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April 11th, 2017, 06:22 AM   #21
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The points are locations on a ball, just as points are locations on a line. The points are not the ball. They are an imaginary ball.

If it's not a real ball, it's not a real paradox.
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April 11th, 2017, 01:34 PM   #22
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Consider a set of n "things. "If I remove m things, but don't change their identifying characteristics, and replace them, it's the same set (order not defined).

Now consider an infinite set. Infinity is not defined. The set doesn't exist, except as words "an infinite set."

On the other hand, if i define a set as a container, the contents don't matter. An egg basket is different than a collection of eggs.

A ball (sphere) as an infinite collection of points is meaningless because infinity is undefined.

A ball as a container of points is independent of the points. You can't remove points to create another ball. All you can do is move the ball or place some points in another, pre-existing, ball.

(A basket is independent of the eggs. You can't remove eggs to create another basket. All you can do is place some eggs in another basket.)

The mathematical gyrations you go through with the points (eggs) is totally irrelevant.
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April 11th, 2017, 01:38 PM   #23
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Quote:
Originally Posted by zylo View Post
The mathematical gyrations you go through with the points (eggs) is totally irrelevant.
Why didn't you raise these philosophical issues when we discussed real analysis?

You're making a really bizarre argument. You're refusing to do math because math doesn't exist. If math doesn't exist, why do you do math in other threads? The concerns you're bringing up apply to all math, not just the B-T theorem. Why would you start a thread about Banach-Tarski just to complain that mathematical 3-space isn't real? That doesn't even have anything to do with Banach-Tarski.

Last edited by Maschke; April 11th, 2017 at 01:45 PM.
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April 12th, 2017, 07:11 AM   #24
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It's not a matter of philosophy, it's a matter of mathematical precision.

You can do anything you want with an infinite set of points. You can take enough points from a pea and create the universe and still have the pea left over.

There is a big difference between is an infinite set of points and contains an infinite set of points.

If the pea contains an infinite set of points (locations actually), I can put an infinite set of points (locations) in every pre-existing object in the universe, and every imaginable geometric figure, and still have an infinite set of points left in the pea.

Now some philosophy. The problem is the continuum. If you did all your thinking in terms of finite number and then let that number become very large, everything would make sense (classical analysis). If you specifically reject making sense, that's fine for you and your little club, but don't take your results to the world and announce a "paradox" as a brilliant intellectual achievement. It isn't. It's just a misconception obscured by smoke.

EDIT
Short and Sweet- You can't create a finite dimension from a 0 dimension.

Last edited by zylo; April 12th, 2017 at 07:52 AM.
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April 12th, 2017, 09:39 AM   #25
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Short and Sweet- You can't create a finite dimension from a 0 dimension.
So you reject real analysis too?
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May 8th, 2017, 09:24 AM   #26
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Dude, math is just rules that are consistent with the context you apply it in.

There's a lot of religion around math. We keep trying to "discover" more things, when we're just really interpreting constraints from pre-existing contexts.

For example. Topology is just the study of geometry with a function applied to everything in the system.
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