The first cause argument

[Polymath257]

No. I want you to explain, how something that is infinite in the first place, can actually expand.
You just admitted that they would be the same volume before and after inflation.

I *did* explain it: any positive number times infinity is equal to infinity. That *is* the explanation. If you don't understand it, please ask for a clarification.

The expansion is the factor in front of the infinity.

In more detail, four dimensional spacetime is a manifold with a metric (in comoving coordinates) that looks something like this:

ds^2 = -dt^2 + a^2 (t) [dx^2 +dy^2 +dz^2 ]

This is for a flat spacetime. that factor a(t) in front is called the expansion factor. In models for cosmology, that expansion factor increases with time (t). That is what it means for space to expand.

Next, space at a particular time would have the restricted metric

ds^2 =a^2(t)[dx^2 +dy^2 +dz^2 ]

and since -infinity<x,y,z<infinity, the volume of space at any time is a^3(t)* infinity=infinity.

Space expands because the distance between points *at rest* (that's what it means to be comoving) increases over time.

 

[Kfox]

First, even defining what that means is problematic.

The situation is that you have a 'boundary' (like the walls of the house) that divides space into an 'inside' and an 'outside'.

Q1: How do you distinguish between inside and outside?

Really? The enclosed area vs the exterior.

For example, suppose I draw a circle on the Earth. That circle will divide the surface of the Earth into two pieces. Which do you designate as the inside and which as the outside?
.

Neither. If the drawing is on the surface of the Earth, both sides are exterior. The only way to get into the interior is by digging a hole into the surface

 

[Kfox]

Of course it matters, otherwise saying it's smaller on the inside has no meaning. You've got to compare a boundary (which is all you see from the outside) with a volume it encloses. Unless you have some expectation of how the boundary relates to the volume it encloses (which needs geometry) then you have no way to say it must be smaller.

I think you are over thinking all of this. You don't have to understand geometry to realize the inside of a structure is always gonna be smaller than the exterior; all you need is a little common sense.

 

[Polymath257]

I think you are over thinking all of this. You don't have to understand geometry to realize the inside of a structure is always gonna be smaller than the exterior; all you need is a little common sense.

Maybe, if someone reasonably intelligent says something that you think violates common sense, you might want to ask whether they know something you don't.

 

[muhammad_isa]

I *did* explain it: any positive number times infinity is equal to infinity. That *is* the explanation..

You're having me on?
If the volumes are the same, there has been no expansion of volume.
I don't care what intricate mathematics you wish to show.
If infinity[before] = infinity[after], then no expansion has occurred.

 

[Polymath257]

You're having me on?
If the volumes are the same, there has been no expansion of volume.
I don't care what intricate mathematics you wish to show.
If infinity[before] = infinity[after], then no expansion has occurred.

And that is simply incorrect. Your understanding of the properties of infinity is poor. And that is harming your understanding of other things in this discussion.

The distances between points at rest increased: that is what expansion means in this context.

 

[muhammad_isa]

Your understanding of the properties of infinity is poor..

Is that the best you can do to explain it?

The distances between points at rest increased: that is what expansion means in this context.

Ah, so the distance between "points" increases, while the volume of space doesn't change .. is that it?

Technically, neither space nor objects in space move. Instead it is the metric (which governs the size and geometry of spacetime itself) that changes in scale. As the spatial part of the universe's spacetime metric increases in scale, objects become more distant from one another at ever-increasing speeds.
Expansion of the universe - Wikipedia

No! It is not it. The inflation/expansion of the universe affects the whole matrix of space-time.

According to inflation theory, during the inflationary epoch about 10^−32 of a second after the Big Bang, the universe suddenly expanded, and its volume increased by a factor of at least 10^78.
Expansion of the universe - Wikipedia

That says that the volume of space actually increases.
Suggesting that [infinity * 10^78] is greater than [infinity] is obviously incorrect.

You have already agreed that they are equal.
i.e. the volumes would be equal after inflation, if the universe.started out with infinite volume.

 

[Polymath257]

Is that the best you can do to explain it?

Would you like me to start a thread about the different types and uses of infinity?

Ah, so the distance between "points" increases, while the volume of space doesn't change .. is that it?

More properly, the volume is undefined. Saying it is infinite is just a way of saying the volume doesn't have a finite (real) value.

So, for example, a cube with sides of 1.2 units has a volume of 1.728 cubic units. Those numbers are real numbers (decimal numbers).

The volume of Euclidean space is larger than any such finite volume and so is said to be infinite. But, unlike cardinality, we don't say this use of the term 'infinity' is a number. For infinite things, the term 'number' is usually restricted to cardinals and ordinals.

 

[muhammad_isa]

The volume of Euclidean space is larger than any such finite volume and so is said to be infinite. But, unlike cardinality, we don't say this use of the term 'infinity' is a number. For infinite things, the term 'number' is usually restricted to cardinals and ordinals.

Yes, you seem to be "backtracking" now.

Earlier, you stated: "I'm not sure what you mean by 'absolute infinity' as opposed to simply 'infinite'. But I said how I am using the term 'finite' above."
and "if you inflate an infinite volume (or deflate it), the volume is still infinite. 5*infinity=infinity"

Are you still claiming that the volume before inflation is infinite,
but is less than the volume after inflation, which is also infinite?

..or are you saying that the volumes are equal?
Which is it?

 

[Polymath257]

Yes, you seem to be "backtracking" now.

I am not backtracking. I am defining the terms involved in a way that is hopefully more clear to you.

Earlier, you stated: "I'm not sure what you mean by 'absolute infinity' as opposed to simply 'infinite'. But I said how I am using the term 'finite' above."
and "if you inflate an infinite volume (or deflate it), the volume is still infinite. 5*infinity=infinity"

Are you still claiming that the volume before inflation is infinite,
but is less than the volume after inflation, which is also infinite?

..or are you saying that the volumes are equal?
Which is it?

First, I was clear that we do not know whether the volume before or after were infinite or not.

I said that if the volume before was infinite (larger than any finite amount), then the volume after was also infinite (larger than any finite amount) and vice versa,

If the volume is larger than any finite amount, we say the volume is infinite. That is simply another way of saying 'larger than any finite amount'.

To say the volumes are 'equal' means you need to understand the technicalities of the phrase 'two infinite volumes are equal'. The *only* thing that it means is that both are larger than any finite volume.

Does that make sense?

And, if it is the case that the volume is larger than any finite amount, then multiplying it by 10 (or dividing it by 10), it will *still* be larger than any finite amount.

 

[Polymath257]

Is that the best you can do to explain it?


Ah, so the distance between "points" increases, while the volume of space doesn't change .. is that it?

Technically, neither space nor objects in space move. Instead it is the metric (which governs the size and geometry of spacetime itself) that changes in scale. As the spatial part of the universe's spacetime metric increases in scale, objects become more distant from one another at ever-increasing speeds.
Expansion of the universe - Wikipedia

No! It is not it. The inflation/expansion of the universe affects the whole matrix of space-time.

According to inflation theory, during the inflationary epoch about 10^−32 of a second after the Big Bang, the universe suddenly expanded, and its volume increased by a factor of at least 10^78.
Expansion of the universe - Wikipedia

That says that the volume of space actually increases.
Suggesting that [infinity * 10^78] is greater than [infinity] is obviously incorrect.

You have already agreed that they are equal.
i.e. the volumes would be equal after inflation, if the universe.started out with infinite volume.

Yes, any *finite* part of the volume increased by that factor.

 

[Kfox]

Maybe, if someone reasonably intelligent says something that you think violates common sense, you might want to ask whether they know something you don't.

Or perhaps if someone says something that violates common sense, you shouldn't just assume they are reasonable intelligent.

 

[ratiocinator]

I think you are over thinking all of this.

It doesn't look to me as if you're thinking about it at all.

You don't have to understand geometry to realize the inside of a structure is always gonna be smaller than the exterior; all you need is a little common sense.

But you haven't thought about what that means. You're applying a property of space to try to tell be that space has no properties. Because everybody is familiar with Euclidean space, they think it's just 'common sense'.

The subject of characterising the properties of space is called geometry: "[Geometry] is concerned with properties of space that are related with distance, shape, size, and relative position of figures."

Or perhaps if someone says something that violates common sense, you shouldn't just assume they are reasonable intelligent.

The problem is that we know that "common sense" (intuition) doesn't apply at the most basic level of physics (and why should it?) General relativity (which you are effectively dismissing) is a well established theory, whose every prediction, that we have been able to test, has been correct. The GPS system has to account for the curvature of space-time:-

"The satellite clocks are moving at 14,000 km/hr in orbits that circle the Earth twice per day, much faster than clocks on the surface of the Earth, and Einstein's theory of special relativity says that rapidly moving clocks tick more slowly, by about seven microseconds (millionths of a second) per day.

Also, the orbiting clocks are 20,000 km above the Earth, and experience gravity that is four times weaker than that on the ground. Einstein's general relativity theory says that gravity curves space and time, resulting in a tendency for the orbiting clocks to tick slightly faster, by about 45 microseconds per day. The net result is that time on a GPS satellite clock advances faster than a clock on the ground by about 38 microseconds per day."

-- Einstein's Relativity and Everyday Life​

And if you think GR lacks "common sense" you should take a look at quantum mechanics, yet it is vital in many engineering applications (it's not just pure science) including the design of semiconductor devices. Hence the device you are using access this forum directly relies on theories that defy "common sense".

 

[muhammad_isa]

I said that if the volume before was infinite (larger than any finite amount), then the volume after was also infinite (larger than any finite amount) and vice versa,

If the volume is larger than any finite amount, we say the volume is infinite. That is simply another way of saying 'larger than any finite amount'.

To say the volumes are 'equal' means you need to understand the technicalities of the phrase 'two infinite volumes are equal'. The *only* thing that it means is that both are larger than any finite volume.

They cannot be both equal and less than or greater than simultaneously.
I think it is you who are confused. Either you refer to an infinity which is 'absolute', or you refer to the 'transfinite' notion where one can be greater than the other.
In other words, you are saying that 5 * [infinity before] = [infinity after].

These concepts are to be strictly differentiated, insofar the former[transfinite] is, to be sure, infinite, yet capable of increase, whereas the latter[absolute] is incapable of increase and is therefore indeterminable as a mathematical concept.
-G. Cantor-

To look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i.e., we are able to "produce" bigger and bigger sets.
-A. Fraenkel-

 

[ratiocinator]

They cannot be both equal and less than or greater than simultaneously.
I think it is you who are confused. Either you refer to an infinity which is 'absolute', or you refer to the 'transfinite' notion where one can be greater than the other.

You really are getting bogged down in irrelevances and tying yourself on knots over the concept of infinity. What the expansion of space means, is that if two objects are not somehow bound together (say by gravity) and are stationary in the comoving coordinate system used in cosmology, then the (proper) distance between them will increase, see: Comoving and proper distances - Wikipedia.

If the total volume of the universe is infinite, then it will remain infinite.

 

[muhammad_isa]

If the total volume of the universe is infinite, then it will remain infinite.

That is a contradiction in terms, as far as I'm concerned.

A volume that can increase is not truly infinite, if it can be increased.
One can only say that it is a "very large" volume, which implies a finite volume.

The concept of an infinity which can be increased is a mathematical construct, and cannot physically exist. It is only a mathematical tool.

 

[ratiocinator]

A volume that can increase is not truly infinite, if it can be increased.

I didn't say it could, that's the point. The expansion of space is defined by increasing proper distances within it. It would only apply to the total volume if it were finite. If you take a finite volume within space, then it will get larger.

 

[Polymath257]

That is a contradiction in terms, as far as I'm concerned.

A volume that can increase is not truly infinite, if it can be increased.

Again, we don't give an actual number to the measure of the volume. It is simply more than any finite volume.

One can only say that it is a "very large" volume, which implies a finite volume.

No, that is NOT the same. Being larger than any finite volume is a possibility.

We say that the volume is infinite if it is not finite. It is that simple.

So, if the volume of the universe is not finite, and it expands, then it is not finite after the expansion.

If you feel better about it, simply say that the measure of the volume is undefined but that it is not finite.

The concept of an infinity which can be increased is a mathematical construct, and cannot physically exist. It is only a mathematical tool.

You seem to be convinced of this. But I see no reason for that conviction.

We have described how to interpret this, but you fail to grasp it.

A volume is infinite if it is not finite. That's it. And, if it is not finite before an expansion, it is not finite after.

Seems simple enough to me.

 

[Polymath257]

They cannot be both equal and less than or greater than simultaneously.

It can be greater than any finite amount.

I think it is you who are confused. Either you refer to an infinity which is 'absolute', or you refer to the 'transfinite' notion where one can be greater than the other.
In other words, you are saying that 5 * [infinity before] = [infinity after].

I am saying that if it is not finite before, then expanding by a factor of 5 will give something that is also not finite.

These concepts are to be strictly differentiated, insofar the former[transfinite] is, to be sure, infinite, yet capable of increase, whereas the latter[absolute] is incapable of increase and is therefore indeterminable as a mathematical concept.
-G. Cantor-

To look at the universe of all sets not as a fixed entity but as an entity capable of "growing", i.e., we are able to "produce" bigger and bigger sets.
-A. Fraenkel-

Cantor and Fraenkel were talking explicitly about the sizes of sets (cardinality). That is a different notion from volume.

For an expanding universe, we are interested in a volume. And volumes are NOT cardinalities.

 

[Polymath257]

Or perhaps if someone says something that violates common sense, you shouldn't just assume they are reasonable intelligent.

There are a great many truths about the world that violate common sense at first.

For example, a man as smart as Aristotle thought it to be common sense that heavy things fall faster than light things. But it is false.

that is why we need to question *all* of our ideas, especially those we think to be 'common sense'. The most common place to make a mistake in reasoning is right after the word 'obviously'.