Dividing by zero

Originally posted by googlefudge


All the integers, and all the positive integers are both examples of 'countable infinities' and are thus equal and equivalent.
.

two 'countable infinities' can only "equal and equivalent" in the narrow sense that they belong to the same order of infinity, NOT, as I think he implied in his post, that they contain the same number of elements as each other! Because they don't! That is because there is infinity in each and infinity is not a number!
(we often say "an infinite number" even though we don't mean to imply from that that infinity IS a number! It is just the way we say it, that's all )
Two infinities belonging to the same order of infinity doesn't logically imply they have the same number of elements/members nor that the ratio/proportion of one to the other is the same!

I was aware there are different classes ("orders" to be more precise ) of infinity and what that means.
I did some very advanced mathematics at university although I don't pretend to be an 'expert' at maths.

Originally posted by humy
How can you have "...is as least the numbers of..." if you are talking about infinities, which are not numbers?
I don't see how.
I think that is partly why I think your conclusion of "They are exactly as many" is incorrect; as I just explained why in my last post, the proportion of one to the other is undefined and this is proven by proof ...[text shortened]... e you can sometimes prove there are infinitely more elements in one infinite set than the other.

I'm not saing anything. Cantor is!
He says: "If an infinite set is countable, it has the same cardinality as the natural numbers."

The set of every integer, negative together with positive ones, is an infinite and countable set. Therefore it has the same cardinality as the natural number. In laymen terms they are equally many.

I know that this seems to be counter-intuitive, and hard to get in.

Originally posted by vivify
Isn't pi infinite? 3.14..........?

Pi is a member of a class of real numbers that are known as "irrational" numbers. These numbers, as opposed to rational numbers, can NOT be written as the division of two integers, say p and q. If a number in decimal representation has a repeating part, e.g. 0.142857142857... then it can always be written as such a fraction (in this case, 1/7). Since pi is irrational, when you write it in a decimal representation it will never start repeating itself and so you can continue indefinitely with finding new decimals (there is an infinite number). However, the number itself is not infinite.

Originally posted by FabianFnas
I'm not saing anything. Cantor is!
He says: "If an infinite set is countable, it has the same cardinality as the natural numbers."

The set of every integer, negative together with positive ones, is an infinite and countable set. Therefore it has the same cardinality as the natural number. In laymen terms they are equally many.

I know that this seems to be counter-intuitive, and hard to get in.

I got twhitehead's comment confused with your comment and I apologize for that. I thought you were saying the opposite.

I have studied this branch of mathematics and have come to believe the conventional wisdom that you can correctly say that to infinite cardinal sets having the same order of infinity means that they have “the same number of elements” is not only highly misleading but plain wrong else it just leads to unresolvable paradoxes proving it total nonsense. For this reason, I would assert that whether two orders of infinity have the same number of elements or what the ratio is of the number of elements in one infinite set to the other, completely contrary to what is asserted in many websites on this matter, is simply undefined i.e. no such ratio and/or equality exists between them.
I think it is best to not talk about two infinite sets with the same order of infinity “having the same number of elements” as each other but rather simply say they have the same “order of infinity" and then emphasize that doesn’t mean the same thing as having the “same number of elements". And I bet it isn't only me who thinks that. But I also bet some here will disagree with me But I can give a mathematical proof of my assertion here on request.

Originally posted by humy
I got twhitehead's comment confused with your comment and I apologize for that. I thought you were saying the opposite.

I have studied this branch of mathematics and have come to believe the conventional wisdom that you can correctly say that to infinite cardinal sets having the same order of infinity means that they have “the same number of elements” is not ...[text shortened]... here will disagree with me But I can give a mathematical proof of my assertion here on request.

Without Cantor, I wouldn't believe it either.

Two sets of countable infinitive numbers are always in a one-to-one relationship. Doesn't that mean that for every element in one set, there exists an element in the other set, and vice versa. Doesn't that say that the number of elements are as many in both sets?

Originally posted by humy
I think it is best to not talk about two infinite sets with the same order of infinity “having the same number of elements” as each other but rather simply say they have the same “order of infinity" and then emphasize that doesn’t mean the same thing as having the “same number of elements". And I bet it isn't only me who thinks that. But I also bet some here will disagree with me But I can give a mathematical proof of my assertion here on request.

That is why the word 'cardinality' is used and not simply 'number of elements'. So yes, others do agree with you.

Originally posted by FabianFnas
Two sets of countable infinitive numbers are always in a one-to-one relationship. Doesn't that mean that for every element in one set, there exists an element in the other set, and vice versa. Doesn't that say that the number of elements are as many in both sets?

No, the conclusion doesn't follow. That is the danger of playing with infinities. Infinity isn't a number so the sets do not have a 'number of elements' at all. They are infinite sets.

Think of it this way:
The graph of x^2 goes up in the y direction. The graph of x^3 also goes up. For every point on the graph of x^2 you can find a point on x^3 that is at the same y value. However it would be wrong to say that the maximum value of x^2 must therefore be the same as the maximum value of x^3 ie that they both 'top out' at the same point. They do not have maximum values and do not ever 'top out'.

Originally posted by twhitehead
Think of it this way:
The graph of x^2 goes up in the y direction. The graph of x^3 also goes up. For every point on the graph of x^2 you can find a point on x^3 that is at the same y value. However it would be wrong to say that the maximum value of x^2 must therefore be the same as the maximum value of x^3 ie that they both 'top out' at the same point. They do not have maximum values and do not ever 'top out'.

Are x and y real numbers or do you make a coordinate system with only integers?

This is important because of this reason:
The number of rationals is a one-to-one correlationship to integers. This I know, so says Cantor. "Every infinite countable sets have the same cardinality."

If every infinite set of reals have the same cardinality I don't know. (I can guess, but I cannot prove.)

Originally posted by FabianFnas
Are x and y real numbers or do you make a coordinate system with only integers?

It doesn't matter for my explanation. There is still no maximum and thus no 'number'.

If every infinite set of reals have the same cardinality I don't know. (I can guess, but I cannot prove.)
No, not every set of reals has the same cardinality. Integers are after all, a set of reals.
However, there is a hypothesis that there is no set of reals with cardinality greater than that of the integers but less than that of all reals:
https://en.wikipedia.org/wiki/Continuum_hypothesis

Originally posted by FabianFnas
Without Cantor, I wouldn't believe it either.

Two sets of countable infinitive numbers are always in a one-to-one relationship. Doesn't that mean that for every element in one set, there exists an element in the other set, and vice versa. Doesn't that say that the number of elements are as many in both sets?

Without Cantor, I wouldn't believe it either.

I would assert that Cantor was at least partially wrong! The ratio of one infinite set and another infinite set of the same order is always undefined and completely meaningless.

Two sets of countable infinitive numbers are always in a one-to-one relationship.

I would say two sets of countable infinitive numbers can be always described in terms of a one-to-one relationship. But that doesn't say anything about how number of elements in each set compare with each other in terms of finite ratio or finite proportions; only whether they belong to the same order of infinity, which isn't quite the same thing.

Doesn't that mean that for every element in one set, there exists an element in the other set, and vice versa

It can be arbitrary described that way. But I would assert that, contrary to conventional wisdom, that doesn't imply there is the same number of element in both! In fact, it just leads to paradoxes and contradiction if you assume there is either equal number or any definable finite ratio between one infinite set and the other and I can give a mathematical proof of that on request if anyone is really interested.

Doesn't that say that the number of elements are as many in both sets?

No. And I can give a mathematical proof of that on request if anyone is really interested.

Originally posted by humy

Without Cantor, I wouldn't believe it either.

I would assert that Cantor was at least partially wrong! The ratio of one infinite set and another infinite set of the same order is always undefined and completely meaningless.

Two sets of countable infinitive numbers [b]are always in a one-to-one relationship.

I ...[text shortened]... te]
No. And I can give a mathematical proof of that on request if anyone is really interested.[/b]

I googled and I found sources that shows the same that I've writen before. Yes, contra-intuitive but many things are contra-intuitive concerning infinite sets.

I will not argue over it. My world doesn't change depending on who is right in this question.

Originally posted by FabianFnas
Are x and y real numbers or do you make a coordinate system with only integers?

This is important because of this reason:
The number of rationals is a one-to-one correlationship to integers. This I know, so says Cantor. "Every infinite countable sets have the same cardinality."

If every infinite set of reals have the same cardinality I don't know. (I can guess, but I cannot prove.)

My dictionary defines "cardinality" as "the number of elements in a group or set, as a property of that grouping." The even numbers (for example) can be put into one-to-one correspondence with the integers; this shows that the two sets are of equal cardinality. To say that the two sets have the same number of elements must, however, not be interpreted to mean that infinity is a number like 153. Any set which can be put into one-to-one correspondence with the integers constitutes a denumerable infinity (EDIT and has the same cardinality END EDIT).

The set of rational numbers cannot be put into one-to-one correspondence with the integers. The set of rational numbers (fractions) is a non-denumerable infinity. It might be colloquially thought that a non-denumerable infinity is larger than a denumerable one, but this might lead to nonsense or paradox. It is better to say that, for any given term in a denumerable infinity, there is always a next term (1,2,3, or 2,4,6, etc. ), whereas for a non-denumerable infinity this is not so: for any two terms, 1/2 and 1/3, let us say, there are infinitely many terms between (13/24, 27/48, 55/96, etc. and between any of those two there are infinitely many more etc.).

PS Don't think of infinity as a quantity--not even a very large one. If you do, you'll either ask silly questions ('is infinity odd or even' ) or you'll just go bonkers. Think of infinity as a property (e.g., limitlessness).

Zero is curious number all round. Not only can you not divide by it, it is neither odd nor even. (Though some states have defined it as even for legal purposes because some states ration gasoline to even-numbered license plates or odd-numbered license plates on certain days.)

Originally posted by moonbus
The set of rational numbers cannot be put into one-to-one correspondence with the integers. The set of rational numbers (fractions) is a non-denumerable infinity.

Zero is curious number all round. Not only can you not divide by it, it is neither odd nor even

Rationals are a countable infinite set.
Zero is an even number.

Originally posted by moonbus
PS Don't think of infinity as a quantity--not even a very large one. If you do, you'll either ask silly questions ('is infinity odd or even' ) or you'll just go bonkers. Think of infinity as a property (e.g., limitlessness).

Zero is curious number all round. Not only can you not divide by it, it is neither odd nor even. (Though some states have def ...[text shortened]... ration gasoline to even-numbered license plates or odd-numbered license plates on certain days.)

Good point about infinity.

Regarding zero being neither odd nor even, couldn't we think of it as even? For example, can you have an odd amount of nothing? Two people can equally have nothing (zero) but they can't have an uneven amount of nothing.