last. Therefore, if the duration is finite, it has a start, and an end. That which precedes the start is the time before, and that which succeedes the end is the time after. Thus, being finite necessitates having a time before and a time after.
Q if P, since if there is a time before the duration, then that time must precede something. That something must be the start of the duration. If there is a time after the duration, then that time must succeed something. That something must thus be the end of the duration.
Therefore, Q iff P. Since we know not-P is true for all of existence due to the Closure Principle, we know that not-Q is also true for all of existence. The negation of finitude is infinitude. Thus, all of existence is infinite in time as well.
The only rebuttal of Part 2 that I can think of is this:
There can be a start and an end, without a time preceeding that start/end. What if before the start and after the end, there is total inexistence (and thus, no time)? If so, all of existence is simply a finite duration that is preceded and succeeded by total inexistence. Basically, this rebuttal just says not-(Q if P), but it does so by being open to the possibility of total inexistence.
However, I think this rebuttal fails. Given this definition of to exist, total inexistence must exist. That does make to exist a paradoxical property, as it applies to its own negation. However, a property doesn't have a physical extension anyways, so the paradoxicality of to exist changes nothing. A property is merely the property-holder's satisfaction of some set of propositions; it's an abstract thing, not a physical object. To exist is simply the property in which the proposition required to be satisfied is that of being something. That is a reflexive definition, but to exist isn't a composite concept, so a non-reflexive definition doesn't exist.
Here's the thing though. This is all logical and all that, but it has an illogical consequence: infinity! There are tons of paradoxes that arise from infinity, yet this argument necessitates that at least something has the property of being infinite. Perhaps these paradoxes only arise when one uses the concept of infinity in certain ways? Perhaps this is the only non-paradoxical application of infinity? Applying the concept of temperature onto individual photons is illogical, but that doesn't mean the concept of temperature is illogical in and of itself. Perhaps the paradoxes of infinity have arisen due to our paradoxical application of it, and not due to it being inherently paradoxical?
I guess that hinges on whether there are any contradictions arising from stating that all of existence is infinite is space and time. If so, then we're at an impasse, where regardless of what we choose, we derive absurdity.
So, that's basically my question. Does saying all of existence is infinite in space and time lead to absurdity?
When I say all of existence is infinite in time and space, I am talking about the parts of existence to which those concepts apply. That all of existence has an infinite amount of content is of course trivial, as there's infinitely many numbers, for example. However, just looking at the parts that have a physical extension, they must exist within an infinitely large space and time.