Applying logic to the question of whether all of existence is infinite or not

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:

Part 1

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.

Part 2

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?

EDIT:

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.

• That everything exists is a position in semantics held by some philosophers, see SEP. However, that all of existence is spatially infinite is trivially false. "Absolutely everything" includes abstract objects, like number 1 or cyclic groups, that are not confined to space and time at all. Applying spatiotemporal predicates to them is nonsensical. This said, paradoxes of infinity do not arise from infinity as such, we have perfectly consistent math theories of infinity. Aug 19, 2022 at 3:48
• If you believe that numbers exist, then it is trivial to prove that "existence" is infinite because there are infinite numbers. Aug 19, 2022 at 3:50
• The ancient Buddhist Chandrakirti (meaning equal-weight of the Moon) famously claimed: There is no chariot which depends on its parts... Once you fully understand this your core claim all of existence is infinite in space and time could go through the Dharma door and its proof and logic involved is not that different from Spinoza's... Aug 19, 2022 at 4:17
• [1/2] @Conifold See my edit. Infinity as a mathematical concept might be consistent, but it is rational? All of classical logic follows from axioms, and they are axioms because they are so rational. Something may be logical in that it is consistent with the framework, but is it logical in and of itself? Or maybe its consequences are consistent within the framework, but are they logical in and of themselves? The paradoxes I am talking about may be consistent features/consequences, but are they logical in and of themselves? I can give an example below: Aug 22, 2022 at 17:49
• [2/2] Lines are one-dimensional, linear rows of zeroth-dimensional points. There's infinitely many of them. You can have lines of different lengths, but there will always be a one-to-one correspondence between their constituent points. Basically, you can take two otherwise identical objects of different size and break them down to their same-size components, and there will be equally many of them. It's a geometric consequence of infinity times a = infinity times b, and it is illogical in my eyes. Not the identity itself, but its geometric consequence. Aug 22, 2022 at 17:55

You define the word existence in a peculiarly nebulous way to be somehow all-encompassing, and you apply questionable reasoning that leads you to conclude that your version of existence is infinite, and then you claim that infinity is an 'illogical consequence'. Later, having reached that conclusion by defining 'existence' in one way, you apply it a question in which 'existence' means something entirely different.

Infinity itself is not paradoxical. It might be hard to come to terms with intuitively, but mathematicians, whose word I am happy to accept, assure us that they have figured out how to incorporate it logically with the rest of mathematics, so your claim is simply false.

Your arguments for time being infinite are based on a self-serving assumption about time. You say that if time had a start there must have been an early time before the start. In the absence of an accepted theory of time, that statement is purely opinion, and conflicts with the premise that time had a start. In any event, it is possible that spacetime is analogous in some way with the surface of a sphere, in that it seems to go on forever but is actually bounded. We currently have no idea what might be outside the Universe, or even whether it is meaningful to talk about anything being outside it.

Your follow-on post contains statements such as

total inexistence is a state in which absolutely nothing exists. This state exists, but it has no instantiation (yet). One might think then that the above rebuttal is true; it cannot be instantiated, because if it were instantiated, it would then contradict itself

and

In fact, it can't really end, because if it ends, it will never have existed in the first place (and thus, it won't have an end).

in which you are simply confusing yourself by your vague use of the word exist.

As far as I understand your definition of "to exist", it applies to abstract things like natural numbers. There are infinite natural numbers, so the infinite clearly exists. Infinity is not illogical at all, but even if it were, your definition of "to exist" seems so broad of a property that there is no concept or object that does not have this property, even "illogical" ones. Paradoxical things are explicitely included, so if we arrive at some paradox in the reasoning, that is simply in the nature of that definition. It carries no information, and in fact is just another way of saying that from falsehood, anything can be concluded.

All of existence is infinite in space and time though seems to make little sense because that seems to assume a physical extension of "all of existence" which you said there is not. So existence may apply to everything part of "all of existence", but time and space are not properties of everything in "all of existence" (things without physical extension don't seem like the concept of space and time applies to them).

So the sentence seems nonsensical in that it combines things that make no sense to combine.

• [1/2] Your second paragraph is answered by my edit. Apologies for my imprecision. As for your first paragraph, you raise good concerns. The explosion principle is contentious I believe, and I find it arbitrary. To say that everything follows from a falsehood is just as reasonabe as to say that nothing follows from a falsehood. It's all just semantics, really, and the choice was made due to pratical reasons, not so much philosophical ones. So disproving anything on the basis that it is trivially true via the explosion principle is quite weak. Aug 22, 2022 at 21:08
• [2/2] The fact is, I start out with a contradiction, which is the concept of the property to exist. From that, my conclusion (and its negation) is already derivable in one, mindless step, assuming the principle of explosion. However, I am able to prove the conclusion through other means. Can you prove a contradictory conclusion through other means from my premise as well? I don't think so. And that's important. I think that this is simply just an arena of thought where the explosion principle, born out of practical concerns, simply isn't applicable. Aug 22, 2022 at 21:11

Newtonian physics is based on the ideas of three infinite spatial dimensions, one infinite dimension of time, and particles and waves that operate within. While this is highly effective for many purposes, such as the design of toasters and bridges, it’s important to remember that it is no more than a model. One might equally well propose that space and time are properties of objects since distance is meaningless unless it’s a distance between two points, and time unless it’s between two events. By that logic, the idea that time has to be infinite carries no more weight than the idea that the abstract concept of a teapot has to be infinite. Set theory allows us to create infinite, paradoxical and nonsensical sets (because it lacks any rules to say that we can’t) but it’s unwise to extrapolate into the real world.

Having defined your use of the words all and existence as you have, and made the inference rules count as "logical" that give you the conclusion you're looking for already, it seems your conclusion does indeed follow: trivially, however.

In fact, it's not clear that the OP reasoning is coherent or even "merely" consistent. Depending on your opinion about the explosion problem in logic, if the OP reasoning isn't coherent, the conclusion follows even more trivially, since every possible conclusion follows, including the desired one. For example, if impossible things exist (as is said early on) but if the OP writer also says:

That is a reflexive definition, but to exist isn't a composite concept, so a non-reflexive definition doesn't exist.

... then are we not confronted with a case where an impossible thing both does and does not exist? For an abstract definition would presumably be a property, on the given account, so it should exist, despite being paradoxical and/or impossible in the sense of never-possibly-instantiated. It might be a false property to ascribe to existence in itself, but it is still a property that exists, isn't it?

Moreover, the references to paradoxical assertions involving the concept of infinity is fairly amorphous. It would not do to expect everyone who asks a sustained question, here, to have to go into exorbitant detail (including to the kinds of citable sources that they might be looking for!) about every presupposition of the question. However, since as it turns out there are both a multitude of paradoxical assertions involving infinity, as well as a multitude of assertions involving infinity that are not paradoxical, it is hard for the general audience to tell why the OP writer thinks that "infinity is paradoxical" and hence a threat to the intended conclusion of the OP argument overall. With respect to Kant, for instance, it is easy to think of the antinomies; among set theorists, we might refer to something in model theory that has been called a paradox, even if it isn't quite such a beast, i.e. the Skolem "paradox" (surely the Skolem problem, at least, though). Does the OP wish to imply all these examples, even such as they are not yet familiar with (or are they familiar with almost all, if not all, major examples?), or only some of them? Because again, it would be one thing to invoke the spirit of Kant, another the spirit of Skolem: for in the train of Skolem's ghost will float forth out of Cantor's abyss the ghosts of many others for whom the transfinite numbers are not so demonic, for whom the abyss is a paradise, no less.

Normally, I don't put much stock in keeping the terms rational and logical too separate in ordinary English. There has been at least a widespread usage slippage for logical from an abstract/esoteric illustration of metaphysical syntax to something close to the same as what rational covers (perhaps differentiable only by the mechanical character of the sound of the word logical vs. the sound of the word rational (c.f. the Koko/Bobo phenomenon)). However, I do wonder if it would help the OP's reasoning to separate the senses of those words more?

P.S. Not to say that I disagree with, "Nothing that exists, exists outside of spacetime." Not at all! However, I do think that there is a "number," V, that is the "number of all other numbers," including all the transfinite ones. So at any rate, I do imagine that there are not only infinite-dimensional possible spacetimes (IDK how far the concept of infinite-dimensional Hilbert spaces can be physicalistically pushed, though), but that difference scales of infinity can be used to cash out the generic description. This leads up to the idea, though, that we might be able to speak sensically about V-dimensional spacetime, which would be outside of relatively infinite spacetimes, even if those were "elementarily embedded" (nontrivially!) into it (or V-spacetime was embedded into them, even). For V is absolutely infinite in scale, so it encompasses all relatively infinite scales. Then everything that exists must exist in some spacetime, but there is a spacetime outside of all other spacetimes as well, and the darker antinomy of the OP mystery is dissolved in a flash of alien light.

The conclusion is true, but a part of the argumentation is false

Specifically, the rebuttal of the rebuttal is false, quoted below:

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.

Total inexistence is a state in which absolutely nothing exists. This state exists, but it has no instantiation (yet). One might think then that the above rebuttal is true; it cannot be instantiated, because if it were instantiated, it would then contradict itself.

No. If it were instantiated, the property of existence itself would cease to exist. Heck, even it's so-called paradoxical instantiation wouldn't exist, along with everything that has ever happened and ever will happen. The past, the future, time itself, every concept and object, and everything itself, doesn't exist.

Thus, its definition is not its own foil, but rather its own fail-safe. Now, we have no reason to rule out that total inexistence may one day come about. We can rule out that it will necessarily come about, as if that was true, we wouldn't exist right now. Thus, we are left with its possible instantiation in the future. We do know, however, that it has never been instantiated before, as then we wouldn't exist either. Thus, by the argument posed in the question, all of existence has existed for an infinitely long time. It may continue to exist for all eternity, or it may end; if it does end however, it will never be preceded by anything again. In fact, it can't really end, because if it ends, it will never have existed in the first place (and thus, it won't have an end).

On Spatial Infinity

"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."

On Time Infinity

"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."

This goes against definition of time. Time is defined by motion.

To have motion you have to have something that can move. If nothing exist then nothing can move. There is no time then at that point.

D - duration of all existence - being finite means time is finite.