see https://en.m.wikipedia.org/wiki/Actual_infinity

But it’s impossible for to construct a smallest possible distance (1/infinity) - we can merely construct successfully smaller distances in a process that tends to but never reaches 1/infinity. That’s the definition of potentially infinite. I asked for an example from nature that is actually infinite... — Devans99

There's no "constructing" here, space is just infinitely divisible. There's no such thing as a smallest possible distance.

Well, actually in physics, space does not seem to be infinitely divisible.

I never look up links referenced during a discussion. This is for two reasons: 1. Any nonsense can be posted on the web, and often is. 2. If someone has an argument to make, then they should be able to state it in their own words.
When a mathematician lays out an infinite series, the mathematician is not stating that this is a process that goes on forever, but, rather, is actually conceiving things as if all calculations for the series have been done. As an example of how this helps science, just think back to Zeno's paradoxes. The one where one cannot possibly get out of a room, because in order to do so, one must first travel half the distance, then half the distance again, and etc., etc., so motion like leaving a room must be an illusion? The series would be S = 1 + 1/2 + 1/4 + 1/8, which can be thought of as the series from zero to infinity of 1/2 raised to power n as n goes from 0 to infinity. The series adds up to 2, which is finite, and solves the paradox --- the series collapses on a finite number and one can certainly travel a finite distance in a finite amount of time. It's rather basic, but shows how dealing with the transfinite gets us out of puzzles and advances science.

No wonder we cannot mentally conceive the Actually Infinite; it does not exist.

There's no "constructing" here, space is just infinitely divisible. There's no such thing as a smallest possible distance. — MindForged

Well, actually in physics, space does not seem to be infinitely divisible. — LD Saunders

In today's physics space and time are usually modeled as a continuum. This is true for classical mechanics and quantum mechanics and for many other theories. This does not mean that we can say something definitively about the ultimate nature of space and time, or that it even makes sense to talk about such ultimate nature, as if it were uniquely defined. Conservatively, the most we can say is that current physical theories are very effective, and that gives us a good reason for thinking of space and time as a continuum and no good reason for thinking otherwise.

This doesn't mean that future physical theories will not quantize space and time. Some think that quantum physics points in that direction, although to repeat, current theory makes space and time a continuum. And an unbounded (infinite) one at that in all but some cosmological models. Speaking of which, those cosmological models with a finite or semi-infinite spacetime are so violently counterintuitive that I very much doubt that most "infinity skeptics" would be more satisfied with them than with the traditional Euclidean infinite space and time.

that gives us a good reason for thinking of space and time as a continuum and no good reason for thinking otherwise. — SophistiCat

This is more or less what I use as justification. I wouldn't put it forward as unchallengable or something, but insofar as we accept what our best theories say I tend to informally just say they're true. I do however believe there are also arguments for the continuous nature of space that bolster that belief as well.

It's discrete and not a continuum at all.

No wonder we cannot mentally conceive the Actually Infinite; it does not exist. — Devans99

We can conceive non-existent stuff. And even a monistic idealist will allow the existence of the inconceivable.

We can’t conceive of logically inconsistent concepts like Actual Infinity in a logically consistent way.

I’d allow for the existence of the inconceivable only if it where possible. No need to allow for impossibilities like Actual Infinity.

I'm just saying that "exists" and "conceivable" only directly track in maybe some form of subjectivite idealism that nobody has ever actually endorsed.

So your target is set theory. Where did they go wrong in your view?

My target is the use of actual infinity in the physical sciences.

Like how?

It's discrete and not a continuum at all. — LD Saunders

What is?

The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. — MindForged

Care to provide that "clear" mathematical definition of infinity?

Yes; so an object with no start is a non-existent object; — Devans99

That is a misuse of the word 'so'. The word is used after a deduction has been presented, to state the result of the deduction. It is invalid to use it to just state a new assertion that bears no relation to previous assertions, which is what has happened here.

If you consider what you've done here you'll discover that you are using a hidden axiom, which is 'Everything must have a beginning'. Only if we accept that axiom can we deduce your assertion. But accepting the axiom is a matter of taste and I find it completely unintuitive, as well as lacking in any aesthetic appeal, so I don't accept it.

I never look up links referenced during a discussion. This is for two reasons: 1. Any nonsense can be posted on the web, and often is. 2. If someone has an argument to make, then they should be able to state it in their own words. — LD Saunders

Amen, comrade!

The only exception that I find worth making is when the link is not to an argument but to statistics that are hosted on the site of a credible, impartial authority, that are relevant to the discussion.

Physicists can give a very different answer to the binary question of whether spacetime is "fundamentally discrete" or "fundamentally continuous". They would say that quantum theory argues that it is neither. At base, it is vague or ambiguous. And then the classical binary distinction of discrete vs continuous is what emerges due to sufficient stabilising contextuality. You get a division into distinct events happening within a connected backdrop once a quantum foam has expanded and cooled enough for that to be the case.

For example:

While almost all approaches to quantum gravity bring in a minimal length one way or the other, not all approaches do so by means of “discretization”—that is, by “chunking” space and time. In some theories of quantum gravity, the minimal length emerges from a “resolution limit,” without the need of discreteness. Think of studying samples with a microscope, for example. Magnify too much, and you encounter a resolution-limit beyond which images remain blurry. And if you zoom into a digital photo, you eventually see single pixels: further zooming will not reveal any more detail. In both cases there is a limit to resolution, but only in the latter case is it due to discretization.

In these examples the limits could be overcome with better imaging technology; they are not fundamental. But a resolution-limit due to quantum behavior of space-time would be fundamental. It could not be overcome with better technology.

http://www.pbs.org/wgbh/nova/blogs/physics/2015/10/are-space-and-time-discrete-or-continuous/

So the key shift in metaphysical intuition is to see reality as wholly emergent from raw potential. And that then means the infinite is always relative.

The classical way of looking at it is that either the discrete is the fundamental - you start with some atomistic part and then are free to construct endlessly by the addition of parts - or the continuous has to be fundamental. You would start with an unbroken extent that you could then freely sub-divide into an unlimited set of parts.

Note the presumption. It is all about a mechanical act, a degree of freedom, that can proceed forever without constraint. If you have a unit to get you started, there is nothing stopping you adding more units to infinity. Or if you have a line you can slice, there is nothing stopping you slicing it finer forever.

It is a wonderfully simple vision of nature. But it is way too simple to match the material reality. So no matter how wonderfully maths elaborates on this naive constructionist ontology, we already know that it is too simplistic to be actually true.

The alternative view is that individuation or finitude is context dependent. It is a resolution issue. Both the continuous backdrop and broken foreground swim into definiteness together. The more definite the one grows, the more sharply defined becomes the other.

So it is like counting clouds in the sky. And beginning in a thin mist. While everything is just a generalised mist, it is neither one thing nor the other - neither figure nor ground, object nor backdrop. It is sort of sky, sort of cloud, but in completely unresolved and ambiguous fashion.

Then the mist starts to divide and organise. It gets patchy. You start to have bits that are more definitely actual cloud, other bits that are actual sky. Keep going and eventually you have some classically definite separation. There is a nice tight fluffy white cloud that sticks out like a sore thumb against an empty blue background. The finitude and discreteness of the cloud emphasises the infinity and continuity of a sky that now goes on forever. You arrive at a state of high contrast. And it is difficult to believe that it could ever be any other way.

Of course, physicists now know just how much of an idealisation this is. They even have the maths to model the actuality in terms of fractals. Real life cloud formations better fit a model which directly encodes the fact that individuation is a balance of a tendency towards discreteness and a tendency towards continuity. The holism of material systems means they have equilibrium properties, like viscosity.

So in the connected world of a weather system clouds are generally bunched or dispersed according to some generalised ratio. They never were these classical objects with definite edges marking them off from the continuous void that surrounds them. All along, they were just a watery transition zone with a fractal balance and hence a fractal distribution in space and time. If you want to model the actual world of the cloud, you have to accept that this grave sounding metaphysical question - is the cloud discrete or continuous? - is pretty bogus.

The actuality is that cloudiness is a propensity being expressed to some degree of definiteness. It can be in a state of high resolution, or low resolution, but it is always in some state of resolution - a balance between two complementary extremes. We imagine a reality that is polarised as either sky or cloud. Everything would have to be one or the other. Yet now even the maths has advanced to the point that we can usefully model a reality which is always actually in some fractional balance, always suspended between its absolute limits.

The next step for fundamental physics is to apply that holistic metaphysics to our notions of spacetime themselves. And that is certainly what a lot of quantum gravity theories are about. The traditional classical metaphysical binaries - like discrete vs continuous and finite vs infinite - lose their power as it is realised that they are the emergent limits and not the fundamental starting options. Instead, where things begin is with simple vagueness or indeterminism. You have a quantum foam or some other new model of a world before it gains any definite organisation via the familiar classical polarities.

It’s a problem I agree but I can think of a way past 2 above: imagine as you get closer to the edge of the universe time slows down and right at the edge time stops. So it’s impossible to poke a spear through the edge of the universe because there is no space time in which to poke the spear. — Devans99

That's an interesting idea. If I'm reading you correctly, you're suggesting that there is some point in the universe, call it C (for centre), such that, as we approach a certain number of km from C, we find our movements increasingly constrained and, as we continue, increasingly slowly, we asymptotically become paralysed. It's like there's some kind of sticky force field in the universe that grows stronger and stickier as we move away from C.

It sounds like a great premise for a fantasy novel, a bit like the waterfall at the rim of Terry Pratchett's Discworld. There's no logical reason why it could not be the case.

For me, I find Occam's Razor demands that I prefer an unbounded or hyperspherical universe to this, as they are both much simpler. They can be explained in terms of science we already know, whereas the sticky force-field universe relies on the existence of some sticky force that we have never observed, are unable to test for, and have no reason to believe exists.

So I concede that a finite, non-hyperspherical universe model doesn't have to run into Aristotle's poke-a-spear challenge. But it does require taking on a whole bunch of extra metaphysical hypotheses. I suppose it depends on how determined one is to not have any actual infinities, as to whether that seems attractive.

...imagine as you get closer to the edge of the universe time slows down and right at the edge time stops. So it’s impossible to poke a spear through the edge of the universe because there is no space time in which to poke the spear. — Devans99

Continuing on the "resolution limit" approach now being taken, this would be modelled relativistically in terms of holographic event horizons. So you could imagine "poking your spear" into the event horizon surrounding a black hole, or across the event horizon that bounds and de Sitter spacetime.

In a rough manner of speaking, your spear would suffer time dilation as you jabbed it into the black hole. It would start to take forever to get anywhere.

Or if you poked it across the event horizon that marks the edge of the visible universe, then it would disappear into the supraluminal realm that exists beyond.

So relativity itself already tells us that there is a radical loss of the usual classical observables when we arrive at the "edge" as defined by the Planck constants of nature. There is a fundamental grain of being, a grain of sharp resolution, which the constants define. Then if we try to push beyond that, then the customary classical definiteness of things begins to break down in ways the theory predicts. The distinctions that seemed fundamental dissolve away.

The conventional way of thinking about spacetime is that it must exist in some solid and substantial fashion. It is just there. So the metaphysical issue becomes how can a backdrop begin and end? By definition, a backdrop just is always there ... everywhere. So spacetime simply has to extend infinitely to meet the criteria.

But the emergent view turns this around. Spacetime as a definite backdrop becomes an emergent region of high coherence. And being bounded or finite is the kind of organisation that has to get imposed to create such a state of being. You need some concrete limit - like the speed of light, the strength of gravity, the fundamental quantum of action - to structure a world. The triad of Planck constants are the restrictions that together form up the thing of a Universe with a holographic organisation and a Big Bang tale of development.

The Universe is essentially a phase transition. Like water cooling and crystallising, it has fallen into a more orderly, lower energy, state. What changes things is not the magical creation of something new - like ice - but the emergence of further constraints that limit the systems freedoms. A solid is a liquid with extra restrictions, just as a liquid is a gas with emergent constraints.

So what lies "beyond" any part of a universe is not simply more of the same. Nor is it something completely different. Instead, the distinction is one of resolving power. If the classical world is about a crystalline coherence, then beyond the edges of any patch of the coherent is simply ... the start of the incoherence.

Crossing an event horizon is just that. It is imagining how things break down now that they are no longer integrated in the usual communicative fashion. Approach the edge and everything just dissolves towards a radical indeterminacy. What seemed definitely one thing or another becomes blurred and confused - a question no longer properly answerable.

It is just like the edge of a cloud. At some point the fabric frays and it is not clear whether it is still largely cloud or now mostly sky. To argue that there has to be a definite answer - as in arguing about whether things are fundamentally discrete or continuous, finite or infinite - is to miss the point. That kind of constrained counterfactuality is the state that must emerge. It is the outcome and not the origin.

I'm pretty sure I did in a previous post, but to take sets

A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals.

That's pretty clear, it's exactly the same reason I can, without knowing the exact number of people in an audience, know that if every seat is occupied, then there's no empty seats (each seat can be paired off with a person).

We can’t conceive of logically inconsistent concepts like Actual Infinity in a logically consistent way.

I’d allow for the existence of the inconceivable only if it where possible. No need to allow for impossibilities like Actual Infinity. — Devans99

Prove it. I've given evidence that we can conceive of the actual infinite by giving a description of it and examples which instantiate it, you just keep begging the question or just asserting what you believe. And of course we have need of the actual infinite. As has been said a few times, several very solid theories make assumptions that include infinity. And as to my original point, if you accept almost any fleshed out mathematics you have to accept that infinity is not a contradictory concept. So to say it's contradictory when applied to reality either makes no sense or you have an unstated argument.

A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals.

That's pretty clear, it's exactly the same reason I can, without knowing the exact number of people in an audience, know that if every seat is occupied, then there's no empty seats (each seat can be paired off with a person). — MindForged

I see no clear definition of infinity here, just a rambling description of a particular type of set, which you call an infinite set. That description doesn't tell me what it means to be infinite, it tells me what it means to be an infinite set.

I see no clear definition of infinity here, just a rambling description of a particular type of set, which you call an infinite set. That description doesn't tell me what it means to be infinite, it tells me what it means to be an infinite set. — Metaphysician Undercover

It is a "particular type of set" which distinguishes the finite sets from the infinite ones by means of a relationship that isn't possible for finite sets. It further allows us to see the exact difference between such sets. A subset is a "proper subset" of a set so long as the members each contain are not all identical, but some are shared. For finite sets, proper subset will always be non-identical and leave some out of the original set. But for an infinite set, this cannot happen, just look (Naturals on the left, evens on the right):

0 - 0
1 - 2
2 - 4
3 - 6
etc.

There's never a point at which the one-to-one correspondence fails to pair up a natural with an even. We know the evens are are proper part of the naturals, as the evens are lacking half the naturals (the odds). And yet they have the same cardinality. That's infinite and it returns exactly the sets of numbers we already intuitively take to be infinite, and (as I said) it gives us a property by which to tell which is which and does not yield any contradictions.

How this is rambling, I don't know. It's literally just lining things up.

As has been said a few times, several very solid theories make assumptions that include infinity. — MindForged

And you have been reminded a few times that these solid theories in fact depend on working around the infinities they might otherwise produce. So it ain't as simple as you are suggesting.

The way to understand this is that modelling seeks the simplest metaphysical backdrop it can get away with. So it is a convenience to treat flatness, extension, coherence, or whatever, as "infinite" properties of a system. If you can just take the limit on some property, it becomes a parameter or a dimension - a basic degree of freedom that simply exists for the system. You don't have to model it as a variable. It is part of the ontic furniture.

So it is for good epistemic reason that physical models appear to believe quite readily in the infinite. If you are going to have a line that extends, it might as well be allowed to extend forever without further question. That way it drops out of the bit of the world that needs to be measured and becomes part of the world that is presumed. As a degree of freedom, it is fundamental.

But the history of physics is all about the questioning of the fixity of any physical degree of freedom. Everything has wound up being contextual and statistical. Newton said space and time were flat and infinitely extended. Einstein said spacetime is instead of undefined curvature and topology. You had to plug in energy density measurements at enough points to get some predictable picture of how it in fact would curve and connect. Newtonian infinity would then emerge as a special case - an exceptional balance point of in fact impossible stability. Some kind of further kluge, like a cosmological constant, would be needed to give a gravitating manifold any actual long-term extension at all.

So if we look at the actual physics, it does seek the "infinities" or taken-for-granted degrees of freedom which can become the "eternal" backdrop of a mechanical description. You've got to find something fixed to anchor your calculational apparatus to. So for good epistemic reasons, it seems that physics is targeting the continuous, the unboundedly extensible, the forever the same.

But does it believe in them? Does it take them literally? Does it say they are metaphysically actual?

By now, that would be a very naive ontology indeed. All the evidence says that nothing is actually fixed. It all just merely hangs together in a self-sustaining structured fashion.

The mathematical notion of infinity is a very misleading one to apply in a physical context these days. The Euclid/Newton paradigm is old hat. Even in maths, geometry has become deconstructed as topology. Space is flat, lines are straight, change is linear, only as the extreme case of a maximal constraint on the possible degrees of freedom in fact. Instead of being fundamental, the perfect regularity and simplicity of a classical geometry is the most exceptional case. It requires a lot of explanation in terms of what removes all the possible curvature, divergence, and other non-linearities.

How this is rambling, I don't know. It's literally just lining things up. — MindForged

To me you have just demonstrated the logical deficiency which the concept of "infinite" introduces into set theory. You have demonstrated that the set of natural numbers is equivalent (in the sense of having the same number of members) as the set of even numbers. That's nonsense, and that's what the concept of infinity introduces into mathematics, nonsense.

It's nonsense because it's a totally useless piece of trivia. Infinite sets have the same number of members as other infinite sets ... a nonsense number ... an infinite number.

And you have been reminded a few times that these solid theories in fact depend on working around the infinities they might otherwise produce. So it ain't as simple as you are suggesting. — apokrisis

That's not true, using an infinity is not the same as a singularity occurring in the theory. Space under relativity is treated as a continuum, but that's not the same as a singularity occurring, it's just part of the geometry.

You seem to think I'm arguing that anytime infinity crops up in our models it ought to be accepted. As I said initially, we have good reasons why we don't do that (the need to get meaningful results being central). But my point was that we still make assumptions (crucial, necessary ones) regarding the existence of infinity in the world as well (relativity and QM both do so), so the notion of an Actual Infinity isn't off the table.

Instead of being fundamental, the perfect regularity and simplicity of a classical geometry is the most exceptional case. It requires a lot of explanation in terms of what removes all the possible curvature, divergence, and other non-linearities. — apokrisis

I certainly haven't said Euclidean geometry is how our universe is actually structured. I said the opposite, in fact.

You have demonstrated that the set of natural numbers is equivalent (in the sense of having the same number of members) as the set of even numbers. That's nonsense, and that's what the concept of infinity introduces into mathematics, nonsense. — Metaphysician Undercover

Nonsense based on what argument? This is what you say but:

It's nonsense because it's a totally useless piece of trivia. Infinite sets have the same number of members as other infinite sets ... a nonsense number ... an infinite number. — Metaphysician Undercover

Like how is the a an actual objection? It's "nonsense because it's useless trivia". Come on, it's literally a property by which we can clearly distinguish one type of set (finite sets) from another type of set (infinite sets). All you're doing is saying infinity is nonsense but you're not actually explaining why.

Nonsense based on what argument? — MindForged

Take any set of a series of natural numbers, 1 - 10, 1 - 20, 2 -40, whatever. If that set has two or more members, then the subset of the even numbers has less members than the original set. This is always the case, and by inductive reason we can state such a law, that this is always the case. The infinite set is specifically designed for no reason other than to break this law, therefore it is unreasonable, nonsense.

As it is an unbounded (open) set, it is not truly a "set", as a collection of objects, it is a boundless collection which is not a collection at all. A collection, or "set" means that the members are collected together in a group. If the collecting is not complete, then the described collection (set) does not exist. To call it a collection, or set, is contradictory nonsense.

The infinite set is specifically designed for no reason other than to break this law, therefore it is unreasonable, nonsense. — Metaphysician Undercover

It's not "designed" to break this "law", it just doesn't apply and it's perfectly obvious that it wouldn't. These sets aren't conjured, they're the numbers we start of learning.

As it is an unbounded (open) set, it is not truly a "set", as a collection of objects, it is a boundless collection which is not a collection at all. — Metaphysician Undercover

Again, what is the non-question begging argument for this? What makes a set a set is not it being bounded. The "set of moments after the present moment" is unbounded but no one gets up in arms about defining such a collection of moments as a set. They share a property in common (their coming after the stipulated moment) so assigning them to the same collection is natural.

A collection, or "set" means that the members are collected together in a group. If the collecting is not complete, then the described collection (set) does not exist. To call it a collection, or set, is contradictory nonsense. — Metaphysician Undercover

They aren't "collected" in a mechanistic process, i.e. going out and declaring "You go in this set" and such. Just sharing a property is enough, and it happens to be perfectly compatible with there being infinite collections.

Hell, let's just show this without reference to numbers as elements of a set and yet it still be infinite:

Let P be the set of all possible English sentences.

It's surely unbounded. I can always add a new word to any English sentence to yield evermore new sentences and it's still going to fall in the set of "possible English sentences". And yet there's no way you can argue it fails to be a set in virtue of being unbounded.