The Identity Function Paradox

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You realize you have no life when you have a serious argument about a single math problem over the course of days. Jeez, lets just say we are all right, and we are all wrong, and move on.

Using logic to point out flaws in a proof is not 'a problem'.

But that only works when there is an actual flaw. In what I posted, there are no flaws unless I made a very minor arithmetic error somewhere. The math is very simple and rock solid.

If you think there is a flaw, please point it out and, more importantly, prove that it is flawed.

The nub of the issue raised in the original post is this: there have to be more points on a plane than there are on a line. And that since the 'number' of points on a line is infinity- then the 'number' of points on a plane MUST BE "infinity squared"!

Thats the essence of the original post.

And i have to admit that that would seem to be the case. But according to mathmaticians it does not work that way. You cant have 'infinity squared' (nor infinity cubed for geometric solids either).

Sorry.

However, according to mathmaticians, some infinities are indeed bigger than other infinities.

But again- it doesnt work the way that you might think it would.

Take the natural numbers- whole integers: 1,2,3,4,... .

The number of all whole integers ofcoures is infinity.

And so is the 'number' of all odd whole integers: 3,5,7,9,...

But since odd numbers are a subset of all whole numbers then you might think that the infinity of all numbers must be greater than the infinity of just odd numbers. But you would be wrong. The cardinality is the same. So they are egual sized infinities.

But the number of ALL numbers ( fractions, decimals, etc) is a greater infinity than the infinity of integers because you cant match them in the same cardinal order.

The infinity of ALL numbers is the same bigger infinity as the infinity of points on a line.

But within this bigger infinity you have the equivalent seeming paradox that the number of all natural numbers is the same as the number of just odd numbers. That being what eric76 explained above-that short line segments have the same size infinity of points as do long line segments ( contrary to your common sense expectation that the number of points on a long segment must more than that on a small segment). Likewise the 'number' of points on a plane is the same as that on a line.

But screw the mathmaticians.

Im with you. The number of points on a plane is "infinity squared" I say!

An I am ignored.

As a matter of principle, I don't argue about math in this place.

My usual answer to questions like "How can I imagine this?" is: "you don't".
You don't need to be able to imagine that 0.999999... = 1. It's possible to prove it, that's enough.
After a few semesters, I've developed an intuition about certain things – that is, I can imagine the proof sketch in my head, but I still need the proof to verify the correctness.

The problem is that a lot of mathematical statements make use of the concept of infinity, and I don't know about any way to imagine infinity.

I'm starting to think that's a wise principle. Mathematics is not debatable. The philosophical "what if's" are anoying.

The math is very simple from the point of view of arithmetical manipulations, not conceptually. The person the proof was aimed at did not have trouble with the arithmetic, but with the concepts.

It was a mistake to try to get you to explain things. I should have explained them myself before she lost interest in getting an answer.

Just how would you have explained it that would have been accurate and any better than the other explanations given?

Something like the explanation Spiderpig gave, but with an intuitive explanation of what a limit is.

This link has a good answer at the top.