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As far as I see, there are only three ways to have infinity apples in the present.

  1. they existed eternally

  2. they materialized at an infinite rate

  3. they were added one apple at a time every second from eternity (infinite time)

However, when contemplating this it seems one is trapped with contradictions.

For example, removing an apple should not decrease the amount, since infinity cannot be decreased. If so, I can remove many apples and create a new infinity from the first. Then, do the same with the second infinity and create a third infinity, and so forth infinitely, which seems absurd.

Is it ever possible to have an actual infinity in the present through infinite finites, or is the whole thing inherently irrational and exists only as a mathematical imagination.

  • I think this is a duplicate of philosophy.stackexchange.com/q/7087/3867 – Paul Ross Jun 23 '13 at 8:09
  • yes, but there it was phrased poorly and closed as not a question. – user813801 Jun 23 '13 at 8:21
  • Well, what is the question here? You're talking about "infinite finites" and about infinities "in the present", but I'm not sure what sense to make of this. Are you asking something like "there aren't infinitely many things at any given point in time, so where does the idea of an object of infinite cardinality come from"? – Paul Ross Jun 23 '13 at 16:21
  • let's say atoms for example instead of "apples". can you have infinite atoms, in light of the above problem – user813801 Jun 23 '13 at 17:14
  • I still think there's something implicitly weird with what you're asking. "There are infinite atoms" makes sense as a true statement if, and only if, a correct, infinitary, mathematical theory of physics says there are. I don't see what sense you make of the idea of "removing an atom from the completed set of atoms" if not purely mathematically, and the sense of the mathematics here is related to what conceptual resources are necessary to do the physics. Maybe the infinitary language of physics lets you split infinities up into sub-infinities. If so, you'd just need to deal with it. – Paul Ross Jun 24 '13 at 12:40
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I think the issue here is your definition of infinity. You are correct in your last sentence that infinity is a mathematical imaginary convenience. It it's not possible to have an infinity of anything in the real world. In mathematical terms, infinity is not considered a "real" number. It is a useful concept to help conceptualize certain otherwise impossible operations.

For example, 1 divided by 0 is technically undefined because you can't divide something into no segments. However, this case comes up frequently when dealing with many math forms, so the concept of infinity is useful. As you divide 1 by smaller and smaller numbers, the result is a larger and larger number. Dividing 1 into any real number of segments will yield some real amount in each segment. But you can get zero in each segment if you have an unreal infinity of segments. So, technically you would say that 1/0 is undefined, but it approaches infinity.

So, to answer your question, you can make an infinite amount of infinities out of a single infinity. Mathematically, 2 times infinity is just infinity. All of the contradictions go away once you let go of infinity being a real thing.

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I think there are two sides to your question: whether the "contradictions" you state actually pose a problem, and whether it is even physically possible to have an infinity of something.

To the former, the answer is that though your description of what one could do with infinite apples "seems absurd," it is nonetheless perfectly valid. If indeed I have an infinity of apples (for simplicity let us assume it is the smallest infinity, countably many apples - a set of cardinality aleph-null), then I could take countably many apples from that set and make as many infinite sets of apples as I want - in theory. It is just a mathematical fact that one can remove countably many numbers from a countably infinite set, and if one does it right, one will get two infinite sets. And one can do that countably many times, getting an infinite number of infinite sets of apples.

However, I think the philosophically relevant issue in your question is whether this could even be possible in practice. Sure, theoretically infinitely many apples isn't a logically incoherent concept, but you want to know if it can ever happen.

Well, the shape of the Universe is presently an open question, but what looks most likely is that it is flat and infinite in extent. It is a physically coherent (and presently quite popular) theory to suggest that the Universe is infinite in size, and by corollary (assuming the cosmological principle of homogeneity and isotropy holds), that there is an infinite amount of matter in existence.

However, it's important to note that this is not a useful fact. Because of limitations such as the speed of light and the Planck scale, which both limit the amount of information we can receive, it really isn't possible to have, within one's range of observation, infinitely many things. This is a twofold argument. First, if the objects have finite size, they must take up infinite volume and thus extend to infinitely far away. Thus, in finite time (which is the only sort of time we have), we will never have information about all of the objects (in fact, we will always have information from almost none of them, mathematically speaking) since there will never be enough time for light to carry their information two us. Second, if the objects are infinitesimal small (if that is even possible), they will be effectively unobservable due to the way the Planck scale works, so we won't even see them.

So the summary is that infinity is physically coherent but not useful on any remotely familiar scale.

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