If the whole is limitless it has no bounds. Then you introduce bounds and impose limits. What?
Maybe I can get it on a second try but why?

Nope, not getting it. W1 is no longer the whole you started with.

You are proving an unlimited whole is not affected by bounds....okay.

If the whole is limitless it has no bounds. Then you introduce bounds and impose limits. What?
Maybe I can get it on a second try but why?

Nope, not getting it. W1 is no longer the whole you started with. — Mark Nyquist

What I am trying to show is that there are two cases where what you consider as the whole is either limited or limitless. If $W_1$ is limitless we reach our goal otherwise $W_1$ is limited and it is bounded by something else, let's call it $B_1$. Etc.

Okay. Worth exploring. It takes time to understand it.

The problem I'm seeing with your approach is that you don't identify the whole as a concept. Its origin is a mental abstraction. Limits are mental definitions. If you apply limits to infinity you no longer have infinity.

I covered this in my Universal Form post not long ago if you want to understand why I object to your method.

To show this let's assume that the whole is limited, let's call the whole W1. This means that W1 is bounded by something else, let's call this B1. — MoK

This doesn't follow, unless by "limited" you mean just that: being limited by something else. But that would make your argument a simple and uninteresting tautology.

The problem I'm seeing with your approach is that you don't identify the whole as a concept. Its origin is a mental abstraction. — Mark Nyquist

By the whole, I mean whatever that exists, spacetime, material, etc.

Limits are mental definitions. — Mark Nyquist

No, it can be physical as well.

If you apply limits to infinity you no longer have infinity. — Mark Nyquist

Correct. But what is the implication of this to my argument?

I covered this in my Universal Form post not long ago if you want to understand why I object to your method. — Mark Nyquist

Could I have a link to your post?

This doesn't follow, unless by "limited" you mean just that: being limited by something else. — SophistiCat

By limited I mean restricted in size. Think of spacetime for example. If spacetime is restricted in size then we can reach its edges by moving in straight lines (of course if spacetime is not a closed manifold). The problem is what is beyond the edges. It cannot be nothing since nothing does not have any geometry and occupies no room. So, whatever is the beyond edges of spacetime is something. Therefore, what I said follows.

I'm not good at links on my smart phone but go to the main page here and it is listed as:

Is Universal Form a good tool?

The last entry as of now.

Ok, I found the thread. I am however confused with what you are trying to say. I will read it more and post my opinion in the thread if I understand something.

By limited I mean restricted in size. Think of spacetime for example. If spacetime is restricted in size then we can reach its edges by moving in straight lines (of course if spacetime is not a closed manifold). The problem is what is beyond the edges. It cannot be nothing since nothing does not have any geometry and occupies no room. So, whatever is the beyond edges of spacetime is something. Therefore, what I said follows. — MoK

OK, so you are basically reprising Lucretius' javelin argument:

For whatever bounds it, that thing must itself be bounded likewise; and to this bounding thing there must be a bound again, and so on for ever and ever throughout all immensity. Suppose, however, for a moment, all existing space to be bounded, and that a man runs forward to the uttermost borders, and stands upon the last verge of things, and then hurls forward a winged javelin,— suppose you that the dart, when hurled by the vivid force, shall take its way to the point the darter aimed at, or that something will take its stand in the path of its flight, and arrest it? For one or other of these things must happen. There is a dilemma here that you never can escape from.

Oh, I didn't know about him and his argument. I am glad you mentioned it.

By limited I mean restricted in size. Think of spacetime for example. If spacetime is restricted in size then we can reach its edges by moving in straight lines (of course if spacetime is not a closed manifold). The problem is what is beyond the edges. It cannot be nothing since nothing does not have any geometry and occupies no room. So, whatever is the beyond edges of spacetime is something. Therefore, what I said follows. — MoK

What if space is closed? As in, a loop where going in a certain direction for enough time sends you back to where you started. The world would then not be limitless, but still unbounded.

A mind-boggling property of this universe is that it is finite, yet it has no bounds. — https://www.astronomy.com/science/what-shape-is-the-universe/

Edit: You might reply that it is limitless because it does not have an edge. But then your original argument is that there is a W=W1+B1+B2+ that goes to infinity. That would be untrue if the universe is closed, as you don't need to add anything for it to be limited. Yet, if you redefine limitless to mean edgeless, your argument becomes either tautological (the universe with edges is surrounded by something) or a discussion on nothingness (the universe with edges is surrounded by nothing).

What if space is closed? As in, a loop where going in a certain direction for enough time sends you back to where you started. The world would then not be limitless, but still unbounded.

A mind-boggling property of this universe is that it is finite, yet it has no bounds.
— https://www.astronomy.com/science/what-shape-is-the-universe/ — Lionino

I think if space is closed then it is embedded in a hyperspace.

I don't know what hyperspace is, neither how closedness of the universe implies one.

I don't know what hyperspace is, neither how closedness of the universe implies one. — Lionino

By hyperspace I mean a space of more than three dimensions. Why does the closedness of the space imply a hyperspace? Because any closed manifold has a local curvature. To help the imagination think of a closed 2D space, a sphere for example, instead of a 3D one. Each point on this sphere has a curvature at any given point which means that the surface of the sphere bends at each location on the sphere. The fact that the sphere bends at each location on its surface requires a higher dimension space, in this case minimally 3D space, where the sphere is embedded within otherwise the sphere cannot bend at each location on its surface and we cannot have a curvature.

What you are saying makes sense, however:

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work. — Does space curvature automatically imply extra dimensions?

Also

Nope, spacetime curvature says nothing about the dimensionality. Your intuition here is probably wrong because human imagination needs 'some dimension to bend into' in order for something to be curved (i.e. an embedding in a higher-dimensional space). This is just our lack of imagination showing, though.

What you are saying makes sense, however:
No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work.
— Does space curvature automatically imply extra dimensions? — Lionino

Here I am not talking about intrinsic curvature in spacetime that is caused by a massive object locally but extrinsic curvature which tells us what is the global geometry of space.

Here I am not talking about intrinsic curvature in spacetime that is caused by a massive object locally but extrinsic curvature which tells us what is the global geometry of space. — MoK

The closedness of spacetime does not require extrinsic curvature. If anything, your argument would require infinite dimensions, as each time you evaluate the extrinsic curvature of a dimension another one would be in order.

The closedness of spacetime does not require extrinsic curvature. — Lionino

It requires as I illustrated.

If anything, your argument would require infinite dimensions, as each time you evaluate the extrinsic curvature of a dimension another one would be in order. — Lionino

Not all hyperspaces that I am talking about are necessarily closed so we could deal with finite hyperspace dimensions which accommodate everything. You are however right that we need infinite dimensions if all hyperspaces are closed. I don't see any problem with hyperspace which has infinite dimensions though.

It requires as I illustrated. — MoK

Where did you show that a closed universe requires extrinsic curvature?

Not all hyperspaces that I am talking about are necessarily closed so we could deal with finite hyperspace dimensions which accommodate everything. You are however right that we need infinite dimensions if all hyperspaces are closed. — MoK

Whether something is closed refers to intrinsic curvature.

I don't see any problem with hyperspace which has infinite dimensions though. — MoK

It is a physically-unfounded belief about the empirical world arrived at using a priori syllogisms in a natural language.

Seems to me you have reinvented the universal set. As such what you have shown is that there is nothing outside of W, not that W is limitless.

Or more simply the surface of a sphere. Without limit or boundary. certainly in the very ordinary sense, finite.

I'm still having trouble understanding what you mean by the whole. Is it a philosophy term? As the whole is the sum of its parts, the whole universe in the physical sense, mathematics or what I thought at first, a concept of the whole being something limitless or infinite.

I can make some progress on your argument but then the conversation goes in another direction such as the physical universe which was never stated.

I'm thinking if it's the physical universe we can't impose our own mathematical model on it without knowing what it is.

Where did you show that a closed universe requires extrinsic curvature? — Lionino

Here. Moreover, considering that you return to the point you started (if you move on a straight line closed manifold) requires extrinsic curvature. How could you return to the point you started if the global geometry of space was flat? You get a square if you cut a balloon and put it on a flat surface. You reach a dead end if you move in a straight line on the square. So a square is not a proper representation of a sphere.

Whether something is closed refers to intrinsic curvature. — Lionino

Actually, after some thought, I realized that even intrinsic curvature also requires a higher dimension. How the lines could deviate towards or away from each other if the geometry of space is flat. This figure is very illustrative:

https://en.wikipedia.org/wiki/File:Cassini-science-br.jpg

It is a physically-unfounded belief about the empirical world arrived at using a priori syllogisms in a natural language. — Lionino

It is physically necessary if it is logically necessary. Moreover, we don't know what is the right curvature of spacetime. Spacetime could simply be limitless if its geometry is flat globally.

Seems to me you have reinvented the universal set. As such what you have shown is that there is nothing outside of W, not that W is limitless. — Banno

But W could not be limited as I argued.

I'm still having trouble understanding what you mean by the whole. Is it a philosophy term? As the whole is the sum of its parts, the whole universe in the physical sense, mathematics or what I thought at first, a concept of the whole being something limitless or infinite.

I can make some progress on your argument but then the conversation goes in another direction such as the physical universe which was never stated.

I'm thinking if it's the physical universe we can't impose our own mathematical model on it without knowing what it is. — Mark Nyquist

By whole I mean whatever that physically exist.

Hello again MoK!

Let me take your abstract into a thought example for a minute. Lets say that in the universe, only a single grain of sand exists. Now we claim that is the whole, but what is the definition of the whole? Usually 'the whole' is seen as 'everything'. But then you add in something outside of the whole as binding the whole. I'm confused here. What is outside of the grain of sand that is binding the sand?
It would seem that the bind to me is the internal limitation of the sand's matter.

But let me explore your other line of thinking and be charitable where possible. Lets say that the grain of sand is actually bound by 'nothing'. You then note that this binding plus the original whole creates a secondary whole. This doesn't quite work in your variable setup, as W1 and W2 are clearly different concepts here. While a whole indicates 'totality', these are obviously different totalities. So how do I see fixing this?

Perhaps what would make more sense is that some 'thing' is bounded and has limitations where there is 'nothing'. 'Nothing' may bind 'something', but 'nothing' has no limits. Is that more along the line of what you were thinking of?

Here. — MoK

And that argument was already refuted.

How the lines could deviate towards or away from each other if the geometry of space is flat. — MoK

The counter-example I am giving is exactly where space is not flat, where it has positive curvature.

Actually, after some thought, I realized that even intrinsic curvature also requires a higher dimension. — MoK

It doesn't.

No, general relativity is based on something called "intrinsic curvature", which is related to how much parallel lines deviate towards or away from each other. It doesn't require embedding space-time in a higher dimensional structure to work.

In summary, it is important to distinguish between extrinsic curvature, which involves bending through an additional dimension, and intrinsic curvature, which is directly visible on a surface without reference to an extra dimension. — physicsforums

For example, if you draw a triangle on the surface of the paper, the sum of the interior angles of the triangle will be 180 degrees. When you bend the paper or even roll it up into a cylinder nothing will change and the angles will still add to 180.

In order to have intrinsic curvature, you have to look at a manifold with at least two dimensions--for example, a 2-sphere [aka circle]. Then your question can be rephrased as: how is it possible to tell that a 2-sphere is curved, without making any use of an embedding of it into a space with more than 2 dimensions? The answer to that is, by looking at geodesic deviation, which can be measured purely within the surface.

It is physically necessary if it is logically necessary. — MoK

Until physics tells us that our human-made logic is not absolute and that we may have to reframe, as modern physics may make us do.

Spacetime could simply be limitless if its geometry is flat globally. — MoK

It could be, some scientists even believe that. My counterexample is to show that your idea is not logically necessary by showing the possibility of the contrary (◇¬p → ¬□p).

What I am trying to show is that there are two cases where what you consider as the whole is either limited or limitless — MoK

And therein is your flaw. Considering the whole to be limited is simply a mistake in logic. And we already knew that.

Hello again MoK! — Philosophim

Hello Philosophim!

Let me take your abstract into a thought example for a minute. Lets say that in the universe, only a single grain of sand exists. Now we claim that is the whole, but what is the definition of the whole? Usually 'the whole' is seen as 'everything'. But then you add in something outside of the whole as binding the whole. I'm confused here. What is outside of the grain of sand that is binding the sand?
It would seem that the bind to me is the internal limitation of the sand's matter. — Philosophim

First, you need a space as large as the size of the sand to embed the sand within. Now, the question of what is outside of the space is valid.

But let me explore your other line of thinking and be charitable where possible. Lets say that the grain of sand is actually bound by 'nothing'. You then note that this binding plus the original whole creates a secondary whole. This doesn't quite work in your variable setup, as W1 and W2 are clearly different concepts here. While a whole indicates 'totality', these are obviously different totalities. So how do I see fixing this?

Perhaps what would make more sense is that some 'thing' is bounded and has limitations where there is 'nothing'. 'Nothing' may bind 'something', but 'nothing' has no limits. Is that more along the line of what you were thinking of? — Philosophim

Nothing cannot have any geometry or occupy any room so nothing cannot bind the space that the sand is within.