Infinite Sets: Are They All Equal?

[Galteeth]

This is something I was thinking about. Note that I am not a mathematician, so don't get mad if the answer to this is really obvious.

Ok, so define x as the set of all real positive integers. Clearly, this is an infinite set. Now define y as the set of all real positive even integers (or odd, it doesn't really matter). Since y is contained by x, it would seem that x has to be greater then y. But they are both infinite, so they would also seem to be equal. What gives?

 

[Ulagatin]

There are different values of infinity. Sounds counter-intuitive, but it's true.

Think of infinity as a direction in the number line (when talking about reals) rather than a number (this is what I do anyway). It does not behave as a number.

Hope this helps.

Davin

 

[Galteeth]

There are different values of infinity. Sounds counter-intuitive, but it's true.

Think of infinity as a direction in the number line (when talking about reals) rather than a number (this is what I do anyway). It does not behave as a number.

Hope this helps.

Davin

So are you saying the concept of greater than/less than is not applicable? Or it is applicable, just not in the same sense as in a numerical value? Are there real world applications of this principle?

 

[Ulagatin]

You should watch this youtube video on infinity: it should answer your question.

Also, you may like to search for Cantor's theory (on infinite sets). It discusses cardinality (i.e. the number of elements in a set).

So are you saying the concept of greater than/less than is not applicable? Or it is applicable, just not in the same sense as in a numerical value? Are there real world applications of this principle?

To give my own example, are you familiar with complex numbers? A complex number is defined, generally, in the form $$z = a + bi$$ where $$i = \sqrt{-1}$$ and $$a, b \in \Re$$

Clearly, $$z \in C~ \forall z$$ where C is the complex field. If you are not familiar with this logic, in plain english, z (which is a complex number) is an element of the set C which contains all possible complex numbers, many more numbers than simply the real numbers.

Mathematically, $$\Re\subset C$$ which says that the real numbers are a subset (note: not a proper subset) of the complex numbers (ie C > R but certainly not equal to R). In this sense then, R is an infinite set, but C is an even bigger set! An even larger infinity (there are infinitely more complex numbers than real numbers if that makes sense)!

So, after all of that, the concept of greater than/less than still applies, just with differing cardinalities though. So, bigger or smaller infinities, depending on the set you are working within - see the youtube link for more on this.

In terms of applications, I guess it would be used in probability theory, where set theory is extensively used, and, if you could call it an application, in pure mathematics and the teaching of it. I am not a mathematician, just a year 12 student with a passion for mathematics.

Davin

 

[Galteeth]

You should watch this youtube video on infinity: it should answer your question.

Also, you may like to search for Cantor's theory (on infinite sets). It discusses cardinality (i.e. the number of elements in a set).


To give my own example, are you familiar with complex numbers? A complex number is defined, generally, in the form $$z = a + bi$$ where $$i = \sqrt{-1}$$ and $$a, b \in \Re$$

Clearly, $$z \in C~ \forall z$$ where C is the complex field. If you are not familiar with this logic, in plain english, z (which is a complex number) is an element of the set C which contains all possible complex numbers, many more numbers than simply the real numbers.

Mathematically, $$\Re\subset C$$ which says that the real numbers are a subset (note: not a proper subset) of the complex numbers (ie C > R but certainly not equal to R). In this sense then, R is an infinite set, but C is an even bigger set! An even larger infinity (there are infinitely more complex numbers than real numbers if that makes sense)!

So, after all of that, the concept of greater than/less than still applies, just with differing cardinalities though. So, bigger or smaller infinities, depending on the set you are working within - see the youtube link for more on this.

In terms of applications, I guess it would be used in probability theory, where set theory is extensively used, and, if you could call it an application, in pure mathematics and the teaching of it. I am not a mathematician, just a year 12 student with a passion for mathematics.

Davin

Thanks!

 

[Phrak]

This is something I was thinking about. Note that I am not a mathematician, so don't get mad if the answer to this is really obvious.

Ok, so define x as the set of all real positive integers. Clearly, this is an infinite set. Now define y as the set of all real positive even integers (or odd, it doesn't really matter). Since y is contained by x, it would seem that x has to be greater then y. But they are both infinite, so they would also seem to be equal. What gives?

This is all about definitions and axioms. You might come up with some consistent set of rules in which it is a true statement that the number of integers is greater than the number of even integers. I don't know if this is meaningful; I haven't tried it. Mathematics is what you make it to be. Make up your own system; go wild. If, under your system 1+1=2 in one case, but 1+1=4 in another case, people will become very upset and say your system is not very good.

Cantor's particular system of counting seems to be very well liked and doesn't seem to upset most folks who are interested in this sort of thing.

 

[Ulagatin]

Thanks!

No problem, Galteeth, if anything else comes up, just ask me and I'll give it my best shot.

Davin

 

[Fredrik]

This is something I was thinking about. Note that I am not a mathematician, so don't get mad if the answer to this is really obvious.

Ok, so define x as the set of all real positive integers. Clearly, this is an infinite set. Now define y as the set of all real positive even integers (or odd, it doesn't really matter). Since y is contained by x, it would seem that x has to be greater then y. But they are both infinite, so they would also seem to be equal. What gives?

Two sets are equal if and only if they have the same members. It's not enough that they have the same number of members. The set {0,1} is not equal to {1,2}. It is however possible to make sense of the notion of "the same number of members" for infinite sets too. Two sets X and Y are said to have the same cardinality if there exists a bijective function f:X→Y. The set of positive integers, the set of even integers, and the set of rational numbers, all have the same cardinality, but that's a different cardinality than the set of real numbers.

The standard argument to prove that the real numbers have a different cardinality than the integers goes like this: Suppose the interval [0,1] has the cardinality of the integers. Then there exists a numbered list that contains all the real numbers in the interval. It would look something like this:

1. 0.8278256862572434...
2. 0.1252858029758282...
3. 0.1824205737523532...
...

Now consider a number of the form 0.a₁a₂a₃..., where we choose a₁ to be different from 8 (the first decimal of the first number), a₂ to be different from 2 (the second decimal of the second number), and so on. This number won't be equal to any of the numbers in the list, and that's a contradiction. So the assumption we started with (i.e. that [0,1] has the cardinality of the integers) must be false.

 

[SW VandeCarr]

This is something I was thinking about. Note that I am not a mathematician, so don't get mad if the answer to this is really obvious.

Ok, so define x as the set of all real positive integers. Clearly, this is an infinite set. Now define y as the set of all real positive even integers (or odd, it doesn't really matter). Since y is contained by x, it would seem that x has to be greater then y. But they are both infinite, so they would also seem to be equal. What gives?

The integers have the cardinality of infinite countable sets, termed $$\aleph_{o}$$. So do all infinite subsets of the integers such as all even numbers. Having the same cardinality means one set can be mapped one to one into another. The rational numbers also have the same cardinality as the integers although this might seem counter-intuitive. (see Cantor's diagonal argument).

The Continuum Hypothesis states that the real numbers have the cardinality of the continuum C which is strictly larger than $$\aleph_{0}$$. (That is, there are no intervening aleph numbers $$\aleph_{i}$$ between $$\aleph_{0}$$ and C.) This is not proven, but is generally accepted by most mathematicians as far as I know.

 

[Galteeth]

Thanks guys. I have to admit, I am unfamiliar with some of the terms used, so i will go look them up to better understand the answers given, and digest this information before I ask any further clarifying questions.

 

[Redbelly98]

The stuff about different types of infinity seems off-topic here. As Frederik said, the two sets must have the same members in order to be equal. So, just find a number that is in one set but not in the other, and you have shown that the two sets are unequal.

 

[SW VandeCarr]

The stuff about different types of infinity seems off-topic here. As Frederik said, the two sets must have the same members in order to be equal. So, just find a number that is in one set but not in the other, and you have shown that the two sets are unequal.

The question is not so much about set equality. It's about set cardinality. Galteeth seemed to be confused about the two. Frederik simply pointed out the difference. There's not much more to be said about set equality. The set of all odd numbers is not equal to the set of all integers, but both sets have the same cardinality.

 

[Redbelly98]

Rereading the OP, what you are saying makes more sense to me now. Thanks.