So you mean they're identical collections, but their elements aren't equal? :S — fdrake
Only those who (perhaps naively) embrace Cantor's mathematical model of a continuum as isomorphic to the real numbers would affirm this metaphysical implication, thus claiming that space is somehow composed of dimensionless points. Such a collection of distinct objects is a bottom-up construction in which the parts are ontologically prior to the whole. By contrast, I would argue that true continuity is a top-down construction in which the whole is ontologically prior to the parts. Positions are only actual if and when they are marked for a purpose, such as measurement; otherwise, they are strictly potential.Or some hold that space is continuous, implying an actually infinite collection of distinct spacial positions in a unit of space. — Devans99
They don't imply; you infer. They hold, if you're referring to a laymen's Hawking, that time is unbounded, like the surface of a sphere, & etc.For example, many cosmologists hold that time has no start, implying an actual infinity of past time. — Devans99
"Actually" and "actual" two different words.Then there is the axiom of infinity. It states that there exists a set with an actually infinite number of members — Devans99
Your feelings irrelevant. And definitions aren't.I feel a line that has existence outside our minds must be constituted of something - points or sub-segments or such. The most common definition is that a line is a set of actually infinite points. — Devans99
By contrast, I would argue that true continuity is a top-down construction in which the whole is ontologically prior to the parts. Positions are only actual if and when they are marked for a purpose, such as measurement; otherwise, they are strictly potential. — aletheist
They don't imply; you infer. They hold, if you're referring to a laymen's Hawking, that time is unbounded, like the surface of a sphere, & etc. — tim wood
Where did any sane person aver that there were any actual infinities of anything? — tim wood
Do you understand the wordI already gave you an example (Cantor). — Devans99
Do you understand the word
"aver"? Do you understand "theory? Do you understand "assumption"? When did Hawking became a "fringe" cosmologist? And Cantor apparently invoked God: are you prepared to demonstrate what exactly Cantor meant by the term in terms of any reification of his own ideas? — tim wood
I'll try a different form of the question: are you aware of any respectable demonstration that there is any actual infinity of anything? — tim wood
If? If? Wtf if?! Who cares if? If the moon were made of green chase. If frogs had wings. If pentagons had 93 sides.if you believe — Devans99
are you aware of any respectable demonstration that there is any actual infinity of anything? — tim wood
Again they are in a one-to-one correspondence so they are identical collections. — Devans99
We have changed something and it has not changed. — Devans99
Except that you have refused to acknowledge that I explicitly made clear I was not asking for what anyone merely believed. What, may I ask, is your problem? — tim wood
I think you are splitting hairs between 'believe' and 'aver'. — Devans99
No, it would not. If space is truly continuous, then it is not composed of distinct positions. We arbitrarily impose distinct positions on space for various purposes, including measurement. They are entia rationis, creations of the mind, not constituents of reality itself.Assuming for the sake of argument space is a continuum (which I do not believe), when I move my hand from left to right, it would pass through an actual infinity of distinct positions. — Devans99
There can be no infinite collections in reality because an infinite collection of something would leave no room for anything else in reality. The infinite collection would BE reality. Maybe reality is an infinite collection of space-time with finite amount of energy-matter.The point I am trying to make with the bananas is that there can be no actually infinite collections in reality because it leads to contradictions (that something can be changed and yet not change). — Devans99
Are you saying that there is something beyond space and time? You're implying a boundary, but a boundary with what? Space and time are infinite. Energy/matter isn't.I believe time, space, matter/energy are all finite and discrete. — Devans99
No, it would not. If space is truly continuous, then it is not composed of distinct positions. We arbitrarily impose distinct positions on space for various purposes, including measurement. They are entia rationis, creations of the mind, not constituents of reality itself. — aletheist
Are you saying that there is something beyond space and time? You're implying a boundary, but a boundary with what? Space and time are infinite. Energy/matter isn't. — Harry Hindu
I bow to your maths knowledge, but I'd still maintain that there is nothing in our universe to which the axiom 'when it is changed, it does not change' applies. — Devans99
This is the same mistake that Zeno made. Positions 0 and 1 do not exist unless and until we arbitrarily mark them as such, and the same is true of any and all intermediate positions between them. We cannot proceed to mark an actual infinity of those; we can only hypothesize that there is a potential infinity of such positions in accordance with our arbitrary system of measurement--for example, the real numbers. Moreover, the physical distance from 0 to 1 does not matter--whether it is 1 kilometer, 1 meter, or 1 millimeter, the real numbers assign the same multitude of intermediate positions between them; likewise if we change our designation of the second position from 1 to 2 or any other value. All this demonstrates that real space is continuous; we only model it as discrete for practical purposes.If my hand passes from position 0 to 1, then it passes through positions 1/2, 1/4, 1/8, 1/16, ... 1/∞. — Devans99
Positions 0 and 1 do not exist unless and until we arbitrarily mark them as such, and the same is true of any and all intermediate positions between them. — aletheist
No, it only marks positions 0 and 1; marking any individual intermediate positions would require their explicit designation, and we can only ever do that for a finite quantity of them.But the action of movement does mark positions 0 and 1 and all positions in-between. — Devans99
No, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion.We know that our hand actually passed through all those positions in the past, so if space is a continuum then motion actualises an infinity of positions. — Devans99
No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions.As pointed out in the OP, actual infinity is absurd, so therefore space is not a continuum. — Devans99
No, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion. — aletheist
No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions. — aletheist
So the boundary is other universes. Time is relative change. Change in each universe is relative to the other, so time and space would encompass the multiverse, not just one universe. Then, your infinite collection would be universes. — Harry Hindu
No, the only actual intermediate positions are the ones that we individually mark. There is a potential infinity of such positions, but we can only mark (and thereby actualize) a finite quantity of them. Again, continuous motion is the reality, while discrete positions are our invention.If my hand moves from position 0 to 1, it is guaranteed to pass through positions 0.5, 0.25, 0.125, etc... Or are you saying it somehow skips over intermediate positions? That would be discrete movement. — Devans99
Given whatever path your hand actually followed in moving between its initial and final positions, it indeed moved through all possible intermediate positions along that particular path--again, a potential infinity, not an actual infinity. In order to describe that path, we would need to mark various intermediate positions and assign coordinates to them relative to the initial and final positions, taking the distance between them as our arbitrary unit of length. The more positions we mark and measure, the more accurate the resulting description of the motion--but it will never be perfectly accurate, since we will never be able to mark and measure all the possible positions; merely a finite quantity of them, which are the only actual positions.But movement is something that actually happened in the past - my hand in the past moved through all possible positions - so that must be an actual infinity of positions. — Devans99
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