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

[musicgold]

TL;DR

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.

 

[fresh_42]

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.

 

[PeroK]

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.

 

[mathman]

y=2x is a bijection.