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

  • Thread starter Thread starter SimonA
  • Start date Start date
  • Tags Tags
    Infinity
AI Thread Summary
Cantor's exploration of infinity introduces the concept of cardinality, which defines the size of sets through one-to-one correspondence with natural numbers. The discussion highlights confusion surrounding the transition from size to order in comparing infinities, questioning the logic behind declaring one infinity larger than another. Participants express a fascination with complex numbers while grappling with the implications of infinite sets, suggesting a perceived breakdown in logic. The conversation also touches on the nature of mathematical definitions and the acceptance of their consequences, emphasizing that the beauty of pure mathematics can sometimes clash with the complexities of Cantor's theories. Ultimately, the thread seeks clarity on the relationship between size, order, and the concept of infinity in mathematics.
SimonA
Messages
174
Reaction score
0
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 ?
 
Physics news on Phys.org
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.
 
"The square root of -1 is essentially illogical"

No it is not. Why do you think that?
 
Hi HOI

HallsofIvy said:
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 ?
 
arildno said:
"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 ?
 
SimonA said:
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.
 
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. :-p
 
Back
Top