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. — Devans99

What about space? Is space finite? What is there outside of space? Nothing/something. The former is space and the latter requires space.

“Is bananas b-a-n-a-n-a-s?”

Your premise of infinity lacks one major factor. That is, continuity.

Infinite also means non-stop (endless). Even before the addition and division which you mention, the collections must have been constantly progressing in size and, possibly, in as many progressions as is possible e.g. arithmetic, geometric, logarithmic, exponential, etc.

Mathematics works with defined limits, so when it encounters an undefinable 'process' (theoretically), it shows how immune it (the undefinable process) is to any of its defined (limited) operations.

What about space? Is space finite? What is there outside of space? Nothing/something. The former is space and the latter requires space. — TheMadFool

I believe that spacetime started expanding 14 billion years ago at a finite rate, so hence spacetime must be finite currently. I believe there would be pure nothing beyond the boundaries of spacetime - there being no time or space for anything to exist. Pure nothing has no dimensions so it cannot be infinite.

Infinite also means non-stop (endless). Even before the addition and division which you mention, the collections must have been constantly progressing in size and, possibly, in as many progressions as is possible e.g. arithmetic, geometric, logarithmic, exponential, etc. — BrianW

I think you are referring to potential infinity? I don't deny the existence of potential infinities; the argument in the OP is squarely aimed at actual infinity.

From what I get outside of mathematics, the word infinite set or set of infinite 'anything' is an oxymoron. Because infinity or an infinity of anything contradicts with the meaning of set, which implies a kind of definiteness (and a kind of limitation). However, infinite set as used in mathematics is a way of expressing a concept which isn't realizable (factual).

That means, infinity is a concept, whether actual, potential or other.

I think we need to look at where infinity comes from. It is simply a product of a kind of human thinking. Human thought is a response to finding itself in a position of existing in a world with dimensions and duration. When the human mind thinks about this world, inevitably they encounter thoughts of endlessness. When this idea is contemplated ideas of infinity emerge.

None of this says anything about the world we find ourselves in, only our rational response to it. A peculiarity of our own mind. So I agree with the OP.

What about non-standard analysis? Mathematics applies to the world, which is why engineering and physics work

That means, infinity is a concept, whether actual, potential or other. — BrianW

I agree they are concepts, not numbers. Focusing on actual infinity only, it is an illogical concept that can only have existence in our minds. If we assume actual infinity is possible then it leads to contradictions (see the OP) and absurdities (Hilbert's hotel etc...).

I think we need to look at where infinity comes from. It is simply a product of a kind of human thinking. — Punshhh

Indeed, I think that actual infinity is the product of top-down thinking. Bottom-up thinking shows it leads to absurdities. So with top-down thinking, illogical things are possible in our mind. It only becomes apparent that they are illogical through bottom-up thinking - and illogical things cannot have existence in reality. An example is that top-down thinking suggests we can construct a square with the same area as a circle (using ruler and compass). But 1000s of years of bottom-up thinking has finally proved that is impossible.

What about non-standard analysis? Mathematics applies to the world, which is why engineering and physics work — Gregory

It does not really tell us anything about actual infinity - they merely assume such a quantity (and its inverse) exist. It may have applications but telling us about the nature of actual infinity is not one of them.

Did you check out the article above about the "inevitability of infinitesimals"? You should be on forums to learn, not preach

I am merely pointing out that if ∞ is absurd (which it is - see the OP) then so is 1/∞.

There is so much math I could learn and life is too short to learn it all. So I choose to learn the areas of maths that are based on sound axioms and disregard areas that are based on unsound axioms.

Holding an unwavering position on those principles is unwise. There is much paradox in this world and you are not going to get far being narrow-minded like you are (on this subject and the God one too). You know what you want to believe and are using all your mental effort to defend. My mind is like water. I let it go wherever truth pulls it. I don't have any structure whatsoever behind it saying "this is what I want to believe".

Hilbert's hotel, ?

Like Shandy's diary, a veridical paradox, i.e. counterintuitive, yet does not otherwise derive a contradiction.

How is 'we change it and it is not changed' not a contradiction?

How is 'we change it and it is not changed' not a contradiction? — Devans99

We change it in one respect (whether it includes this particular individual member), but it is not changed in another respect (its cardinality as an infinite set). Not a contradiction.

We change it in one respect (whether it includes this particular individual member), but it is not changed in another respect (its cardinality as an infinite set). Not a contradiction. — aletheist

We change the sequence (its a sequence of identical bananas; not a set) to include an extra identical banana and we get back an identical sequence of identical bananas. So both the sequence itself and the cardinality remain unchanged despite us adding one banana.

So both the sequence itself and the cardinality remain unchanged despite us adding one banana. — Devans99

The only basis for claiming that the two infinite sequences are "identical" initially is that they allegedly consist of "identical" bananas in "identical" order. Accordingly, adding another "identical" banana to the beginning of one of them is not really a change, since the new "first" banana is indistinguishable from any other.

But we have added a banana to the sequence so we have definitely changed it. So we have changed the sequence and it has not changed - contradiction.

I think you are trying to defend the indefensible; actual infinity is a logical impossibility. For example, is it possible to construct actual infinity or its inverse mathematically or otherwise? No it is not, each is a task that never ends and never ending tasks are impossible to complete.

Something that goes on forever like actual infinity would just be pure magic and magic is not possible because it defies logic and reality is logical. Is there anything with the structure of the natural numbers in reality? No so aleph-zero is just the invention of a deranged mind.

What are aleph-zero, aleph-one? They are names of patterns formed by imagining the abstract and illogical structure we call infinity. They are not sizes or cardinalities; just names of patterns or organisations. Does it make sense to add one to a snowflake? Or multiply a snowflake by a hexagon? Cantor had it all wrong about set theory just like he had it wrong about God talking to him.

So we have changed the sequence and it has not changed - contradiction. — Devans99

Again, we have changed it in one respect but not in another - no contradiction.

I think you are trying to defend the indefensible; actual infinity is a logical impossibility. — Devans99

We are discussing hypothetical infinity, not actual infinity. We do not have an actually infinite sequence of actually identical bananas, let alone two such sequences.

Again, we have changed it in one respect but not in another - no contradiction. — aletheist

If I add one banana to the sequence, I should get back something that is someway different from the original sequence. The fact that I get the same sequence back is a contradiction.

You seem to be saying it is possible logically and/or in reality to change something and it does not change. Your logic / reality therefore differs from mine.

We are discussing hypothetical infinity, not actual infinity. We do not have an actually infinite sequence of actually identical bananas, let alone two such sequences. — aletheist

Can you explain the difference between hypothetical and actual infinity?

If I add one banana to the sequence, I should get back something that is someway different from the original sequence. — Devans99

Not when all the bananas are stipulated as identical.

You seem to be saying it is possible logically and/or in reality to change something and it does not change. — Devans99

It is possible to change something in one respect without changing it in another respect. If I peel a banana, it is still the same banana, even though I have changed it in one respect.

Can you explain the difference between hypothetical and actual infinity? — Devans99

I honestly do not see what there is to explain. Do you not know the difference between the hypothetical and the actual?

Not when all the bananas are stipulated as identical. — aletheist

But the mass of the sequence must have changed. But the maths says the sequence is identical so it has the same mass. So its a contradiction.

It is possible to change something in one respect without changing it in another respect. If I peel a banana, it is still the same banana, even though I have changed it in one respect. — aletheist

A peeled banana is no longer identical to a non-peeled banana.

I honestly do not see what there is to explain. Do you not know the difference between the hypothetical and the actual? — aletheist

I assume by hypothetical you mean the imaginary structure of actual infinity in our minds? But hypothetical means it might or might not be true. In this case, it cannot be true. It is impossible to actualise infinity in the mind; just dreaming illogically about it is as close as we can get.

I think of actual infinity as both a logical concept and something that could apply to reality. But it is illogical so it can exist only in our minds (where the impossible is possible) and reality is logical so it cannot exist in reality.

If you don't like my proof actual infinity is impossible, how about this reductio ad absurdum proof:

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

So there are 9n balls in the bag for each finite step. There are 0 balls in the bag after actually infinite steps. An infinite number of steps is just the sum of the actions of the finite steps and none of the finite steps result in 0 balls in the bag. Reductio ad absurdum. Actual infinity is logically impossible.

But the mass of the sequence must have changed. — Devans99

A sequence has no mass, since it is a mathematical concept, not anything physical. An actual collection of bananas would have mass, but it would necessarily be finite, such that adding a banana would indeed add mass.

But the maths says the sequence is identical so it has the same mass. So its a contradiction. — Devans99

You have made it clear that you reject the established mathematics of (hypothetical) infinite collections, but please stop pretending that there is no such mathematics, or that it cannot be different from the more familiar mathematics of (actual) finite collections.

A peeled banana is no longer identical to a non-peeled banana. — Devans99

It is no longer identical in one respect--whether it is peeled--but it is still identical in others. For example, it is still a banana, and most people would even say that it is still the same banana.

I assume by hypothetical you mean the imaginary structure of actual infinity in our minds? — Devans99

"Imaginary structure of actual infinity"? Now that is a contradiction.

But hypothetical means it might or might not be true. — Devans99

Hypothetical means logically possible, not necessarily true or even metaphysically possible. A hypothetically infinite collection or sequence is logically possible, while an actually infinite collection or sequence is metaphysically impossible.