## Proof that infinity does not come in different sizes

• 276
If I count 1, 2, 3, 4 ad infinitum, will I reach infinity? One cannot count to infinity, and even if something like a number sequence goes on forever, it will not reach infinity. To call {1,2,3,4,...} an infinite set is to imply that {1,2,3,4,...} consists of an infinite number of numbers. No doubt, even if 1, 2, 3, 4 goes on forever, an infinite number of numbers will never be reached. So the question that must be asked now is whether there is any meaningful difference between 1, 2, 3, 4 ad infinitum and [1,2,3,4,...}.

One might argue that the latter encompasses imagining that the count to infinity is complete, but one cannot imagine such a thing. Perhaps one might argue that there is no count involved with regards to the latter and that it's just a fact that Infinity encompasses an infinite number of natural numbers. But if that's the case, then Infinity also encompasses an infinite number of possible real numbers and possible letters or possible x. But where there is no counting involved, all infinites are of the same size/quantity (or rather, infinity is one quantity as opposed to different quantities). How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached?
• 24k
And yet Cantor.

So you have gone astray somewhere.
• 21.6k
Apropos of which, a splendid Medium publication solely dedicated to just such abstruse considerations

• 489

Any infinite sequence is equal in terms of number of elements to any other infinite sequence, but i do not think they are equal in terms of magnitude or value.

Considering these two infinite sequences:
sequence 1 = {1, 2, 3, 4, ...}
sequence 2 = {.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, ...}

Take for instance the first 4 elements of any 2 infinite sequences, and observe the last number. If the last number of one series is bigger than the last number in the other series then that sequence has a larger magnitude. In this case then sequence 1 is larger in magnitude than sequence 2.

sequence 1 = {1, 2, 3, 4, ...} = 4
sequence 2 = {.5, 1, 1.5, 2, ...} = 2

Therefor sequence 1 is greater than sequence 2 in terms of magnitude or value.

Alternatively, if one selects two arbitrary numbers that are common to both sequences such as the numbers 1 and 4 then sequence 2 is larger than sequence 1 because more numbers are included by sequence 2 between values 1 and 4.

sequence 1 = {1, 2, 3, 4, ...} = 4
sequence 2 = {1, 1.5, 2, 2.5, 3, 3.5, 4, ...} = 7

Therefor sequence 2 is greater than sequence 1 in terms of magnitudes or values represented. It can be said that sequence 2 has higher resolution than sequence 1.
• 1.5k

The size of an infinity is determined by the size of its elements, not the size of the collection (viz., it is not determined by how many elements the collection has but, rather, how many elements, including recursion, the elements have).

As a basic example, it is clear that a set of {1,2,3,...} is smaller than {{1,1},{1,2},{1,3},...}. Likewise, the largest infinity is the one with infinities as its elements all the way down recursively, which I cannot shorthand accurately, due to its nature, than {{*},{*},{*},...} where * represents an infinitely recursive set of collections.
• 3.7k
Off to the races . . . :cool:
• 276
And yet Cantor.

You do not consider the possibility that Cantor is wrong?

Suppose someone brought proof. How will you recognise it?
• 276

but i do not think they are equal in terms of magnitude or value

Suppose two things are travelling at two different speeds. One is faster than the other. Both are set to go on forever. Would you say something like the value of the faster one is greater than the value of the slower one in terms of distance covered? Or would you say they are both of equal value? Or would you say both are set to go on forever but neither will reach an infinite amount of distance covered and so one will have travelled farther than the other if a measurement was to be taken of how much distance it has covered in comparison to the other.
• 761
My understanding is that infinities are undefined. They can not be defined by equations but are a mental concept. So the idea of using logic of some kind on them has problems.

Generally the problem physically exists in this form,

Brain; (Abstraction)

More specifically,

Brain; (Abstraction 1, infinite set 1)
Brain; (Abstraction 2, infinite set 2)

And a relation,

Brain; (Abstraction 3, the relation of sets)

The problem I see is that any element of an infinite set is a finite number and can be reached by finite means. So those numbers that are finite can be subject to logic because they are defined.
Infinity as undefined is off limits to logic.

If you introduce infinities into the elements then again you are using undefined terms.

I do see some logic in the OP arguments.

It seems possible to map a smaller infinity, one to one, on a larger infinity simply by freezing the larger infinity and letting the smaller one catch up.

Since we set imaginary parameters anything goes. This is not based on anything physical at all.
• 276

How would you respond to this:

How would a difference in size be established between two infinite sets when there is no counting involved? And if there is counting involved, how would infinity be reached given that one cannot count to infinity?
• 276

It seems possible to map a smaller infinity, one to one, on a larger infinity simply by freezing the larger infinity and letting the smaller one catch up.

Whilst I believe it's possible for two different things to go on forever, I don't believe it's possible to have two different sized infinities because even if the two things (such as two number sequences) go on forever, infinity will not be reached (we cannot count to infinity). So when you say "freeze the larger infinity" I assume you mean something like stopping it from continuing to go on forever. But my whole argument is that if something goes on forever, it does not make it infinite precisely because one cannot count or expand to infinity.

Since we set imaginary parameters anything goes. This is not based on anything physical at all.

Some things are imaginary, but some things are truths about the nature of Existence/Being (such as triangles have three sides or one cannot count to infinity).
• 761

Actually, I think you have the better grasp of this problem being that the extended nature of infinities is off limits to logic.
• 761

Well, what you call truths I call Abstractions and the parameters can be anything we choose. Again, no physical basis so variation in opinion is expected.
• 761
What I'm saying is that applying logic to infinities is akin to dividing by zero. Something off limits.
• 276

Well, what you call truths I call Abstractions and the parameters can be anything we choose

No one our earth has ever seen a physically perfect triangle (because perfectly straight lines are impossible in our universe as far as I'm aware). Yet we know that the angels in a triangle add up to 180 degrees. What I'm trying to say with this example is that the parameters cannot be anything we choose. If what we choose or say is contradictory (such as triangles have four sides or one can count to infinity) we cannot meaningfully/rationally say it. As for what determines what's meaningful/rational and what's not, I believe that is Existence. If x is true of Existence, it is rational/meaningful (for example triangles have three sides is true of Existence). If x is contradictory or not true of Existence (such as one can count to infinity), it is not true of Existence (as in the nature of Existence is not such that triangles have four sides or that one can count to infinity).
• 761

Okay. Just giving another perspective.

A lot of interesting math in the subject.

I agree the abstractions should conform to the subject matter.
• 761
There are real world problems that involve mapping infinities to finite physical resources.

A couple examples,

Central banks where a potential infinite supply of currency is mapped to a populations finite physical resources.

Zelenski-ism where infinite military wants are mapped to a coalitions finite resource base.
• 14.9k
How would a difference in size be established between them when there is no counting involved?

See Cantor's diagonal argument, which proved that there are higher-order cardinal numbers.

You've already admitted that you're not a mathematician, so it's strange that you think you know mathematics better than Cantor (and Russell).
• 276

I've seen cantor's diagonal argument and the following objection applies to it:

How would a difference in size be established between two infinite sets when there is no counting involved? And if there is counting involved, how would infinity be reached given that one cannot count to infinity?

You've already admitted that you're not a mathematician, so it's strange that you think you know mathematics better than Cantor (and Russell).

Reason is accessible to everyone (not just mathematicians). I try to focus on the argument at hand as opposed to who is doing the arguing.
• 14.9k
I've seen cantor's diagonal argument and the following objection applies to it:

It doesn't. If you were a mathematician then you would understand it. Your question simply shows your ignorance of mathematics. You're really in no position to argue against Cantor.
• 276

Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how? If no, what do we do with the problem of "one cannot count to infinity"?
• 14.9k
Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how?

Cantor's diagonal argument.
• 276

It's like me asking "can you count to infinity?" where the answer should be no, but someone responding with "Jack's diagonal argument" implying you can without actually showing how.
Again, I've seen Cantor's diagonal argument. It does not answer the questions I asked you in my last post to you.
• 14.9k

It is an answer. You just don't understand it because you're not a mathematician.
• 276
Again:

I've seen Cantor's diagonal argument. It does not answer the questions I asked you in my last post to you.
• 276
I leave the following as an open question to anyone who believes in infinite sets of varying sizes:

Can we establish set x as being bigger than set y without counting the number of items in sets x and y? If yes, how? If no, what do we do with the problem of "one cannot count to infinity"?

Peace
• 14.9k
Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how?

Yes, we can establish set X as being "bigger" than set Y without counting the number of items in X and Y. We can establish this by using Cantor's diagonal argument. It is a well-accepted mathematical proof. If you were a mathematician you would understand it.
• 276

Yes, we can establish set X as being "bigger" than set Y without counting the number of items in X and Y. We can establish this by using Cantor's diagonal argument. If you were a mathematician you would understand it.

If you used reason you'd know that you cannot count to infinity and that you cannot say x is bigger than y without some measurement/count involved to compare the sizes of the two.
• 14.9k
and that you cannot say x is bigger than y without some measurement/count involved to compare the sizes of the two.

If you were a mathematician then you would know that this is false.

You're just in no position to argue against Cantor.
• 14.9k
bold
italic
underline
strike
code
quote
ulist
image
url
mention
reveal