Is an Actual Infinity Actually Possible?

I don’t think a true infinite is compatible with reality. More likely would be a potential infinity, which wouldn’t have these warts.

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The universe has a long history of disregarding our intuitions about what is compatible with reality.


In this case I’m not talking about intuitions but logic. Do you know what I meant by actual vs potential infinite?

I know what those terms ordinarily mean, which may or may not be what you mean by them.

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Why not? This can be moved to a different thread.

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So, here is some starting points [not implying endorsement!]:

(@PdotdQ informs me this one is crackpottery)

Infinity in Cosmology: Infinite | Internet Encyclopedia of Philosophy

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From Patterson:

Okay, already I’m inclined to ignore this person.

That definition assumes the conclusion that he wants to reach. It is possible to define, say, the set of real numbers between 0 and 1 through a never completed counting process. It’s also possible to define them as a a complete, infinite set. There is no logical basis for insisting that only the first description is valid.


@glipsnort that was the most informal article of all them, for the non-technical readers. You can skip it if you if you like, also I do not endorse everything in that article. The intention of those links is to prime the pump.

@glipsnort, highly relevant is this thread, Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.

Though we are discussing the size of the universe, there is a related question as to whether or not true irrational numbers (with infinite digits and infinite precision) actually exist or not. A key point, and I think this is important, is that if they do exist, it would seem a lot of work in physics, such as time irreversibility and conservation of information, are no longer coherent.

The best book you can read on this:

I did not see anything in those links that implied the spatial extend of the universe could not be actually infinite.

What you seem to have provided links that argue against, is the position that an actual infinite could be sort of incrementally generated. But I could accept that for the sake of argument, and it would not show that the universe could not be infinitely big. Just that it couldn’t become infinitely big by adding finite amounts together, if it was at some point finite in the past.

But that is not what I am suggesting. I am not claiming the universe began with (or at some point was) a finite size, and then grew to be infinitely big by successive addition of finite elements. Rather, my hypothetical here is that the universe is infinitely big, and that there was never a time at which it was not.

You could envision for example that our local cosmic expansion (hereafter defined as the visible universe) is just one such out of an infinite number of similar cosmic expansions, all connected to each other(we could imagine here being at the furthest observable distance from our current position on Earth, and from that position being able to see further still into another such cosmic expansion). And the total number of such cosmic expansions could each be represented by one of the infinite number of positive and negative integers. Thus the total number of them would be actually infinite, and thus an actual infinite would exist.


They exist, in the mathematical sense of exist. They do not exist in the ordinary sense of exist.

For that matter, the number 1 exists in the mathematical sense of exist, but not in the ordinary sense of exist. That’s the viewpoint from mathematical fictionalism.

The numeral “1” (the pencil marks on paper) can exist. But the number is not the same as the numeral.

You have to be careful with this. When you are working with the wave equation, time looks reversable. When you are looking at the evolution equation (at thermodynamics) time looks irreversible.

The conservation of information was never coherent (in my opinion).

Yes, I pretty much agree.

Perhaps he finds some parts of mathematics are a bit difficult to understand. His mistake is to assume that there is a problem with the mathematics itself.

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I tend to believe we don’t understand causality or that somewhere “at the edges” our intuitions are probably wrong.

This is not true. Time irreversibility in physics is not attacked at all by the existence of infinities. Indeed, there are many (actually, most) formulations of the arrow of time that is formulated with real numbers and works perfectly fine in infinite universes. Further, once an arrow was picked, there is no motion that one can do to go backwards in time (locally); this is a straightforward theorem from special relativity.

The same thing goes with conservation of information, which in quantum mechanics just refers to the unitarity of physical processes.

I am not sure if @swamidass is aware, but Patterson is a well known crank.

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Well I’m not aware, but I do take your word for it and I stand corrected. Thanks.

You can travel for an infinite length across the surface of a globe without ever going over the edge. Would this count as an infinity of sorts?

My physics-fu is weak, but could we also apply this concept to a black hole where space and time run into infinities, or are those infinities supposed to tell us something is wrong with relativity?

You touched on something very important in physics.

This property is called “geodesic completeness”. In a space with geodesic completeness, any “straight” line can be followed indefinitely. A theorem (Hopf-Rinow) states that any compact manifold (of which the globe is an example) is geodesically complete.

It’s actually the opposite: a singularity (or at least he most common notion of it) in general relativity is when a straight line terminates! If a straight line terminates (i.e., you cannot follow it indefinitely), this is because it hits something bad. We call this bad thing “singularity”.

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What do you think about a Platonist position (in the vein of Van Inwagen,, where actual infinities, like actual numbers and shapes, exists, but that these are abstract things, as opposed to concrete things?

Thank you, that makes more sense. I did run into this figure, and it does mention infinities in the caption. I’m not saying you are wrong by any means, but I continue to try and wrap my head around it:

Figure 13.30 The space distortion becomes more noticeable around increasingly larger masses. Once the mass density reaches a critical level, a black hole forms and the fabric of space-time is torn. The curvature of space is greatest at the surface of each of the first three objects shown and is finite. The curvature then decreases (not shown) to zero as you move to the center of the object. But the black hole is different. The curvature becomes infinite: The surface has collapsed to a singularity, and the cone extends to infinity. (Note: These diagrams are not to any scale.)