Cantor and Numbers Larger than Infinity

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So… ∞
Exists as something far far far larger than our tiny universe by comparison and yet it can fit within something far smaller than a marble.
 
Ok, you are not going to get this right if you do not start accepting a couple of basic facts.

First of all, infinity is NOT a number. You continue to show examples where you are treating it as a number, and you conclusions cannot be correct if you do that. For example: you say " If I count to ∞ by 1/2’s
{.5,1,1.5,2,2.5,3…∞} ". I can see you still think of infinity as a number because you include it in the set and you say “if I count to infinity” . You cannot count to infinity it is not a number. You can count indefinitely, but that is not the same as counting to infinity.
You say if I count to infinity by 1/2s you count twice as many numbers as if you count by 1. But that is wrong. Since you cannot count to infinity, it actually makes zero sense. What matters is, and I am going to use actual mathematical terminology, that there is a one-to-one AND onto mapping betweetn the set of natural numbers and the set of 1/2 numbers (for lack of a better term of counting by 1/2s). Being one-to-one means for each and every element in the set of natural numbers, there is exactly one element in the set of 1/2 numbers. And, being onto means that it works both ways, or rather that for each element in the 1/2 numbers there is exactly one element in the set of natural numbers that it maps to it. So no, if you count the set of natural numbers (note, I did not say I am counting to infinitiy, but I am counting indefinitely), it is the same as counting every 1/2 number. The sets are of equal size. Period, end of story. If you do not want to take my word for it, go read some Countable set - Wikipedia, the first sentence says exactly what I have been telling you.
catholicray said:
No if we are being reasonable then we would say that ∞ is the same number no matter how you count to it
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No, you are not being reasonable. Since the very beginning of this thread people have told you that infinity is NOT a number. And as I stated above, since it is not a number, you explanation completely falls apart. Yes, if you count to 100 by halfs or by ones you will count 200 vs 100. But you CANNOT count to infinity. Hence if I count 1/2s or count 1’s either way I will count forever, so how many times I count a value does not determine the cardinality of the sets.

Read the Wikipedia article.

The second thing you need to realize is that there are two sizes of infinite sets: countable and uncountable. The first is the size of the set of natural numbers, the second is the size of the set of real numbers. You give examples showing only rational or decimal numbers, and talk about infinite infinities. That term is not precise, has no meaning, but it is close to the idea of uncountable infinity: ie there are infinite real numbers between any two real numbers.

tbd:
 
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cont.:

So, I have tried to be patient. Let me give you some advice. Again you are to be credited for trying to figure this out on your own. A while back I said you were getting close. I now believe I was wrong. I know that is blunt, but example after example you give is not true. Any mathmatician would laugh at you.

So convince yourself that infinity is NOT a number and remember you cannot make any assumption in your examples or proofs that it is a number (eg “we would say that ∞ is the same number no matter how you count to it”).

Secondly, go and understand the different types of sets of numbers that are useful for thinking about, discussing, and understanding these issues: in particular natural numbers, rational numbers, and real numbers. Make sure you know the difference. pi is a real number, it is not a rational number. 2/3 is a rational number, even though it cannot be represented by decimal notation.

Finally, understand what a one-to-one and onto mapping between to sets and what it means. Contrast it with just a one-to-one mapping, a one-to-many mapping, or just an onto mapping. When you do, this whole concept of countable infinite sets all being the same size will make sense to you.

Sorry, you need to understand these things the way mathematician have defined them. Centuries of really smart people have come to a consensus on this stuff. They are right, you are not.

ETA: to the moderators and users of this forum: I apologize if the forum should not have been used for a purely mathematical discussion.
 
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catholicray said:
So… ∞
Exists as something far far far larger than our tiny universe by comparison and yet it can fit within something far smaller than a marble.
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As many infinities as you find between 0 and 1, Cantor found more.

The example of counting by halfs is instructive. Your sequence:
(.5,1,1.5,2,2.5…)
can also be written as
(1/2, 2/2, 3/2, 4/2, 5/2…)
While it may seem like there are twice as many numbers, the numerators repeat the natural numbers, and so are equal to them. At any point there are twice as many, but at infinity they are the same.

Halfs or wholes, both sets are countable, and so are equal at infinity. Just compare
(1,2,3,4,5…) to (1/2, 2/2, 3/2, 4/2, 5/2…)
When would the second pass the first?

We can tak the inverses of the rational numbers between 1 and 2 and another infinity is added.
1/(1+r) where r is a rational number less than one. This is the same size as the rational numbers between 1 and 2, but it fits between 1/2 and 1.

Infinity is VERY BIG.

But all of these infinities are the same size. They can be counted. And there is an EVEN BIGGER infinity between each pair of points between zero and one. Numbers that cannot be expressed as fractions are an larger group.
 
You are coming to a different conclusion than they. You or they are wrong, even if you don’t say so. There is no paradox that you describe. Further, you base your conclusion on at least one assumption that they do not make. Either the way you use infinity, as a number, or they, as only a cardinality, is wrong.

Go back to your examples and remove the assumption that infinity can be treated as a number, and you will find they do not work.

Again, it’s great that you have such an interest in this, but you have to educate yourself. IIRC a good book on Discrete Mathmatics should do so. Go to a library and look at a couple and see if the cover set theory and infinity. If so, it’s probably only a couple if chapters you have to study.
 
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catholicray said:
Philosophically I would assert that, depending upon how you looked at it, 1 candy bar is infinitely more candy bar than 0 candy bars.
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Well, I think I’ve learned something from this thread. But with respect to candy bars, I will have to say that I don’t think there is an infinite amount of candy in there. There’s only a finite number of candy particles. Which is a shame.
 
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And I understand I have yet to consider -1,-2,-3…
But not assuming I’m right because I don’t mind being wrong… which count are we agreeing to observe the numbers by and why? And why do we have to agree at all?

I guess that’s what I need to know.

Also note the above is only useful if we agree on what a count of 1 is in the first place. But the pattern holds after we agree upon a count of 1
 
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You posted picture does not make sense to me. You seem to be equating the size of sets with the value of the sets. Sets are, by definition, unordered. Of course, when we discuss countable sets, we are essentially saying we can order the elements in a way such they can be counted. But what do you mean by a count of 2 or a count of 1? Makes no sense. When we discuss counting elements in a set, what we are doing is ordering the elements such that they can be counted with the set of natural numbers. Or to be more precise, a countable set is a set that has a one-to-one and onto mapping from the set of natural numbers to the elements of the set being counted (note: for finite sets, we are obviously not mapping the entire set of natural numbers, just those natural numbers <= to the size of the set being mapped). At any rate, a count of 2 makes no sense.
So, lets rework your example using proper notation and terminology:

{1,2,3,4} != {2,4,6,8} That means the sets are not equivalent, they do not contain exactly the same elements. But, using the correct notation: |{1,2,3,4}| = |{2,4,6,8}| . |S| denotes the size of the set (ie its cardinality). One cannot say {1,2,3,4} = 4. That statement makes zero mathematical sense. One can say |{1,2,3,4}|=4. That statement says the cardinality is 4.

So, then if we extend this sets to infinity:
{1,2,3,4…} != {2,4,6,8…} The sets are not equal, ie they do not contain the same elements
|{1,2,3,4…}| = |{2,4,6,8…}| The cardinality of the sets IS equal, they are the exact same size: countably infinite. The function: f(n)= n*2, where n >0, provides an exact one-to-one and onto mapping. For each and every natural number greater than zero, I can find exactly one, and only one member of the right hand side set.

There is obviously not a paradox between saying the size of the sets are equal, but the elements of a set are not the same.
catholicray said:
And why do we have to agree at all?
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Well, I assume you started this thread because you wanted feedback on some of your mathematical speculations. I am trying to help. We don’t have to agree, you can go through life probably just fine misunderstanding of these concepts, provided you do not want to be a mathematician or a computer programmer or you never have to pass a course in college of certain math subjects (eg discreet mathematics).
 
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I’m not trying to ruin your mathematical construct. Your not even beginning to listen 😭
Nevermind been nice chatting with you.
P.s. I actually have programming experience just never had a professor teach me.
 
catholicray said:
I’m not trying to ruin your mathematical construct. Your not even beginning to listen
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Who is not listening to whom? I have over and over again read everyone of your examples and pointed out what is wrong. I went line by line on your last example and restated it properly, which resulted in a different conclusion.
Meanwhile, I have, multiple times and very patiently, tried to explain to you some basic concepts, such as how countability works, how infinity should be treated as a cardinality of a set and not as an element of the set or a number that can be counted. You have completely ignored everyone of these concepts in your repetitive examples. Again, consider who is not listening. From my side, it looks rather obvious. It appears you had no interest in truly “listening to any rebuttal” as you stated in your very first sentence of the thread. I am not for sure you are capable of listening at all.
 
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(1, 2, 3, 4…∞)
(2, 4, 6, 8…∞)
The second set can be written as:
(12, 22, 32, 42…∞)

Is there ever a point where the formula x*2 does not apply in the second set? Then the first set ‘counts’ the second. Even though the set is 2, it has the same number of elements. For that reason, we do not say ∞2, but just infinity. It has the same number of elements.

At any point in this series where we stop, the value will be twice, but the number of elements the same. ∞ means we do not stop, so the number of elements remain the same.

This is the nature of infinity. It is boundless to the widest space, and dense to the narrowest. It is not like a number, but bigger and broader.

This is why we use infinity to describe God. God is boundless, and contains every infinity, surpasses every infinity. All the infinities times infinity are less than God. All the manageable infinities, the ones we can count, do not come close to the uncountable immensity of God. Even the uncountable immensity of Real numbers are less than the singular reality of God.
 
Dovekin said:
Is there ever a point where the formula x*2 does not apply in the second set?
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To answer this question I have to ask you a question first. What is the true value of 1 as it relates to 0?

Perhaps I’m just dumb so I apologize if the question is stupid.
 
Infinity is not a set number. It is ongoing such that any number you can come up with, infinity (by default) is always the greater.
 
One is a thing in itself. Two is things in relations to each other.

One is the most important number. Without it there are no other numbers. One is God, within everything and greater than all together.
 
I definitely appreciate your interpretation of 1.
I guess where I am at with it is that the value of 1 on an infinite number line is 1/∞
Your interpretation is as good as any then.

I think fundamentally we’ve never really been taught the value of 1. We were shown and pattern and never told to question it.
 
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