Cantor's Infinity and the Concept of Sets: Understanding the Comparison of Sizes

[SimonA]

I have tried to understand Cantors ideas of infinity, but they still don't make sense to me. If you use the mathematical concept of sets to investigate something like this - a 'value' that can't be written because it has no end - then surely the size of the set is what you are evaluating ? How does changing 'size' to 'order' suddenly make a comparison of individual component s result in something different from size ?

There seems to be a fundamental breakdown of logic. I don't mind that at all in mathematical terms. The square root of -1 is essentially illogical, but the reasoning behind it is solid. I love the very concept of complex numbers. But different sizes/orders of infinity seems plain crazy unless 'orders' describes rate of growth in terms of an individual comparison of members of the set. I honestly find it difficult, though it may be because I'm stupid, to understand how such a comparison of members can lead to a conclusion that one type of infinity is bigger than another. There seems to be an unwarrented assumption of an end point that makes no sense in terms of infinity. Infinity seems to be an absolute platonic concept. It just seems plain wrong to convert that concept into a relative one by cutting corners. Pure maths seems beautiful to me - Cantors infinities seem ugly.

What is it that I'm missing here ?

 

[HallsofIvy]

Perhaps that this has nothing to do with physics? And is it really homework?

Cantor's basic point is the we count things by going "1, 2, 3, ..."- that is by asserting a one-to-one correspondence to a set of natural numbers. All he did was use that as a definition of "counting" (strictly speaking "cardinality") and accept the consequences of that definition. If you don't like the consequences, suggest a different "counting" concept that still corresponds exactly to what we do for finite sets.

 

[arildno]

"The square root of -1 is essentially illogical"

No it is not. Why do you think that?

 

[SimonA]

Hi HOI

Perhaps that this has nothing to do with physics? And is it really homework?

Not at all - where should I post these questions ? Though it is related to physics.

Cantor's basic point is the we count things by going "1, 2, 3, ..."- that is by asserting a one-to-one correspondence to a set of natural numbers. All he did was use that as a definition of "counting" (strictly speaking "cardinality") and accept the consequences of that definition. If you don't like the consequences, suggest a different "counting" concept that still corresponds exactly to what we do for finite sets.

Is there an end to the number of members in any set with infinite members ? If not - how is countability relevant to a totalal quality of a set ? Or do you deny that there is any perceived relationship between size and cardinality ?

 

[SimonA]

"The square root of -1 is essentially illogical"

No it is not. Why do you think that?

I actually said that I thought there was value in thinking of the square root of -1. The fact it is essentially illogical is not a problem to me - in fact I think its brilliant. But I'm using reason - in purely logical terms it is illogical. Unless you know of some natural form of logic where a quantity multiplied by itself can be negative ?

 

[arildno]

Unless you know of some natural form of logic where a quantity multiplied by itself can be negative ?

Sure, depends on what you mean with "quantity" and "multiplication", though.

 

[Hurkyl]

But I'm using reason - in purely logical terms it is illogical. Unless you know of some natural form of logic where a quantity multiplied by itself can be negative ?

Have you asked yourself why you think a quantity multiplied by itself cannot be negative? Can you give a logical reason why that should not happen?

The logical reason a real number multiplied by itself cannot be negative is because you can prove that based upon the axioms we've set forth that define the real numbers.

Not all of those axioms are true for the complex numbers.