## Musings On Infinity

• 169
I believe there is constructivism - a minority view in maths - which rejects actual infinity.

I see that there is a lot of misunderstanding about constructivism meaning the rejection of actual infinity.

Constructivism is not about the rejection or acceptance of actual infinity, but it's about choosing computable functions as a fundamental logic concept (that is implemented the rules of logic), as opposed to the more abstract idea of functions that is implied by the classical axiom of choice.

But this, in my opinion, is more a question of practical convenience in simplifying the proofs (mainly in topology), rather than a philosophical point of view: you can always reason about computable functions by using the internal logic of a "topos" in category theory, even taking non constructive logic as fundamental. And you can easily axiomatize the standard functions (the one corresponding to the classical set theory with the axiom of choice) by using constructivist logic. Only that, in my opinion, the latter choice is conceptually simpler and easier to use, at least for the branches of mathematics directly related to topology.

If you choose logical rules to represent computable functions, you get constructive logic.
If you choose logical rules to represent abstract functions (in the sence of input-output correspondence, not necessarily computable), you get the standard non constructive logic.

In summary, it's only about the choice of which kind of functions you choose to be "fundamental".
• 2.7k
???

Not when they're speaking with precision. For example, it's acceptable to refer to all of a certain types of animals on a farm as chickens, or cows. But better not confuse a rooster with hen, or worse, a bull for a cow. You are simply conflating (not to be confused with confusing) usage acceptable, if wrong, in some contexts, with other contexts where the same usage is unacceptable - for the completely simple and obvious reason that the terms do not mean the same thing.

The definition of Aleph-naught is contradictory:

1. Aleph-naught is the size of the set of naturals
2. Sets contain a positive number of whole items only
3. So Aleph-naught must be a natural number
4. But there is no largest natural number
5. So Aleph-naught cannot exist (or be larger than all the natural numbers)

1. Aleph-0 is the how-many of the set
2. Not quite, needs a better spec. Wrong as it stands.
3. Who said? Maybe that's your personal problem. In any case it is defined as the first transfinite cardinal. That is, not an integer. And not a natural number.
4. Close. Whatever natural number you have, you can always add at least one to it.
5. Your conclusion simply ignores definition. And you know this perfectly well because you've been told it again and again by persons whose telling is a heck-of-a-lot more authoritative than my telling.

But here's a proof for you: 1) everyone is ignorant about lots of things, always. But 2) it is possible to learn about some things. And 3) having learned about a thing, one is no longer ignorant about that thing. But 4) sometimes some people will not learn and insist on persistently persevering in their ignorance notwithstanding available knowledge that counters their ignorance, as if insisting that a bull is a cow. 5) Such persistence is called - defines - stupidity, and the person thus persistent, stupid. 6) Devans99 persistently perseveres in his ignorance against plain understanding. Therefore... QED.

Are you stupid, Devans99? The argument says you are. Mere denial won't avail you - that would just be more stupid. Instead, without any (more) Ifs or questions, prove here and now that the mathematics of transfinite numbers and their definitions are just wrong.
• 2k
I have a question:

"computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space"

https://en.wikipedia.org/wiki/Computable_function

It strikes me that it is impossible to specify a function that a computer could execute that would result in actual infinity as the output? No amount of successive addition or multiplication yields actual infinity.
• 2k
Tim you are a f**king idiot.
• 169
Yes, no doubt it's impossible. But nobody says that only computable functions exist.
• 2k
3. Who said? Maybe that's your personal problem. In any case it is defined as the first transfinite cardinal. That is, not an integer. And not a natural number.

Any size of a set must be a natural number. So the definition of Aleph-naught as 'not a natural number' is plainly contradictory.
• 169
Here's a video that may interest you:

Goodstein's theorem is a theorem about a computable function that cannot be proved without assuming the existence of actual infinity.

P.S. Don't ask me why: I don't understand it either.. :smile:
• 2.7k
Tim you are a f**king idiot.

Well now, that's a nice response. Full of so much substance I scarcely know how to respond. I better just plead guilty! But at least I try not to be stupid - not always successful, but I'm a closet Pelagian.
• 2.7k
Any size of a set must be a natural number. So the definition of Aleph-naught as 'not a natural number' is plainly contradictory.

What is the natural number that describes how many natural numbers there are that are larger than ten?
• 2k
I’d like to show that space must be discrete, but my maths is not so great. Maybe one of our resident mathematicians can comment?

I’m considering the number of points between 0->1 and 0->2. I’ll denote these as functions: points(0,1) and points(0,2).

One of the following must hold true:

1. points(0,1) = points(0,2)
2. points(0,1) < points(0,2)
3. points(0,1) > points(0,2)

I think by common sense, we can eliminate [3] and focus on [1]:

points(0,1) = points(0,2)
points(0,1) = points(0,1) + points(1,2)

If there is a finite number of points in each interval, space must be discrete, so I will assume an infinite number of points in each interval:

$\lim_{n\rightarrow\infty} n = lim_{n\rightarrow\infty} n + lim_{n\rightarrow\infty} n$
$\lim_{n\rightarrow\infty} n = 2 lim_{n\rightarrow\infty} n$
(1+1+1+1+…) = 2 x (1+1+1+1+…)
1 = 2

This last step, dividing though by ∞, is not conventionally allowed, presumably because it’s argued that there are different kinds of infinity. I would have thought (wrongly?) that because I am dividing though by an identical kind of infinity to the infinity in the expression then it should be allowed?

So [1] does not seem possible, that just leaves [2]:

points(0,1) < points(0,2)

Which makes sense, but it implies that space cannot be a true continuum. There are more points in the larger interval implying that the larger interval is structurally different to the smaller interval, whereas for a true continuum, they must be identically structured.

It could be that a point is not the fundamental unit of space, but the argument I think would be identical for a segment.
• 169
If there is a finite number of points in each interval, space must be discrete, so I will assume an infinite number of points in each interval:

If you want to show that continuum space implies a contradiction, you should start by assuming as hypothesis that "space is continuum AND one of the three points is true".

So how do you show that "space is continuum and point 1 is true" implies a contradiction?
The argument that you are making with the limits assumes that space is infinite but discrete, right?
How do you rule out that "space is infinite and continuum" ?
• 2.7k
One of the following must hold true:

1. points(0,1) = points(0,2)
2. points(0,1) < points(0,2)
3. points(0,1) > points(0,2)

Because you're navigating transfinite arithmetic, you have to be careful of your definitions. And you have to keep track of exactly what you're working with.
Lim(n)(as n->aleph-c) is aleph-c.

Aleph-c = λ(aleph-c), λ being pretty much anything.

Corollary: the number of points between zero and one inches on a number line is the same as the number of points in a cube one mile on each side - or one thousand miles on each side.

Proof. Assign to each point a unique number. That number can be uniquely mapped onto the other and vice versa. Therefore both sets are the same size.

In dividing by (1+1+...+1...) you're taking it as a finite number, but it is not. The finite number is (1+1+...+1). See the difference?

And again and as usual, Devans99, I accuse you of knowing this perfectly well - at the least it's an easy Google look-up. So what are you about?
• 2k
The argument that you are making with the limits assumes that space is infinite but discrete, right?

A piece of space is discrete if it allows only a finite number of possible positions (points). So I've assumed an infinite number of possible positions - if its not discrete, it must be continuous. From Wikipedia:

"Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound."

So all continua are (in the above sense) alike in that they can be subdivided forever, so we can write:

points(0,1) = points(0,2)

At this point, there is a one-to-one correspondence between the left and right side so they have the same 'size' so the equals sign seems maybe justified.

points(0,1) = points(0,1) + points(1,2)

Now there is a still a one-to-one correspondence between the left and right side, however, when it is written like this, it appears to part from common sense. I know this corresponds to the convention ∞=∞+∞. I also know that if two things are identical and you change one of them, then they cannot be identical anymore.

I just cannot get my head around continua, they just seem impossible. It is possibly the use of the equals sign to represent a one-to-one correspondence that is the problem. It maybe that it is invalid logically to write:

points(0,1) = points(0,2)

It comes back to Galileo's paradox - the above are equal in the sense of a one-to-one mapping but at the same time, one is clearly twice the other. I think that it is not valid to compare the size of two infinities (as Galileo believed) - they are fundamentally undefined so have no size and cannot be compared. If something never ends, then it can never have a size and never be fully defined. I do not believe Cantor has added anything our the understanding of infinity - he has detracted from it - Galileo was on the right lines.

So coming back to my original starting place, my assumption that one of the following must be true:

1. points(0,1) = points(0,2)
2. points(0,1) < points(0,2)
3. points(0,1) > points(0,2)

seems incorrect. It seems I should instead have written:

1. UNDEFINED != UNDEFINED
2. UNDEFINED !< UNDEFINED
3. UNDEFINED !> UNDEFINED

So I think that maths cannot model actual infinity or continua. Does that mean these things do not exist in the real world? I think that maybe the case. If continua exist, then that implies that the informational content of 1 light year of space is the same as the informational content of 1 centimetre of space - in the sense that both 'containers' record the position of a particle to an identical, infinite, precision. This flaunts 'the whole is greater than the parts'. I trust that axiom more than I trust Cantor's math.
• 169
A piece of space is discrete if it allows only a finite number of possible positions (points). So I've assumed an infinite number of possible positions - if its not discrete, it must be continuous. From Wikipedia:

"Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound."

But to be "discrete" does not mean to be "finite" (https://en.wikipedia.org/wiki/Discrete).

Discrete means that is made of parts that are distinct from each other: it can be finite, but not necessarily. The set of natural numbers is discrete but infinite. In fact, I think discrete is probably always synonym of "countable": it mean that you can associate each element of a discrete set with a natural number, so there are as many elements as there are natural numbers.

A linear continuum instead is made of "non separable" elements: you can shrink or expand the length of a segment, but you can't "separate" it's points from each other: the elements of a continuous set are not "countable".

So all continua are (in the above sense) alike in that they can be subdivided forever, so we can write:

points(0,1) = points(0,2)
OK

It comes back to Galileo's paradox - the above are equal in the sense of a one-to-one mapping but at the same time, one is clearly twice the other. I think that it is not valid to compare the size of two infinities (as Galileo believed) - they are fundamentally undefined so have no size and cannot be compared. If something never ends, then it can never have a size and never be fully defined. I do not believe Cantor has added anything our the understanding of infinity - he has detracted from it - Galileo was on the right lines.

Galileo's paradox is about positive integers, not about continuous sets. In fact, I believe that the idea of a "continuous set" had not even been invented in XVII century. For what I know, Euclidean geometry never speaks about a line being a set of points: for Euclidean geometry, 1-dimensional objects (lines) are a completely different kind of things then discrete (countable) objects. And Galileo does not even consider the idea that a line can be made of a set of distinct objects. For what I know, the idea of continuous (uncountable) sets was invented after Cantor, 200 years after Galileo.

So I think that maths cannot model infinity or continua. Does that mean these things do not exist in the real world? I think that maybe the case. If continua exist, then that implies that the informational content of 1 light year of space is the same as the informational content of 1 centimetre of space - in the sense that both 'containers' record the position of a particle to an identical, infinite, precision. This flaunts 'the whole is greater than the parts'. I trust that axiom more than I trust Cantor's math.

So, if I understand correctly, you are trying to prove that the set of points of a line is finite (not countably infinite). Is it right?

If that's what you are arguing (that a line is made of a finite set of points), the obvious question is: how many points are there in a given segment?
• 2.7k
I just cannot get my head around continua, they just seem impossible.

Just think about the decimal expansion of π or the square root of two. They just go on forever. And of course they're very real. If you still have trouble with it, you're in good company.
• 2k
Discrete means that is made of parts that are distinct from each other: it can be finite, but not necessarily. The set of natural numbers is discrete but infinite.

In this instance, I am referring to finite distances of space. If a finite distance of space is also discrete then it cannot be decomposed into infinite sub-sections.

Galileo's paradox is about positive integers, not about continuous sets. In fact, I believe that the idea of a "continuous set" had not even been invented in XVII century. For what I know, Euclidean geometry never speaks about a line being a set of points: for Euclidean geometry, 1-dimensional objects (lines) are a completely different kind of things then discrete (countable) objects. And Galileo does not even consider the idea that a line can be made of a set of distinct objects. For what I know, the idea of continuous (uncountable) sets was invented after Cantor, 200 years after Galileo.

I think Galileo's paradox is applicable to continua. For example, to establish a one-to-one mapping between two continua:

{0, 1} maps to {0, 2}
{0, 1/2, 1} maps to {0, 1, 2}
{0, 1/4, 1/2, 3/4, 1} maps to {0, 1/2, 1, 3/2, 2}
etc...

At the same time we can note that 0->1 is half the length of 0->2. So Galileo's paradox appears to apply - the continua are equal in terms of a one-to-one mapping but unequal in terms of one being twice the length of the other.

So, if I understand correctly, you are trying to prove that the set of point of a line is finite (not countably infinite). Is it right?

If that's what you are arguing (that a line is made of a finite set of points), the obvious question is: how many points there are in a given segment?

Yes I suspect that space is discrete... proving it would be nice (so would winning the lottery!).

If reality is discrete and a line segment (in reality) is composed of points (or fixed sized line segments) then the length of each point/sub-segment is some number greater than zero. The Planck length is often mentioned in this regard.
• 2.7k
The Planck length is often mentioned in this regard.

No. It isn't. One's a creature of math, the other of physics.
• 169
OK. So you are arguing that space is made of indivisible pieces, and every piece of space is made of a finite quantity of indivisible pieces. I think a more appropriate term for this is "atomic" (from the Greek world "atomos" that means indivisible). I didn't understand because before you wrote: "So all continua are (in the above sense) alike in that they can be subdivided forever".

But this means that points in geometry should have a size that is not zero, right? So, other question: what's the shape of a point? a 3-dimensional sphere? But you cannot fill the space with 3-dimensional spheres (there would be gaps between them). They should be rather like cubes.
In 1 dimension it could work: finite segments can be attached to each-other with no gaps. But if you try to build 2-dimensional geometry with squares in the place of points everything becomes much more complex:
- two "intersecting" lines not necessarily must have a point in common
- space cannot be isotropic (meaning: the same in all directions)
- the concept of straight line becomes very difficult to define, if the direction of the line respect to the grid is not a rational number.

Basically, you loose most of the symmetries of space. And symmetries seem to be a fundamental fact of nature. How do you deal with this problem?
• 2k
I think it is not necessary to fill space (as in a space-filling polyhedron like a cube). I am more imagining a grid of zero dimensional points in space. The particle, which has a non-zero dimension, would be centred on one of the grid points. If there are two neighbouring particles, they would not be in contact with each other, so space is not filled. Particles would move from point to point in the grid rather like the electron performs a quantum jump from one orbit to another - not passing between any intermediate space.

With QM, we have waves (and I suspect a particle is just a compressed wave) and so the waves would be centred on one of the grid points.

All of this would take place down near Planck length, so the universe would appear completely continuous to us.

Loop quantum gravity - the competitor of string theory - has space as discrete.
• 82

Are you familiar with these concepts? Let me know what you think of the videos.
• 2k
Thanks. I am not sure what size the actual universe is - very large but I suspect it must be finite in both time and space. An idea I've been playing with:

Imagine a backwards travelling, counting, eternal, time traveller. Assume that past time is infinite. Then the traveller, should have, from our perspective, counted every number! If this is not an actually completed infinite process, I’m not sure what would qualify. But it is impossible to count all numbers - no matter how many you count, you are always 0% of the way to completion. So it seems that even given an actual, completed infinity of time, actual infinity is not realisable/completable. Which makes me think an actual infinity of time is impossible.
• 82
Why would we have expected him to have counted every number? When did he start going back from? The scenario doesn't make sense. The scenario uses the existence of infinite to argue against its existence. If I have to assume that past time is infinite then there is no way to count every number because their are infinite ones. No one can count infinite, that's the point.

Hell, neither of us in our lifetime could ever write enough zeros to write a googolplex properly, 10^10^100. It's not a real fixed number sure, but it is a concept within our universe of discourse and we don't know if the universe is of a finite or infinite size. Can't know is more accurate here really.
• 2k
So he is eternal and travels back in time forever counting. From our perspective - it has all actually happened - it is history - every number has been counted - a completed infinity

But as you mention, it's impossible to count to infinity (from the traveller's perspective, its impossible to get more than 0% of the way to counting to infinity).

I guess to me infinity does not make sense as a concept, perhaps this scenario brings that out.
• 82
I think it does make sense to you, you just don't see the value in the concept as it's just a fantasy of pure mathematics.

The reality is that numbers as we understand them will always be finite as long as we are for each new high number that is written in sequence can consider itself discovered by we who use numbers and are counting them out. However, inbetween 0 and 1 you could also infinitely count out decimal places without ever reaching 1. Numbers are infinitely divisible. with the exception of zero.
• 82
I personally agree with you. I think there isn't much that can be Known about the concept of infinite so I don't like to play around with it too much. I could never know if the universe is infinite because it's too big for me to ever find out if it is true or not.
• 2k
I think that 'potential infinity' is a very useful concept in science/maths. It is 'actual infinity' that bothers me.

The universe expanding and the universe being actually infinite seem to me contrary.
• 82
Contrary to what?
• 2k
If something is bigger than everything possible, then it expands, then it cannot have have been bigger than everything possible to begin with.

I am a finitist and I do not believe in different sizes of infinity - a minority viewpoint.
• 82
Why do you believe that? Is it intuitive or does the belief come with an argument?

Who said anything about different sizes of infinite? Are you just trying to be provocatively edgy without a logical argument to back up the belief? Because you come off as incredibly arrogant when you do this.

Our best observations to date strongly suggest that the universe has no spatial curvature. It may be expanding in time, but the geometry of space at any given time, is Euclidean.

The simplest topology that corresponds to Euclidean geometry is that of flat, infinite space. So by Occam’s razor, we can conclude that in the absence of evidence to the contrary, the universe appears infinite.

Now the desire to engage with these ideas is great, but I sincerely hope that if I'm taking the time to recommend reading to you (Like Cohens preface to logic, for the 3rd time) then you are at least going to look it up. If you aren't going to listen and are just here to waste time with edgy nonsense for the sake of placating your ego then I will no longer interact with you. Here to learn and help others learn.
• 169
I think it is not necessary to fill space (as in a space-filling polyhedron like a cube). I am more imagining a grid of zero dimensional points in space. The particle, which has a non-zero dimension, would be centred on one of the grid points. If there are two neighbouring particles, they would not be in contact with each other, so space is not filled. Particles would move from point to point in the grid rather like the electron performs a quantum jump from one orbit to another - not passing between any intermediate space.

With QM, we have waves (and I suspect a particle is just a compressed wave) and so the waves would be centred on one of the grid points.

But the wave functions in quantum mechanics are not supposed to have the size of the particle that they represent. For example, electrons are supposed to be point-like (or at least smaller than any detectable size), but the relative wave functions inside atoms are around $10^{-10}$ meters, and are even visible using scanning tunneling microscopes.
And there's even worse: in quantum mechanics if you make the space variables discrete, the momentum variables become continuous and periodic: a zero size for both position and momentum is forbidden by Heisemberg's uncertainty principle.

Loop quantum gravity - the competitor of string theory - has space as discrete.

I don't know loop quantum gravity (except from some vague descriptions), but I am quite sure that it makes use of differential equations defined on continuous domains, like in all physics. If you use discrete values for the space variables, you cannot use derivatives on space variables.
Anyway, in any theory compatible with quantum mechanics events must be associated with a probability amplitude (https://en.wikipedia.org/wiki/Probability_amplitude), that is a complex number. And complex numbers are continuous. And you even need variables for fields, energy, momentum, etc.. I don't think there is any realistic theory of physics that has been defined without making use of continuous variables.
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