I don't think we will reach agreement on this point.

Or another way. Let' suppose you do pass through an infinity of points, an uncountable infinity of points. How long does that take? Consider the problems you have with that!

It takes say 1 second for my hand to pass through 1 meter of space and thereby pass through, if space is a continuum, an actual infinity of intermediate points. This is what is referred to as a supertask. I believe all supertasks are impossible - they lead to paradoxes such as:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
https://en.wikipedia.org/wiki/Thomson%27s_lamp

Paradoxes indicate we have a wrong assumption somewhere, in this case, the assumption that it is possible to complete an actually infinite number of steps in a finite time is suspect. So I doubt that true continuity is possible.

The one-to-one correspondence thing is a hoax I think. They just line up a few terms and say "it goes on forever so they are the same size". You can do that with ANY infinity whatsoever.

They would have to see the infinities side by side in their infinity to know they are the same size. Cantor says he can find numbers that aren't in the countable group by his diagonal "proof". But 2 is not on the odd numbers, yet the odd numbers are the same as all the whole numbers? They haven't worked this out properly

Paradoxes indicate we have a wrong assumption somewhere, — Devans99

The operative word here being "somewhere."

So I doubt that true continuity is possible. — Devans99

A 'true" continuity? What in the world is that? And while you're at it, please provide an epistemological ground for your "doubt." You seem to think it and other of your feelings are substantive with respect to any of your claims.

And what does "continuity" have to do with it? There are plenty of discontinuous points for your illustration without worrying about any continuum. Keeping in mind you have already been educated about the distinctions between continuity and continuum. You seem determined to remain ignorant. but there's another more accurate if less kind word for that.

I recommend you construct for yourself something like a wind-driven Tibetan prayer wheel, on which you can inscribe your thoughts and contemplate them as they go 'round in circles.

Gregory, meet Devans99. A marriage made in Heaven - maybe it's Heaven. Might not be.

He is obsessed with proving God, so he ain't in my Heaven :)

Paradoxes indicate we have a wrong assumption somewhere, in this case, the assumption that it is possible to complete an actually infinite number of steps in a finite time is suspect. So I doubt that true continuity is possible. — Devans99

The wrong assumption in this case is that the true continuity of space would require your hand to complete an actually infinite number of steps by passing through an actually infinite number of intermediate positions. As I have explained repeatedly now, the only individual positions that exist are whatever finite quantity of them we explicitly mark. If you still want to insist that real space is discrete, then make your case, but please stop pretending that this particular objection to its continuity is valid.

Every video and article I read about one-to-one-correspondence is garbage. They arbitrarily move infinity, place the first units together, send them off into infinity (without proving anything yet about uncountable vs countable) and exclaim "they are the same!". Nevermind ALL infinities are composed of units and you can do this trick with ALL infinities

And while you're at it, please provide an epistemological ground for your "doubt." — tim wood

If space is continuous then my hand moves through an actually infinite number of intermediate positions. But actual infinite leads to contradictions. So I doubt that it can exist.

Also there is an information based argument. True continua would be structurally identical no matter what the size. So a millimetre of space would have the same structure as a light year. Suggesting the same information content. That is hard to swallow.

Why are you on about this? Everyone agrees with you and no one disagrees. As was said in my parochial school to the girls, do you want a medal or a chest to pin it on?

Every video and article I read about one-to-one-correspondence is garbage. They arbitrarily move infinity, place the first units together, send them off into infinity (without proving anything yet about uncountable vs countable) and exclaim "they are the same!". Nevermind ALL infinities are composed of units and you can do this trick with ALL infinities — Gregory

As explained above, you can use mathematical induction to see that all the bananas in each sequence are in one-to-one correspondence. So for all finite numbers n, the bananas are in one-to-one correspondence. At 'actual infinity' are the bananas in one-to-one correspondence? I think not because actual infinity is not a well defined concept. It is an illogical concept that cannot occur in reality IMO.

Why are you on about this? Everyone agrees with you and no one disagrees. As was said in my parochial school to the girls, do you want a medal or a chest to pin it on? — tim wood

I'm glad we are in agreement. I think though that not everyone agrees with us. For example, the bedrock of maths:

1. Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'.

2. Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible.

I believe that the universe is both finite and discrete (=free of actual infinities). Plenty of people disagree so I think there is still a discussion to be had.

If space is continuous then my hand moves through an actually infinite number of intermediate positions. — Devans99

False. Again, if space is a continuous whole, then it is not composed of individual and distinct positions.

Suggesting the same information content. That is hard to swallow. — Devans99

I know you saw this, from @fishfry in the bijection thread:

(5) "Cantor's surprise." For any positive integer n, the n-dimensional Euclidean space Rn has the same cardinality as the real numbers.

Cantor originally thought that the real numbers had cardinality ℵ1; and the plane R2 had cardinality ℵ2, and Euclidean 3-space R3 had cardinality ℵ3, and so forth. [In math, n dimensional space just means the set of all n-tuples of real numbers, with pointwise addition and scalar multiplication by reals, just as with the usual x-y plane and x-y-z space....

He was surprised to realize that in fact all finite-dimensional Euclidean spaces have the same cardinality. Here's the proof. We'll show that the open unit interval and the open unit square have the same cardinality. That is, we'll show a bijection between the real numbers strictly between 0 and 1, and the set of ordered pairs in the x-y plane each of whose coordinates are strictly between 0 and 1.

Suppose (x,y) is a point in the open unit square with decimal representations x=.x1x2x3... and y=.y1y2y3... respectively. We map the pair (x,y) to a single real number by interleaving the digits to get .x1y1x2y2x3y3....

It's clear that you can reverse this process. Given any real number you can de-interleave its digits to get a pair of real numbers. We can extend the result from the unit interval to the entire real line via the tangent/arctangent. Of course this bijection is highly discontinuous, it has no nice properties at all.

You can clearly interleave n-digits this way, and that's the proof. When Cantor discovered this result he wrote to his friend Dedekind: "I see it but I don't believe it!" — fishfry

(Thank you, fishfry.)

No need to swallow anything, just learn!

1. Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'. — Devans99

You've already had it explained to you by others that you're completely out of court, here. There's no two in reality, are you prepared to argue that two does not exist?

2. Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible. — Devans99

"Should?" Clearly you do not understand what the simple English means.

Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'. — Devans99

This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual.

Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible. — Devans99

This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible.

In summary, mathematics is the science that draws necessary conclusions about strictly hypothetical states of affairs. That includes its application to infinity--never actual infinity, always potential infinity.

I know you saw this, from fishfry in the bijection thread: — tim wood

You are missing the point I'm making. I believe that the naturals and reals are purely mental constructs. They exist in our minds only (where the impossible is possible). They have the same status as talking trees and levitation - illogical/impossible things can exist in our minds but they cannot exist in reality.

In the instances of the axiom of infinity and axiom of choice, maths departs radically from reality and that departure leads other folks astray (physicists, cosmologists, philosophers). That is why I raised this thread.

If you disagree, then please give an example of something that has the structure of the naturals from nature.

This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist

An actual infinity of naturals (IE a set with a greater than any number of elements) is impossible.

This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible. — aletheist

It is not logically possible to complete a task that has no end.

I believe that the naturals and reals are purely mental constructs. They exist in our minds only (where the impossible is possible). They have the same status as talking trees and levitation - illogical/impossible things can exist in our minds but they cannot exist in reality. — Devans99

All of these are logically possible, just not metaphysically possible.

An actual infinity of naturals (IE a set with a greater than any number of elements) is impossible. — Devans99

No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity.

It is not logically possible to complete a task that has no end. — Devans99

Incorrect--it is logically possible, just not metaphysically possible.

All of these are logically possible, just not metaphysically possible. — aletheist

They are not logically possible as you can see from the argument in the OP - assuming that they are logically possible leads to a contradiction. Or if you don't like that argument, see:

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

So the logical assumption of the existence actual infinity leads to paradoxes/contradictions. Paradoxes/ Contradictions indicate a logical error has been made, in this case the assumption that actual infinity is a logical concept.

No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity. — aletheist

A potential infinity is like a limit - something approaches but never actually reaches that limit. Actual infinity is equivalent to the claim that the natural numbers exist - the axiom of infinity says they actually exist - not potentially. That leads to the conclusion that there is a set that exists with a greater than any number of elements.

Incorrect--it is logically possible, just not metaphysically possible. — aletheist

It is not logically possible to reach the end of something that has no end.

You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual impossibility does not entail logical impossibility. Mathematical existence is not metaphysical actuality. The infinity of the natural numbers is potential, not actual. Continuity of space does not require an actual infinity of distinct positions.

Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist

Question: Do you think that "5 is prime" is true? Or merely logically possible? Or a complete fiction made up by evil set theorists?

I think "5 is prime" presents a challenge to those who say that math isn't true. "Actual" as you put it. Is "5 is prime" actual or not? And if not, what is it?

You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual impossibility does not entail logical impossibility. Mathematical existence is not metaphysical actuality. The infinity of the natural numbers is potential, not actual. Continuity of space does not require an actual infinity of distinct positions. — aletheist

But logical impossibility, which actual infinity is (as demonstrated by the paradoxes/contradictions) does imply actual impossibility.

The set of natural is defined in maths as an actual infinity:

'The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers' - http://mathworld.wolfram.com/AxiomofInfinity.html

A set containing all natural numbers exists both logically and actually... but that leads to logical contradictions... so such a set is not logically possible.

Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist

Do you think that "5 is prime" is true? — fishfry

Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. Do you think that either of these terms denotes something actual?

The set of natural is defined in maths as an actual infinity
'The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers' — Devans99

No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an extreme platonist, claims that there is an actual set containing all the natural numbers.

No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an extreme platonist, claims that there is an actual set containing all the natural numbers. — aletheist

Cantor did claim actual infinity exists:

"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

How exactly can the set of naturals be potentially infinite? That would suggest it is a partially defined set, IE an undefined set. It is defined as an actual infinity (all sets are actual).