Infinite Bananas

Devans99

We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. So the two collections of bananas are therefore identical:

{b, b, b, b, …}
{b, b, b, b, …}

We add one banana at the start of the second collection and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence so they are still identical collections.

We then remove every second banana from the second collection and then shift all the bananas in the second collection to the left so they are lined up with the first collection again. Again they are in a one-to-one correspondence so they are identical collections.

What we have just done with bananas is of course:

∞+1=∞
∞/2=∞

So if you believe in actual infinity, there exists objects that obey the following axiom:

When it is changed, it is not changed

Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?

So I think we have to conclude that actual infinity is not part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).

--------------------
(above is my revised argument. Below is my original post... which has errors...)

We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets.

We add one banana at the start of the second set and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence again - so Cantor would claim they are identical sets.

We then remove every second banana from the second set and then shift all the bananas in the second set to the left so they are lined up with the first set again. Again they are in a one-to-one correspondence - so Cantor would claim they are still identical sets.

What we have just done with bananas is, of course, the transfinite arithmetic:

∞+1=∞
∞/2=∞

So in Cantor’s dream world (the alternative/virtual reality of infinite set theory), there exists objects that obey the following axiom:

'When it is changed, it is not changed'

Surely this statement is a contradiction? Surely in our reality, when something is changed, it changes?

So I think we have to conclude that actual infinity cannot be part of our reality (or indeed any logical form of reality), it is just an illogical concept that exists in our minds (along with concepts like levitation, talking trees and square circles).

fdrake

Bijections between sets only show they're of equivalent cardinality, not that they're identical sets. Sets are identical when and only when they share all and only the same members. So - the odd numbers:

{1,3,5,...}

And the odd numbers without 1

{3,5,7,...}

are not the same set, the second is a proper subset of the first, but they're the same cardinality (the bijection is f(x) = x+2 where x is an odd number).

Devans99

↪fdrake

You will see in the OP that I did state it was two sets of identical (IE unnumbered) bananas.

Devans99

↪fdrake

You can use mathematical induction to see the two sets of bananas are identical:

1. The first pair of bananas is in one-to-one correspondence
2. If the nth pair is in one-to-one correspondence so is the nth+1 pair
3. So all bananas are in one-to-one correspondence
4. So the sets are identical

Yohan

Kinda left of center observation.

fresco

This is all just a semantic game regarding the word 'identity'. The fact that 'two' or more bananas can be considered functionally or materially 'identical', belies the fact that they necessarily 'differ' in their
instantaneous existential locations with respect to an observer....whence the ensuing word salad (or fruit salad :wink: )

god must be atheist

Kinda left of center observation. — Yohan

I love this. Accusing math of being a Democrat.

god must be atheist

↪fresco

Not quite.

The two rows are different in location, but the bananas are identical to each other in every other aspect but location.

Try the experiment now.

Devans99

I have made a mistake with this argument... sorry.

A set is a set is a well-defined collection of distinct objects, so my identical bananas cannot be said to form a set.

I still maintain however that my argument highlights the absurdity of actual infinity.

Harry Hindu

We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets.

We add one banana at the start of the second set and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence again - so Cantor would claim they are identical sets.

We then remove every second banana from the second set and then shift all the bananas in the second set to the left so they are lined up with the first set again. Again they are in a one-to-one correspondence - so Cantor would claim they are still identical sets. — Devans99

This is another one of those philosophical "problems" that is a misuse of terms.

How does one add a banana at the beginning of a row of infinite bananas? There is no beginning, and therefore no second banana, in a infinite row of bananas. There is no beginning or end with infinity. You're simply misusing terms.

Devans99

How does one add a banana at the beginning of a row of infinite bananas? There is no beginning, and therefore no second banana, in a infinite row of bananas. There is no beginning or end with infinity. You're simply misusing terms. — Harry Hindu

What I mean is:

{ b, b, b, b, ... }
^
new banana is inserted here (at the start).

So the infinite collection has a start but no end.

Harry Hindu

↪Devans99

An infinite collection has no start and no end.

Devans99

↪Harry Hindu

What about the collection/set of natural numbers? They start at 1 and go on forever.

Harry Hindu

↪Devans99

Sure, numbers can go on forever, but a row of bananas? Impossible. One is imaginary while the other isn't. You're confusing the two.

Devans99

↪Harry Hindu

Numbers exist in our heads only, bananas exist in reality. The point of my argument is to demonstrate that actual infinity can exist in our heads but not reality.

So I think we are in agreement?

Harry Hindu

↪Devans99

I thought you were talking about bananas, not reality. You're moving the goalposts. Reality can be infinite, but if bananas were infinite is reality just bananas?

Devans99

↪Harry Hindu

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).

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

fdrake

There are two miscomprehensions in your post. The first is based on how sets are defined. The second is based on a conflation of an equivalence (of sets) with strict identity (of sets).

The first one:

We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets. — Devans99

Firstly, if all the elements of the sets are identical, then they just have one element. Sets are defined by what distinct elements belong to them; a set is a collection of distinct objects. If x is in X, there's only one copy of it. If you want to consider set like objects that allow multiple copies of identical elements in them, that's a multiset.

Reveal

Don't get hung up about sequences like {1,1,1,1}, these are formally distinct objects; they're sets of ordered pairs; {1,1},{1,2},{1,3},{1,4}. Sequences are functions from sets of natural numbers to other objects, you can write it out like a set because the reading order from left to right represents the sequence order neatly.

The second one:

3. So all bananas are in one-to-one correspondence
4. So the sets are identical — Devans99

Two sets being in a one to one correspondence says nothing about whether they are identical sets. The odds are in a one to one correspondence with the evens, but even numbers are necessarily not odd. However, if you stipulate the definition:

(C) Two sets $X,Y$ are related in sense C when and only when there is a bijection between them. When a bijection exists between the two sets, write $X \cong Y$ .

You'll see that (C) is an equivalence relation. The equivalence classes of (C) are sets of the same cardinality. IE, insofar as C is concerned:

$\{1,2\} \cong \{3,4\} \cong \{5,6\}$

But

$\{1,2\} \neq \{3,4\} \neq \{5,6\}$

!

You'll also see that if two sets are identical, a bijection exists between them (the identity function), so a set has the same cardinality as itself. That is $X = Y \implies X \cong Y$ , but the reverse does not hold. The reverse implication is precisely what you require to go from 3 to 4 in your argument, and you're obtaining it by equivocating between two sets being identical and a bijection existing between two sets.

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Devans99

Firstly, if all the elements of the sets are identical, then they just have one element. Sets are defined by what distinct elements belong to them; a set is a collection of distinct objects — fdrake

You should have really read the posts above. I did acknowledge my mistake here:

https://thephilosophyforum.com/discussion/comment/367343

And I've also updated the argument in the OP accordingly. Sorry.

Two sets being in a one to one correspondence says nothing about whether they are identical sets. The odds are in a one to one correspondence with the evens, but even numbers are necessarily not odd. — fdrake

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

fresco

↪god must be atheist

The point I am making is that 'identity' lies in the eye of the beholder (or beholders by agreement). Classical logic ignores that point and any shifting of contextual state transitions. IMO The whole argument is an example of Wittgenstein's language on holiday.

Devans99

Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood

For example, some folks believe that past time is actually infinite; implying an actually infinite collection of moments in the past.

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

Deleted User

This user has been deleted and all their posts removed.

fdrake

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

Assume what you're saying is true. Then {1} = {2}, then 1=2, contradiction, then what you're saying is false.

Devans99

↪fdrake

I'm not claiming {1} = {2}. I am saying that {1} = {1}. And that {1,1}={1,1}. And that, by induction, {1,1, ...}={1,1, ...}.

Devans99

A little more rigour, please. First, what "some folks believe" is no standard for anything (than perhaps that some folks may believe anything). — tim wood

For example, many cosmologists hold that time has no start, implying an actual infinity of past time. All those past moments actually happened, so it would count as an instance of actual infinity, regardless whether you hold a presentist or eternalist viewpoint.

Then there is the axiom of infinity. It states that there exists a set with an actually infinite number of members (the natural numbers). As you have pointed out, no such set exists (outside our minds where the impossible is possible).

Back to the line segment. It's just a line. Is there "an actual" infinity of points? Depends on your purposes and definitions - but then you're beyond what it is. — tim wood

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.

Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood

Do you consider Cantor sane? If so:

"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

jgill

Must we pick at the bones of this troubled man?

Wiki: Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[33] The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.

Profound set theory can be harmful to one's health, Devons99!

Devans99

↪jgill

Cantor has had a huge impact on the way we regard actual infinity. It is due to his work that actual infinity has gained legitimacy in maths and that legitimacy has spread to other fields like physics, cosmology and philosophy - to the detriment of those fields IMO.

fishfry

{b, b, b, b, …}
{b, b, b, b, …} — Devans99

Inaccurate notation, since by the axiom of extensionality, the set {b,b,b,b,b,...} is the exact same set as {b}. Perhaps if you notate it $\{b_1, b_2, b_3, \dots \}$ your argument will be more clear.

Devans99

↪fishfry

I called the groups of bananas 'collections' rather than 'sets' to get around the issue of sets having to be composed of distinct objects. Hope that is clear. Sorry for any confusion caused by the use of '{' and '}'.

fishfry

I called the groups of bananas 'collections' rather than 'sets' to get around the issue of sets having to be composed of distinct objects. — Devans99

I don't think that helps any. You're starting out with bad notation and that's leading to incorrect conclusions. By collection do you mean a proper class of bananas? That's a lot of bananas.

Perhaps you mean what's called in computing a multiset, or (older terminology) a bag. If so say that. If you mean something else, say that. That's what notation is for: to engender clarity of communication.

Infinite Bananas

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