Cantor and Numbers Larger than Infinity

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catholicray

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I am not convinced there is a number larger than the whole number (∞). I am most ready to listen to rebuttal but please hear me out.

The Concept of 1
First the value of the whole number 1 as it compares to 0 is infinite. Allow me to explain a little. If we divide 1 by 10 between 0 and 1 there is a subset of 10 numbers. Divide 1 by 100 and we find that there is a subset of 100 numbers between 0 and 1. If we divide 1 by ∞ (or 0) we would be dealing with a subset of numbers between 0 and 1 that is infinite. Here I would suggest that the value of the whole number 1 is relative to the subset you are examining. From this I would conclude that each whole number has a value potential of (∞). Philosophically I would assert that, depending upon how you looked at it, 1 candy bar is infinitely more candy bar than 0 candy bars.

Now if I’m examining the infinite subset of whole numbers our number line would be as follows:
0,(1∞),(2∞),(3∞),…(∞) I would argue that it is on this number line that the value of (∞) is actually represented. I would go a step further and suggest that the value of (∞) does not change. Why? Because each whole number actually represents a value potential of (∞). On this number line (∞) represents an infinite infinities. Yet even when we are not examining subsets of numbers (∞) has the value potential of infinite infinities.

What about evidence like Hilbert’s hotel? I think Hilbert’s hotel has the potential to be false because it assumes we can represent (∞) on a linear number line properly.
 
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Transcendental numbers are continuous; they differ by infinitessimals. Since transcendentals therefore cannot be expressed in a one-to-one correspondence with integers, there are more than aleph-null transcendental numbers. See the diagonal theorum.
 
You have made the incorrect assumption that infinity is a number. It is not a number, but an expression of boundlessness.

You cannot divide by infinity. You can divide by larger and larger numbers without bound.

You can not multiply by infinity. You can multiply by larger and larger numbers without bound.

You cannot add infinity. You can add larger and larger numbers without bound.

You cannot add to infinity. That just doesn’t make sense because it is equivalent to saying there is a number that increases without bound and then increases some more (or a bit less, if you subtract from infinity).

Nor can you divide infinity by infinity, or subtract infinity from infinity. The explanation is left as an exercise to the reader.
 
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This is why I said (or 0) which has the potential to spark an entirely different argument but I stand by my own reasoning here that, regardless of whether we should or not, you can divide by zero.
 
You can divide by smaller and smaller numbers approaching zero, and the result is larger and larger numbers, but it makes no sense to divide by zero. Infinity is not the answer. There is no answer to division by zero.
 
Again I think this evidence has the potential to be wrong because it assumes that ∞ can be represented on a linear number line. I would suggest that ∞ should be represented on a multi dimensional number line where our infinite linear number line represents merely the space between 0 and 1 in a higher dimension.

Wait a minute…

Okay I would be satisfied with concluding that Cantor is absolutely right and infinitely wrong.
 
I have a limited understanding of higher mathematics, but my understanding is that it might be possible for OP to create a self consistent mathematical model such as he describes, but that unless it proves to be more useful than the standard models, nobody but the OP will much care about it.

I believe this is because the appropriate question to ask about mathematical models is not ‘is this true’, but, rather, ‘is this useful’.

Perhaps some people here who know more about the subject than I do could comment on this point.
 
Look up surreal numbers. Discovered by John Conway, this number system formalizes the idea of infinite quantities.

Calculus is actually a veiled extension of the real numbers into something called hyper real numbers, a subset of the surreals.
 
TechieGuy said:
I believe this is because the appropriate question to ask about mathematical models is not ‘is this true’, but, rather, ‘is this useful’.
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How can you tell? Crazy ideas sometimes lead to unexpected breakthroughs, and usefulness is most easily determined in hindsight.
 
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The problem with talking about various kinds of infinity as if it were a number like numbers you use every day is that it is not like that. Infinity as a concept is best viewed, not as a number, but as a characteristic of a set. This characteristic is called the cardinality, and is like the count we normally associate with finite sets, that is, the number of elements in the set. So in that sense, the number “5” is realized as the cardinality of a set of 5 apples.

The next interesting thing we do with cardinalities is to compare them. How do we decide if one set has a higher cardinality than another set? What we do in finite sets is we count the number of elements in each set, and then compare where those two numbers appear in number line and see which one appears first. That method just does not work with infinite sets because the cardinality of infinite sets does not appear in the number line at all. So we must devise a different method of comparing cardinalities.

Another method that works with finite sets does not involve counting. Instead it involves associating each element in one set with an element in the other set. Suppose I have a bottle of vitamin C tablets and a bottle of vitamin D tablet. I want to find out which bottle has more tablets. Instead of counting them, I can dump them out side by side on the table and carefully take away one tablet from each pile. If I end up with a few vitamin C tablets and no more vitamin D tablets, I can say that there were more vitamin C tablets than vitamin D tablets, even though I didn’t bother to keep track of how many there were in each set. That is the kind of comparison we can use when comparing one infinite set with another infinite set. We don’t take away one element at a time from each set because that would take forever. But we can write one formula that all at once makes all the associations we need between the two sets. If there are elements left over, not associated with anything in the other set, then the set with the left over elements might have more elements.

The reason I have to say “might have more” instead of “does have more” is that it is possible to find two different association rules that each say the other set is larger. In that case, we say the two infinities are equal in size. The only time we can say that one infinity is definitely larger than the other is if there exists an association rule that maps the “smaller” set into the “larger” set, but there is no association rule that maps the “larger” set into the “smaller” set. The aforementioned diagonal proof is one such instance of showing that the set of rational numbers is a smaller infinity than the set of real numbers.
 
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LeafByNiggle said:
Infinity as a concept is best viewed, not as a number, but as a characteristic of a set.
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Cardinalities and infinite quantities are distinct. Calculus is an extension of the reals that allows us to deal with infinity in some generic way, but there are systems such as the Surreal numbers, which have much more depth.
 
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I’d like to add one more train of thought…
Earlier I suggested that the value of ∞ on our linear number line had the potential value of infinite infinities. I further suggested that this value would represent the value between 0 and 1 on a number line one dimension above ours.

It occurs to me then, given we inverse our thinking here, moving to dimensions lower than our own that we might be able to say that the largest possible ∞ can be accounted for in the “space” between 0 and 1…
 
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Many moons ago, I studied quite a bit of advanced math in graduate school. I am certainly out of practice and I am not knowledgable on the more esoteric mathematic theoris, eg wheel theory. But I am having trouble with your concept of infinity as a number, I do not see how that works. As to the “space between 0 and 1”, that is the same as the space between any two real numbers, regardless how close together the two numbers are. There is an infinite amount of space between two numbers, that is the continuum, which is why the set of real numbers is not countable, and bigger than the set of whole numbers, which is countable.

So I suppose you might be talking about the two sizes of infinite sets: the size of a countably infinite set and the size of an uncountable infinite set. Neither of these two values assist in the idea of dividing by zero.
 
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tafan2 said:
So I suppose you might be talking about the two sizes of infinite sets: the size of a countably infinite set and the size of an uncountable infinite set. Neither of these two values assist in the idea of dividing by zero.
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Sizes of sets isn’t the same as infinite quantities. Infinite quantities have already been definied, just not in the way OP suggests. And further you’re right that they don’t aid in defining division by 0. Infinitesimals , not 0, are the inverse of infinite numbers.
 
I wasn’t saying they were the same. I was trying to figure out how catholicray was using the term infinity as a number.
 
Oh man think about this let us agree we have a number line that counts 1,2,3,4… ∞

Let us agree to start at ∞ and count backwards. By the time we reached 0 we would have counted a finite ∞

I’ll try to explain more later but the way I see it the problem with dividing by zero would be that the answer is (1 and ∞)

I am still processing this so bear with me.
 
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Excluding ng some esoteric mathematical system, dividing by zero does not give you 1 or infinity, it is an undefined operation.
 
I know that and I am not saying the value of dividing 1 by 0 would be 1 or ∞ but rather (1 and ∞) but let me think about all of this…
 
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