I An infinity of points on two unequal lines- an intuitive explanation?

AI Thread Summary
The discussion centers on the challenge of intuitively understanding the concept of infinity, particularly in relation to Cantor's set theory, which asserts that a one-foot line segment contains the same number of points as a two-foot segment. Participants explore the difficulty of grasping dimensionless points, suggesting that our inability to visualize them leads to misconceptions. The analogy of a rubber band is used to illustrate that length can be doubled while retaining the same quantity of points. Additionally, it is noted that the Banach-Tarski paradox highlights similar intuitive challenges regarding points. Ultimately, the mathematics of bijections, such as y=2x, demonstrates that these intervals can be mapped one-to-one, reinforcing the concept of equal cardinality despite differing lengths.
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How do you train your intuition to accept the fact that a 1 foot long line segment has the same amount numbers /points as a 2 foot long?
I am familiar with Cantor's work on the concept of infinity and his use of the set theory to explain various types of infinities. Having said that my intuition never seems truly grasp/accept it.

Is there a way to train my mind to see this seemingly contradictory situation as a fact? This is the opposite of an illusion.
 
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musicgold said:
Summary:: How do you train your intuition to accept the fact that a 1 foot long line segment has the same amount numbers /points as a 2 foot long?

Is there a way to train my mind to see this seemingly contradictory situation as a fact?
Imagine it like a rubber. You can double its length and still have the same rubber in hand.

This image isn't as weird as it may sound. The actual problem with intuition here is the concept of a point. We cannot imagine a dimensionless object, so we associate a tiny ball. Both are wrong. While points are factually dimensionless, we still can work with them. E.g. collect enough of them to form a line. So there must be something, regardless of the fact that it has no dimension.

This system immanent misconception of points is also the major obstacle in the Banach-Tarski paradox, more than the axiom of choice is.
 
musicgold said:
Summary:: How do you train your intuition to accept the fact that a 1 foot long line segment has the same amount numbers /points as a 2 foot long?
You can't measure the number line in feet and inches. The interval ##[0, 1]## and the interval ##[0, 2]## are sets of points. One is a proper subset of the other, but there exists a 1-1 mapping between them. Both of these statements are elementary to prove. That's mathematics. You don't have to train any intuition.
 
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y=2x is a bijection.
 
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