We add 1 banana to the sequence (=it should change quantitatively and qualitatively).
But is does not change quantitatively(∞+1=∞) or qualitatively(still identical rows of identical bananas). — Devans99

It does change. The problem is in your definition of identicalness.

1. Logical identicalness. I'll use examples to make it clear.

Charles Lutwidge Dodgson is identical to Lewis Carroll. There's only ONE object but with different names. Carl Lutwidge Dodgson and Lewis Carroll can occupy the same space at the same time. There's absolutely no difference between them.

2. Xerox identicalness. Identical twins or two instances of the same car model. Identical twins or two instances of a car model cannot occupy the same space at the same time. There's a difference there.

Your bananas are not type 1 identical because then there would be only ONE banana. Ergo, your bananas are type 2 identical but then there's a difference between each instance of such identicalness by virtue of their inability to occupy the same space at the same time. It's this difference that produces the change in your two sets.

but then there's a difference between each instance of such identicalness by virtue of their inability to occupy the same space at the same time. — TheMadFool

Each banana has a different spatial position I agree, but the two sequences, ignoring their space time position are identical (same mass, same number of bananas, all bananas in one-to-one correspondence). The definition of a sequence (similar to a set) does not include their relative spacial positions - so the two sequences of bananas remain identical whilst they are changed. Which is a contradiction. Hence actual infinity cannot exist.

Each banana has a different spatial position I agree, but the two sequences, ignoring their space time position are identical (same mass, same number of bananas, all bananas in one-to-one correspondence). The definition of a sequence (similar to a set) does not include their relative spacial positions - so the two sequences of bananas remain identical whilst they are changed. Which is a contradiction. Hence actual infinity cannot exist. — Devans99

You can't ignore their space-time positions because it's critical to your argument. Why are there infinite bananas? Because they occupy different spaces? If they occupy the same space there would be only one banana.

You can't ignore their space-time positions because it's critical to your argument. Why are there infinite bananas? Because they occupy different spaces? If they occupy the same space there would be only one banana. — TheMadFool

I think the two sequences are identical in that if there is a banana at spacial position 1 in the first sequence, then there is also an identical banana at spacial position 1 in the second sequence.

If you don't buy my proof via contradiction that actual infinity is impossible, what about a proof via reductio ad absurdum:

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

So we have an infinite bag and we add ten balls and remove one. We repeat that an actually infinite number of time. At each finite step, there are 9n balls in the bag. At actual infinity, there are zero balls in the bag. Reductio ad absurdum, actual infinity is impossible.

that is not a clear cut paradox. There is much about infinity we don't know. You are making this too simple. However, if you think it is simple, then how is it that objects are infinitely divisible and therefore have infinite parts?

Objects are only infinitely divisible in our minds. The process of infinitely dividing an object goes on forever so it is not possible to complete - so it is an example of potential rather than actual infinity.

Things don't potentially have parts. They actually have parts

Things don't potentially have parts. They actually have parts — Gregory

And those parts are discrete and finite:

1. The parts can't be size zero or size undefined as then they could not constitute the whole
2. The parts can't be size 1/∞ because they would not constitute the whole and because ∞ leads to contradictions
3. So the parts must have a finite, non-zero size. IE discrete

You need to understand the difference between what is possible in your mind (where sure you can go on dividing forever and trees can also talk) and what is possible in reality.

So we have an infinite bag and we add ten balls and remove one. We repeat that an actually infinite number of time. At each finite step, there are 9n balls in the bag. At actual infinity, there are zero balls in the bag. Reductio ad absurdum, actual infinity is impossible. — Devans99

step 1, 9 balls numbered 2, 3, 4, 5, 6, 7, 8, 9, 10
step 2, 18 balls numbered 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
step n, 9n balls numbered n+1, n+2, n+3,...10n

The bag isn't empty because at the nth step even if the nth ball is removed, the balls numbered n+1, n+2,...10n are still in the bag.

But all balls numbered less than ∞ have been removed from the bag at the end of actually infinite steps. So there are zero balls in the bag.

Mathematical induction leads to 9n balls in the bag at each finite step (where n belongs to the natural numbers). You can't use mathematical induction for the infinite part of this problem as it applies for all n belonging to the natural numbers only and ∞ is not a natural number.

You need to understand that what the mind thinks geometrically of an object actually applies to it. How many parts does a banana ACTUALLY have? Don't say one because I can split it in half. And if I was all-powerful I could split it up infinitely. Objects are both infinite and finite at the same time. Logic proves this

You need to understand that what the mind thinks geometrically of an object actually applies to it. — Gregory

No it does not. My mind thinks of levitating dogs on a regular basis. No dogs levitate in reality. The mind is fundamentally illogical so the impossible is possible in the mind (with the aid of fuzzy, top-down, thinking). What is possible in reality is a completely different question from what is possible in the mind.

How many parts does a banana ACTUALLY have? Don't say one because I can split it in half. And if I was all-powerful I could split it up infinitely. Objects are both infinite and finite at the same time. Logic proves this — Gregory

If you were all powerful you could not split up a banana indefinitely - you would never finish the process so its impossible.

If we grant you were beyond time (timeless), you'd still need a hypothetical continuous substance to subdivide. But no such substance is possible as I pointed out here:

https://thephilosophyforum.com/discussion/comment/368770

Continua are mathematically impossible to define so in all likelihood, they do not exist in reality.

Discreteness does even mean anything. Does the discrete have parts? If not it's zero and has nothing to do with an object

Discreteness does even mean anything. Does the discrete have parts? If not it's zero and has nothing to do with an object — Gregory

Fundamentally everything is composed of parts and the parts cannot be zero sized or infinitely small. So they must have non-zero finite size. This has been our experience with matter (molecules, atoms, quarks etc...).

Something discrete is a part and it does not have any sub-parts. It is indivisible. A pixel on your computer screen is an imprecise analogy. In reality it is a particle - a packet of discrete energy, probably taking on a wave form. It is not dividable into subparts.

"everything is composed of parts and.. must have non-zero finite size."

Then you keep thinking of the division. The numbers of parts go on forever. So infinity does exist

Do you potentially have a hand, or do you actually have one? How can something have parts only potentially? How can something exist yet not have parts? These are all non-sensical statements.

But all balls numbered less than ∞ have been removed from the bag at the end of actually infinite steps. So there are zero balls in the bag.

Mathematical induction leads to 9n balls in the bag at each finite step (where n belongs to the natural numbers). You can't use mathematical induction for the infinite part of this problem as it applies for all n belonging to the natural numbers only and ∞ is not a natural number. — Devans99

Just check the math. In the 1st step the 1st ball is removed but there are more than 1 ball. In the 2nd step the 2nd ball is removed but there are more than 2 balls. Ergo at the nth step then nth ball is removed but there are more than n balls.

Then you keep thinking of the division. The numbers of parts go on forever. So infinity does exist — Gregory

∞ leads to contradictions so cannot exist. So neither can the inverse (1/∞) exist.

Do you potentially have a hand, or do you actually have one? How can something have parts only potentially? How can something exist yet not have parts? These are all non-sensical statements. — Gregory

I think that my hand exists in actuality and is composed of discrete parts that move through spacetime in discrete steps.

Just check the math. In the 1st step the 1st ball is removed but there are more than 1 ball. In the 2nd step the 2nd ball is removed but there are more than 2 balls. Ergo at the nth step then nth ball is removed but there are more than n balls. — TheMadFool

That applies for all n belonging to the natural numbers. But the proof is about what happens at the point of actual infinity, which is not a natural number. The proof is all about showing that actual infinity is impossible.

Does the discrete part have parts. If it doesn't, why isn't it zero?

Does the discrete part have parts. If it doesn't, why isn't it zero? — Gregory

I think it is likely that reality has a nature akin to a computer screen made out of pixels. So each discrete part has a size of 1 (say). It makes sense then (in the mind) to talk of size 1/2, but such cannot exists in reality; it is merely a mental construct (in the mind, the impossible is possible).

The world is physical, which is made of infinite parts. If it's more like a simulation, than why are you elsewhere arguing for a God?

The world is physical, which is made of infinite parts. If it's more like a simulation, than why are you elsewhere arguing for a God? — Gregory

I think the world is made of finite parts; an actual infinity of anything is impossible; see the argument in the OP. What would an actual infinity of things be? It would be a set with greater than any number of elements - nonsensical - such an aberration can only exist in our minds, where the impossible is possible.

I doubt it is a simulation, but if it is, I believe God is the ultimate cause of that simulation.

Are the parts non-zero? Do they have a front and back? Uh, the front and back are parts! This is the paradox started by Zeno. YOU don't have the solution

Are the parts non-zero? Do they have a front and back? Uh, the front and back are parts! This is the paradox started by Zeno. YOU don't have the solution — Gregory

Does a quark have a front or back? You can imagine it having so in your mind, but in reality it is an indivisible whole and it is not possible to address its front or back; only the whole unit.

A valid solution to Zeno's paradoxes is the universe is discrete and actual infinity does not exist.

It either has a back and front, or it doesn't. That is, it is either real or zero

t either has a back and front, or it doesn't. That is, it is either real or zero — Gregory

How do you tell the difference between front and back? You could fire photons at the discrete object. But the minimum wavelength of a photon is much larger than the object. So you cannot detect front or back. Front or back are concepts that exist in your mind due to your experience with everyday macro objects. The microscopic world is different and quarks do not have fronts or backs. You can only imagine front and back of a quark because your mind (incorrectly) associates macro level attributes to micro level objects.

"discrete object" dont exist. Or maybe the do, but they are nonsensical.

https://www.academia.edu/343657/Inevitability_of_infinitesimals

If God (assuming he exists) focused in on a discrete object, what would he see?