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October 13th, 2018, 04:47 AM   #1
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Math Philosophy

What the differences between actual infinity to potentially infinity?
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October 13th, 2018, 09:18 AM   #2
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When a mathematician asks what is the difference between A and B he means

What is (B-A) ?

(Potential infinity minus actual infinity) was never defined by the ancient Greeks who introduced them. Nor was any arithmetic involving infinity.

Modern concepts of infinity are quite different from these in any case.
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October 13th, 2018, 10:16 AM   #3
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Quote:
Originally Posted by studiot View Post
When a mathematician asks what is the difference between A and B he means

What is (B-A) ?

(Potential infinity minus actual infinity) was never defined by the ancient Greeks who introduced them. Nor was any arithmetic involving infinity.

Modern concepts of infinity are quite different from these in any case.
So if I asked you what's the difference between a vector and a scalar, is that what you'd tell me? That the Greeks never defined (vector minus scalar)? Jeez man.
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October 13th, 2018, 12:57 PM   #4
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Originally Posted by shaharhada View Post
What the differences between actual infinity to potentially infinity?
Aristotle was the one who originally made the distinction. A potential infinite is like the counting numbers: 1, 2, 3, 4, ... The sequence never ends, since given any number you can always identify the next one. But there is no "actual" infinity. The process of taking "the next one" never ends. There's no point where you can say, "We're all done, we have ALL the counting numbers in a box."

Actual infinity is putting all the numbers in a box. Finishing the process of taking successors. Aristotle said that potential infinity was possible and actual infinity was impossible.

Fast forward to Cantor, who said there was a "set" that contained ALL the counting numbers. Reasoning from that, he built out an endless hierarchy of infinite sets. That's actual infinity.

In modern math, the difference is characterized by the axiom of infinity, which says there's an infinite set. It's not true or false in any meaningful sense. If you accept it, you can build almost the entirety of modern math on top of set theory. If you reject it, you get the Peano axioms and a lot of number theory, but you can't get the theory of the real numbers off the ground. So mathematicians accept the axiom of infinity because it's useful.

https://en.wikipedia.org/wiki/Axiom_of_infinity

To sum up, potential infinity is the sequence 1, 2, 3, 4, ... Actual infinity is the set {1, 2, 3, 4, ...}. The curly braces say that you have ALL of the counting numbers inside a single object, called the "set" of natural numbers. What that really means is for the philosophers. Mathematicians find the concept useful so they use it.

There is one bit of confusion around the term actual infinity. People sometimes ask if there is an "actual infinity" in the real world. That's a different meaning of the phrase. In math, actual infinity is simply the set of natural numbers, or in general any infinite set. An infinite set has no physical existence, it's only a conceptual abstraction.

But if we're talking about physics, the question arises as to whether anything in the universe is actually infinite. An infinite number of stars as opposed to merely a very large finite number of them.

When you see online discussions about "actual infinity," you'll invariably see a lot of confusion on this point. Do they mean the actual infinity of math? Or an actual infinity in physics? These are two entirely different concepts.
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October 15th, 2018, 08:55 AM   #5
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Potential Infinity: Start walking.
Actual Infinity: You never get there.
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October 15th, 2018, 01:03 PM   #6
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Potential Infinity: Start walking.
Zeno is in the hizz-house!

-Dan
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October 15th, 2018, 01:05 PM   #7
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Zeno is in the hizz-house!

-Dan
I think the kidz call it the hizzy.
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