# Actual infinity vs Potential infinity

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Lately I've been arguing with one of William Lane Craig's drone lackeys on his Facebook page. On the one hand this person is reciting the common mantra that 'infinities cannot exist in reality', and on the other hand he is claiming to hold to a classical view of space-time. When I pointed out to him that these two views are incompatible since if space truly is classical in nature then there would be an actual infinity quantity of spatial locations within space-time, he retorted that that is 'just a potential infinity, not an actual one!'

Now, I think I know enough about calculus to be pretty sure that he is simply misusing terms here, but on the off chance that I am the one who is mistaken, I just wanted to ask for a second opinion.

My understanding is that a potential infinity refers to the limit of a function over time that goes to infinity. For example, if someone asked you how many numbers can be counted (i.e. 1, 2, 3, 4, 5, 6... etc. ad infinitum), that would be an example of a potential infinity, since you couldn't actually count to infinity. However, if someone asks how any possible numbers actually exist, that would be an actual infinity.

Similarly, if we assume that space-time is classical in nature (which assumes that space is infinitely divisible), the number of spatial locations that you could actually count, or attain through a function that continuously halves the number of points, etc. would be a potential infinity, but that if you asked how many spatial locations/points would actually exist given a classic view of space-time, that would be an actual infinity by definition.

Am I more or less understanding this properly?

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Lately I've been arguing with one of William Lane Craig's drone lackeys on his Facebook page. On the one hand this person is reciting the common mantra that 'infinities cannot exist in reality', and on the other hand he is claiming to hold to a classical view of space-time. When I pointed out to him that these two views are incompatible since if space truly is classical in nature then there would be an actual infinity quantity of spatial locations within space-time, he retorted that that is 'just a potential infinity, not an actual one!'

Now, I think I know enough about calculus to be pretty sure that he is simply misusing terms here, but on the off chance that I am the one who is mistaken, I just wanted to ask for a second opinion.

My understanding is that a potential infinity refers to the limit of a function over time that goes to infinity. For example, if someone asked you how many numbers can be counted (i.e. 1, 2, 3, 4, 5, 6... etc. ad infinitum), that would be an example of a potential infinity, since you couldn't actually count to infinity. However, if someone asks how any possible numbers actually exist, that would be an actual infinity.

Similarly, if we assume that space-time is classical in nature (which assumes that space is infinitely divisible), the number of spatial locations that you could actually count, or attain through a function that continuously halves the number of points, etc. would be a potential infinity, but that if you asked how many spatial locations/points would actually exist given a classic view of space-time, that would be an actual infinity by definition.

Am I more or less understanding this properly?

Well, potential infinity and actual infinity go along with their descriptors with the fact that something has the potential to be infinite while the other is actually infinity.

For example, there is a difference between a set that has the potential to be infinite and a set that is actually infinitely large. One has the capabilities to be infinite because no constraints have been put in place for that set to not be infinite while a set that is actually infinitely large does have infinite sets.

A limit would be a mediocre example because though it does seem to imply such comparisons, it doesn't do a very good job of it. For example, Zeno's paradox would not be a good example.

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

This is an example of a limit because it shows that if we were to finitely calculate the distance he travels he would never reach the tortoise. However, if a limit is used then he does reach the tortoise. The potential infinity stage is that Achilles has the potential to reach the tortoise because of the equation behind his differences of distance between him and the tortoise. However, with actual infinity he does reach the tortoise.

The difference lies within possibility and actuality.

Now, I wanted to address something you wrote in the post:

On the one hand this person is reciting the common mantra that 'infinities cannot exist in reality', and on the other hand he is claiming to hold to a classical view of space-time. When I pointed out to him that these two views are incompatible since if space truly is classical in nature then there would be an actual infinity quantity of spatial locations within space-time, he retorted that that is 'just a potential infinity, not an actual one!'

I don't know what he meant by "infinities cannot exist in reality" because they do exist in nature, however not in the ways we would expect them in mathematics. As we know of, black holes either have a really, really, REALLY large density or their density is infinitely large because of how they work. But, I think the person misrepresented Craig's arguments since I have watched some of the debates between him and Lawrence.

Craig's argument deals with the idea of an infinite lapse of time at the beginning of the Universe. His argument states that if there is no creator of the Universe and that the Universe has always existed then the beginning of time has, in fact, no "beginning" because the Universe has existed for an infinite amount of time and, therefore, there is no way to determine beginning relative to a certain aspect in time.

However, I think Lawrence's rebuttal to the argument was that since space-time at the singularity did not allow time to pass, therefore this would account for such an issue but I do not remember so don't quote me on this.

Just to note, this post is not in favor of either Craig's or Lawrence argument. I simply giving a clarification.

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Lately I've been arguing with one of William Lane Craig's drone lackeys on his Facebook page. On the one hand this person is reciting the common mantra that 'infinities cannot exist in reality', and on the other hand he is claiming to hold to a classical view of space-time. When I pointed out to him that these two views are incompatible since if space truly is classical in nature then there would be an actual infinity quantity of spatial locations within space-time, he retorted that that is 'just a potential infinity, not an actual one!'

In that case, you are right and he is wrong.

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In that case, you are right and he is wrong.

And the reason is...?

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If you look at integrals or derivatives or really any concept in calculus, doesn't it seem to suggest that there's cases where infinity and "potential" infinity can be treated as the same thing? Like since I have to add literally an infinite amount of boxes to get the exact area, or get literally infinitely close to a point to get a perfectly accurate derivative. The main problem I think is defining where the difference is between mathematics and the way reality actually works. If I walk two meters per second, did some invisible point on an invisible number line actually get infinitely close to some other invisible point that represents my velocity? Not that I see, but that derivative model can still work. If you have space, and you say "there's no limit to space," why not just treat that as an infinite amount of space? Limits are just loop-holes to originally not being able to deal with infinity, they exist just so that we can get a sense of what a situation is actually like when some variable is infinite because we have no real mathematical way of modeling what infinity really is since we can only define it as a man-made concept and not an inherent part of mathematical logic. Like there's no "infinity" on a number line, there's no defined answer if I say "what's 1/infinity?"

However, I think Lawrence's rebuttal to the argument was that since space-time at the singularity did not allow time to pass, therefore this would account for such an issue but I do not remember so don't quote me on this.

Wait, if no amount of time was allowed to pass, doesn't that means there was 0 time that the universe wasn't in existence? How long was the universe non-existent for? Well time didn't exist to count how long the universe wasn't in existence for, so...the universe was never non-existent.

Edited by SamBridge

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Wait, if no amount of time was allowed to pass, doesn't that means there was 0 time that the universe wasn't in existence? How long was the universe non-existent for? Well time didn't exist to count how long the universe wasn't in existence for, so...the universe was never non-existent.

That's the point of potential and actual infinity. There is the potential of the universe's existence through time moving forward, but the actuality of the matter is it has not reached that state. The actual infinity would arise from the universes's existence through time moving forward. Is this what you are asking about?

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That's the point of potential and actual infinity. There is the potential of the universe's existence through time moving forward, but the actuality of the matter is it has not reached that state. The actual infinity would arise from the universes's existence through time moving forward. Is this what you are asking about?

Well if there's no physical space that's not a "potential infinity", it's simply not infinite space. But, we don't really have enough information to determine what type of situation the universe was created from anyway, I just thought it seemed weird to use that explanation to say that the universe wasn't infinite because it would imply the universe was never nonexistent which is sort of paradoxical because then you should have infinite space if there was no amount of time that passed where space wasn't in existence. Is there an actual difference in this scenario between saying "infinite space" and "space extends indefinitely, we keep counting units of distance with no end in sight" since we don't have any reason to put a boundary on space?

Edited by SamBridge

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Well if there's no physical space that's not a "potential infinity", it's simply not infinite space. But, we don't really have enough information to determine what type of situation the universe was created from anyway, I just thought it seemed weird to use that explanation to say that the universe wasn't infinite because it would imply the universe was never nonexistent which is sort of paradoxical because then you should have infinite space if there was no amount of time that passed where space wasn't in existence. Is there an actual difference in this scenario between saying "infinite space" and "space extends indefinitely, we keep counting units of distance with no end in sight" since we don't have any reason to put a boundary on space?

I think the problem lies in the thinking of determining what is and what it could be. For example, potential gravitational energy converts to kinetic energy as an object falls. Though the object hasn't fallen it has the potential of that energy. Therefore, the potential becomes the actual. However, the potential may not always because the actual.

...which is sort of paradoxical because then you should have infinite space if there was no amount of time that passed where space wasn't in existence.

It isn't paradoxical. Just because a seed can grow to be a flower doesn't mean the flower already exists while no time has passed. The seed has the potential, and the predictability of it is, to become a flower.

Is there an actual difference in this scenario between saying "infinite space" and "space extends indefinitely, we keep counting units of distance with no end in sight" since we don't have any reason to put a boundary on space?

There would be a difference, which is why the two forms should be separate, where there is potential and actual. The potential and actual, at one particular time, cannot be equal. However, on a broad scale they are equal. For example, if you were to take a limit n->infinity of 1/n = 0, which is representative of potential, then if we chose a finite value n then it would not equal 0.This fraction has the potential to become 0, where the actual is 0, but the actuality of the fraction is it will only get closer and closer to 0.

Edited by Unity+

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I think the problem lies in the thinking of determining what is and what it could be. For example, potential gravitational energy converts to kinetic energy as an object falls. Though the object hasn't fallen it has the potential of that energy. Therefore, the potential becomes the actual. However, the potential may not always because the actual.

But so what? If it happens it happens, if it doesn't it doesn't.

It isn't paradoxical. Just because a seed can grow to be a flower doesn't mean the flower already exists while no time has passed. The seed has the potential, and the predictability of it is, to become a flower.

Well a flower isn't a big bang or a singularity and doesn't approach infinity in any similar way to those objects. He said it himself, no time passed, so I can extrapolate that therefore, there was zero time that the universe was non-existent for if what he said is actually true.

There would be a difference, which is why the two forms should be separate, where there is potential and actual. The potential and actual, at one particular time, cannot be equal. However, on a broad scale they are equal. For example, if you were to take a limit n->infinity of 1/n = 0, which is representative of potential, then if we chose a finite value n then it would not equal 0.This fraction has the potential to become 0, where the actual is 0, but the actuality of the fraction is it will only get closer and closer to 0.

You keep saying potential as if it's a real thing, but all you're really saying is "we didn't confirm it yet." or "we didn't get to it yet." Limits deal with infinities because infinities and divisions by zero are often indeterminate on their own, they could literally yield any answer, you could literally put 0 into a something else any amount of times you want, but, there's no mathematical property to determine how many times 0 actually goes into something. So, the specific equations that involve those things use limits as a loop-hole to see what happens at infinity and divisions by zero in specific circumstances, that's all they are, loop-holes. And I also don't see the value in what you're saying. So what if a finite number never gets to 0? So what if an infinity small divisor does? The limit and the proposed value agree. We say but can't prove it's infinite, and we also say the limit approaches infinity, so two different engagements agree, so we might as well treat them as the same thing if we have no more evidence to consider. Or what about sin(x)/x? Visually and philosophically, it looks like the ratio approaches 1 when x approaches 0, and mathematically, we define that limit as approaching 1 anyway, so we have some evidence to suggest that sin(0)/(0) = 1 for that specific circumstance if we express those numbers as directly relating to the scenario we proposed. On its own, sin(0) would just be another random constant, but the difference is that we assign meaning to those numbers and what they represent when we use a limit on a visual reference, like angles and ratios.

Similarly, we say space extends indefinitely because we can't find a boundary, but also say it could be mathematically infinite because flat space doesn't loop back in on itself and "nothingness" has no dimensional size. We see no boundary, we keep counting distance with ease, so we just assume it's infinite, we don't have a lot of evidence to suggest otherwise. "Potential" is more like a test, like a limit, a loop-hole to see what the answer likely will be since we can't just plug in values, so it can easily agree with a logical concept or result. Physically, we say based on our observations, the universe appears to get denser back in time. You could say it has a "potential" to be infinitely small because the limit of its density as time approaches present time minus 13.8 billion years approaches infinity, so that combined with our knowledge of black holes, we say the observable matter and energy was some nearly infinitely small, singularity-like sphere or point.

If you talk about "potential energy" that's a whole different thing.

Edited by SamBridge

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You keep saying potential as if it's a real thing, but all you're really saying is "we didn't confirm it yet." or "we didn't get to it yet." Limits deal with infinities because infinities and divisions by zero are often indeterminate on their own, they could literally yield any answer, you could literally put 0 into a something else any amount of times you want, but, there's no mathematical property to determine how many times 0 actually goes into something. So, the specific equations that involve those things use limits as a loop-hole to see what happens at infinity and divisions by zero in specific circumstances, that's all they are, loop-holes. And I also don't see the value in what you're saying. So what if a finite number never gets to 0? So what if an infinity small divisor does? The limit and the proposed value agree. We say but can't prove it's infinite, and we also say the limit approaches infinity, so two different engagements agree, so we might as well treat them as the same thing if we have no more evidence to consider. Or what about sin(x)/x? Visually and philosophically, it looks like the ratio approaches 1 when x approaches 0, and mathematically, we define that limit as approaching 1 anyway, so we have some evidence to suggest that sin(0)/(0) = 1 for that specific circumstance if we express those numbers as directly relating to the scenario we proposed. On its own, sin(0) would just be another random constant, but the difference is that we assign meaning to those numbers and what they represent when we use a limit on a visual reference, like angles and ratios.

They aren't "loop-holes." Loop holes give inconsistencies, which are a no-no in mathematics. The point I was giving was if at any given point, or specific point, you would try to see if the potential and actual were equal this would never occur, but if the potential is measured similarly with the actual then they yield equality.

But so what? If it happens it happens, if it doesn't it doesn't.

It is as if you are saying that any result or conclusion gives nothing to the discussion of hypothesis at hand. We don't just say "If it happens."

If you have heard of Zeno's paradox, you might want to consider that as my argument.

Well a flower isn't a big bang or a singularity and doesn't approach infinity in any similar way to those objects. He said it himself, no time passed, so I can extrapolate that therefore, there was zero time that the universe was non-existent for if what he said is actually true.

I must clarify that his argument went more into depth about how if there was no creator then there is no time of causality of the Big Bang and therefore there are many paradoxes that arise. I'm still getting confused about how you came to the conclusion that he is saying that the Universe never had a time of non-existence.

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They aren't "loop-holes." Loop holes give inconsistencies, which are a no-no in mathematics. The point I was giving was if at any given point, or specific point, you would try to see if the potential and actual were equal this would never occur, but if the potential is measured similarly with the actual then they yield equality.

It is as if you are saying that any result or conclusion gives nothing to the discussion of hypothesis at hand. We don't just say "If it happens."

"If the potential is measured similarly"? I don't know what that means. Are you tying to say the lim(1/x)x->0 = infinity is the same is plugging in 0 in 1/x? Sometimes there's evidene to suggest it is, sometimes there isn't. A sonic boom mathematically gives you sound waves that approach infinite frequency, but you never physically see a sound wave of infinite frequency.

It is as if you are saying that any result or conclusion gives nothing to the discussion of hypothesis at hand. We don't just say "If it happens."

A limit may coincide with results or it may not coincide with results, infinities and divisions by zero are indeterminate on their own, there's no generalization that can be made about what they will "always" do, it depends on the specifics of the situation they are more of a test or a loop-hole to deal with instances where you can't just plug in numbers.

I must clarify that his argument went more into depth about how if there was no creator then there is no time of causality of the Big Bang and therefore there are many paradoxes that arise. I'm still getting confused about how you came to the conclusion that he is saying that the Universe never had a time of non-existence.

Paradoxes don't have to rise if you treat it mathematically, or if you don't think of time as some special dimension that means everything. If before the big bang, time existed, then we just start counting into negative numbers on the timeline before the big bang and say all the matter and energy was some random fluctuation. If you say the big-bang was when time was created instead of just the matter and energy we see which there's no particular reason to believe, it only has paradoxes if you say it started at some arbitrary number like 0 and then imposed out of no where that it wasn't allowed to pass for a certain amount of time, since there's no real explanation for why it wouldn't pass. Everything we know says time does pass, if everything was in a singularity, then everything would have had the same frame of reference and time would have still passed for all the matter and energy inside that singularity. Saying when a mathematical construct (like the number line of time) was created can't really mathematically tell you about what went on before it was created. That's like asking "how did we calculate 1+1=2 before we had 1+1=2" the answer is we didn't, there was nothing to calculate. If time didn't exist prior to the big bang, there was no dimension to constitute any finite duration of time that the observable universe existed as a singularity for, and that's it, and but only comes from saying time had an arbitrary starting point, you're trying to apply it's own counting system to mathematics where it can't count, like saying "can you do something you can't do"? I don't think time likes it when you tease it like that, and on its 13.8 billionth anniversary too.

Edited by SamBridge

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Paradoxes don't have to rise if you treat it mathematically, or if you don't think of time as some special dimension that means everything. If before the big bang, time existed, then we just start counting into negative numbers on the timeline before the big bang and say all the matter and energy was some random fluctuation.

That make sense and I can't really argue against that at the moment.

"If the potential is measured similarly"? I don't know what that means. Are you tying to say the lim(1/x)x->0 = infinity is the same is plugging in 0 in 1/x? Sometimes there's evidene to suggest it is, sometimes there isn't. A sonic boom mathematically gives you sound waves that approach infinite frequency, but you never physically see a sound wave of infinite frequency.

Therefore, there is a potential for that sound wave to reach its point, which is my point. Which is similar to my limit example.

A limit may coincide with results or it may not coincide with results, infinities and divisions by zero are indeterminate on their own, there's no generalization that can be made about what they will "always" do, it depends on the specifics of the situation they are more of a test or a loop-hole to deal with instances where you can't just plug in numbers.

Again, they are not loop-holes.

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Therefore, there is a potential for that sound wave to reach its point, which is my point. Which is similar to my limit example.

Physically, there isn't the potential, there's no physical way to have a sound wave of physically infinite frequency because that would require infinite energy. The limit and the predicted conceptual result of plugging in infinity can differ as much as they agree, you can't generalize 1/0 and 1+x = infinity for all situations, they are indeterminate results and so there's no concise pattern that you can draw, it depends on the situation.

Again, they are not loop-holes.

Well that's what limits were developed for.
Edited by SamBridge

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Physically, there isn't the potential, there's no physical way to have a sound wave of physically infinite frequency because that would require infinite energy. The limit and the predicted conceptual result of plugging in infinity can differ as much as they agree, you can't generalize 1/0 and 1+x = infinity for all situations, they are indeterminate results and so there's no concise pattern that you can draw, it depends on the situation.

Well that's what limits were developed for.

http://www.intuitive-calculus.com/limits-and-continuity.html

Limits were developed because regular algebra can't view the "extra dimension" that exists. For example, 1/0 doesn't make sense. Therefore, we can define some form of it by using a limit to see what would happen. That isn't a loop-hole.

Physically, there isn't the potential, there's no physical way to have a sound wave of physically infinite frequency because that would require infinite energy. The limit and the predicted conceptual result of plugging in infinity can differ as much as they agree, you can't generalize 1/0 and 1+x = infinity for all situations, they are indeterminate results and so there's no concise pattern that you can draw, it depends on the situation.

Physically, there is no see able potential, but mathematically there is.

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http://www.intuitive-calculus.com/limits-and-continuity.html

Limits were developed because regular algebra can't view the "extra dimension" that exists. For example, 1/0 doesn't make sense. Therefore, we can define some form of it by using a limit to see what would happen. That isn't a loop-hole.

Physically, there is no see able potential, but mathematically there is.

http://dictionary.reference.com/browse/loophole

Its the mathematical law that 1/0 is undefined, yet we get close to an apparent answer with limits.

http://www.uiowa.edu/~c22m025c/history.html

Saying "infinitely small number" didn't work, so limits were devised.

If mathematically there is but physically there isn't, then that just proves my point. Limits and real results and logical concepts can agree or disagree at different times, that's it, there's no pattern to generalize.

Edited by SamBridge

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http://dictionary.reference.com/browse/loophole

Its the mathematical law that 1/0 is undefined, yet we get close to an apparent answer with limits.

http://www.uiowa.edu/~c22m025c/history.html

Saying "infinitely small number" didn't work, so limits were devised.

If mathematically there is but physically there isn't, then that just proves my point. Limits and real results and logical concepts can agree or disagree at different times, that's it, there's no pattern to generalize.

But you aren't evading the laws and axioms of Mathematics for limits...

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But you aren't evading the laws and axioms of Mathematics for limits...

But you're still getting around the fact that you can't mathematically divide something by zero. Either way, this is purely a semantic argument, it's just a tool for testing what happens at very large or very small values. Sometimes it works, sometimes it doesn't.

Edited by SamBridge

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But you're still getting around the fact that you can't mathematically divide something by zero. Either way, this is purely a semantic argument, it's just a tool for testing what happens at very large or very small values. Sometimes it works, sometimes it doesn't.

Yes, I think we got way off topic with that one.

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