Two collections of identical objects in a one-to-one correspondence are, by mathematical induction, identical collections. — Devans99

So you mean they're identical collections, but their elements aren't equal? :S

I use the term 'collection' because I'm not a mathematician I'm afraid - I mean a group of (in this case) identical objects. I'm not familiar with multisets but it sounds like the same concept as I'm thinking of, IE:

{b, b, b, b, ... }

Is a multiset where 'b' has multiplicity=∞ I guess?

So you mean they're identical collections, but their elements aren't equal? :S — fdrake

You are confusing me. Every element in both sets is indistinguishable and they are lined up with each other (in one-to-one correspondence):

{b, b, b, b, ... }
{b, b, b, b, ... }

We know all elements are identical 'b's and the collections both have the same infinite cardinality. The first elements are in one-to-one correspondence. If the nth element is in one-to-one correspondence, then so is the nth+1 element. So the two collections are surely equal?

Or some hold that space is continuous, implying an actually infinite collection of distinct spacial positions in a unit of space. — Devans99

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.

For example, many cosmologists hold that time has no start, implying an actual infinity of past time. — 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.

Then there is the axiom of infinity. It states that there exists a set with an actually infinite number of members — Devans99

"Actually" and "actual" two different words.

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

Your feelings irrelevant. And definitions aren't.

I have to conclude one of two things from this exchange: 1) You're incapable of answering, or 2) You refuse to answer. 1) seems unlikely. That leaves 2).

Maybe you do not know what "aver" means. People can and do claim and believe all kinds of things. Neither is to aver them. To aver means to maintain that something is so, to support or prove the proposition that the thing is so. Let's try again:

Where did any sane person aver that there were any actual infinities of anything?

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

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. Every intermediate position is therefore actualised by motion and we are left with the conclusion that the particles in my hand passed through an actual infinity of intermediate positions.

I believe instead that each particle in my hand performs something like a quantum jump down at a microscopic level and that there are no true continua in our universe.

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

You are just choosing one (fringe) theory of cosmology; many theories do assume past time is actually infinite. See CCC by Roger Penrose as an example.

Where did any sane person aver that there were any actual infinities of anything? — tim wood

I already gave you an example (Cantor). You need look no further than the philosophy forum for more examples: many folks on here argue that time has no start or that space is a continua, see https://thephilosophyforum.com/discussion/comment/367423 for example - and my response https://thephilosophyforum.com/discussion/comment/367425.

I 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?

I'll try a different form of the question: are you aware of any respectable demonstration that there is any actual infinity of anything?

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

Lets look at what Cantor said again:

"Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. - Georg Cantor

I've highlighted the relevant piece - he is talking about actual infinities in our universe.

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

Yes, if you believe space is a continuum (which many people do), then simply moving your hand from left to right actualises an infinity of intermediate positions.

Why omit God? Cantor didn't: he thought God mattered. Do you think that's an irrelevant detail? Of course, if you invoke God and that's your test and standard - your evocation -, then you can have anything you want.

if you believe — Devans99

If? If? Wtf if?! Who cares if? If the moon were made of green chase. If frogs had wings. If pentagons had 93 sides.

Again:

are you aware of any respectable demonstration that there is any actual infinity of anything? — tim wood

I think you're not. But can you say so? Asking the same question four or five times taxes my patience, which you have manifestly abused. So answer straight or withdraw. .

Again they are in a one-to-one correspondence so they are identical collections. — Devans99

I'm thinking of them as sequences.

The first sequence, the one in your OP, is the sequence "the constant sequence where every element is b".

The second sequence, the second one in your OP, is the sequence "the sequence derived from the previous constant sequence by deleting every second element". This is also "the constant sequence where every element is b" since the sequence consists solely of the constant b... The constant is b, and every element is b.

The two sequences are identical. They also have the same cardinality. They are not identical because they're in a one-one correspondence, they're identical because all their elements are the same and in the same order.

I have already given you two clear demonstrations that some people believe in actually infinite collections:

- Infinite past time (leading to a belief that an actual infinity of moments has occurred in the past)
- Space is a continuum (leading to a belief that motion actualise an infinity of intermediate positions)

Not sure what else I can say here :sad:.

I have already given you two clear demonstrations that some people believe in actually infinite collections: — 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?

OK. But the problem remains: removing half the elements from the second sequence results in a sequence that is identical to the first sequence. We have changed something and it has not changed. I am not a mathematician as you can tell but to me the maths seems to lead to a straight forward absurdity, so there must be something wrong with our assumptions. I feel the problem is that we have assumed that actual infinity can exist. If we remove actual infinity and conduct the same operations on finite sequences, the absurdity does not occur. Hence my assertion that actual infinity is an illogical concept that can only occur in our minds and is not a feature of the real world. Hence the axiom of infinity should not claim that 'there exists' an actually infinite set - because such a set does not exist in reality.

We have changed something and it has not changed. — Devans99

A function can have fixed points. These are when you get the same thing out as what you get in.

The operation "take a sequence, delete every second element, output the sequence without the deleted elements" is a function on the space of infinite sequences to the space of infinite sequences. It has a fixed point (produces an output equal to the input) whenever:

{a1, a2, a3, a4, a5, a6, a7, a8, ...}={a1, a3, a5, a7, ... }

a1=a1, a2=a3, a3=a5, a4=a7...

whenever all sequence elements are equal.

Notably, the sequence {1,2,3,4,5,6,...} of naturals produces the odds {1,3,5,...} upon applying this function. So, applying the function can produce the same thing (when the input sequence is a constant sequence), or produce a different thing (when the input sequence is not a constant sequence).

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'. Cantor states he has 'firm conviction' in the belief of the existence of the actually infinite. Aristotle believed/averred time had no start because:

- Time had no start because for any time, we can imagine an earlier time.
- Time had no start because everything in the world has a prior cause.

Newton held this belief also and many people still hold this belief. And such a belief leads to a belief in actual infinity. Similarly, the commonly held belief in space/time being a continuum again leads to a belief in actual infinity.

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. Maths may produce alternative realities in which such an axiom may hold but these realities are not our reality.

It is very convenient to regard the naturals, the reals as actually infinite sets, but this is merely a mental convenience to allow us to reason with all the naturals/reals. It does not mean that anything with the structure of the naturals/reals can exist in reality - they are a purely mental construct.

We can imagine all sorts of things in the mind - levitation, talking trees, actual infinity - but only a subset of what we can imagine is possible in reality.

I have a similar reaction to the axiom of choice. How can balls be selected from actually infinite bins? It is not possible to complete such a selection - the selection process goes on forever so it is impossible to complete. So again, we have something that is impossible in reality, but we can imagine it in our minds where the impossible is possible.

I think you are splitting hairs between 'believe' and 'aver'. — Devans99

Then either you don't understand English or you choose not to - and you have certainly ignored the substance of my question. And I have elsewhere challenged you to put your money here your mouth is on the matter of beliefs, and you there chose not to do that. It's clear you're ignorant and confused, but also that you refuse to give up either. That's a personality problem.

And if you don't like being called out, then learn how to discuss a topic, or to make an argument.

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

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.

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

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.

I believe time, space, matter/energy are all finite and discrete. — 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.

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

I don't understand you. If my hand passes from position 0 to 1, then it passes through positions 1/2, 1/4, 1/8, 1/16, ... 1/∞. So movement in a continuum actualises infinity, which is why I think everything must be discrete.

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 think that there maybe pure nothing beyond spacetime - no time at all so nothing can be in any way whatsoever.

Time probably started 14 billion years ago. All expanding universes have a start in time and our universe has always been expanding.

Space has been expanding at a finite rate for 14 billion years so it must be finite in size.

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

I guess it comes down to whether you require that the application of a function to an input necessarily produces a distinct output (IE, no fixed points). Trying to avoid this blocks lots of useful things.

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.

If my hand passes from position 0 to 1, then it passes through positions 1/2, 1/4, 1/8, 1/16, ... 1/∞. — 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.

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

But the action of movement does mark positions 0 and 1 and all positions in-between. 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.

As pointed out in the OP, actual infinity is absurd, so therefore space is not a continuum.

But the action of movement does mark positions 0 and 1 and all positions in-between. — Devans99

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.

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, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion.

As pointed out in the OP, actual infinity is absurd, so therefore space is not a continuum. — Devans99

No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions.

No, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion. — aletheist

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.

No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions. — aletheist

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.

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

Beyond the boundary is nothingness IMO. Nothing cannot be actually infinite because it is nothing. If it is other universes then they cannot be actually infinite because it would lead to the absurdities referenced in the OP. Or see here for another example of the absurdity of actual infinity:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox

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

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.

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

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.