It’s this notion of infinity that makes fundamentalists condemn set theory as un-Biblical. They would like to believe that there is only one infinity: God.

Good presentation but somewhat oversimplified regarding Godel and Cohen’s statement about the Continuum hypothesis (CH). It’s not that one cannot prove or disprove CH at all, but rather, that CH is an example of a logical mathematical statement that can be formulated using only the language of the Zermelo-Frankel axioms (ZF), but cannot be proved or disproved within the mathematical structure those axioms define.

In other words, ZF can be used to state the claim of CH, but is insufficient to prove or disprove that claim. You could postulate a system of mathematics in which CH is true (e.g., ZF+CH), and this would work; or, you could just as easily postulate an alternative system in which CH is false.

This situation is a bit like the discovery of non-Euclidean geometries, in which the parallel postulate is not accepted as axiomatically true.

In fact, Godel’s incompleteness theorems have far greater consequences than that related to set-theoretic claims like CH. His work showed that in ANY axiomatic system of mathematics, it is possible to formulate a logical statement within that system that cannot be proved or disproved using those axioms.

Good presentation but somewhat oversimplified regarding Godel and Cohen’s statement about the Continuum hypothesis (CH). It’s not that one cannot prove or disprove CH at all, but rather, that CH is an example of a logical mathematical statement that can be formulated using only the language of the Zermelo-Frankel axioms (ZF), but cannot be proved or disproved within the mathematical structure those axioms define.

In other words, ZF can be used to state the claim of CH, but is insufficient to prove or disprove that claim. You could postulate a system of mathematics in which CH is true (e.g., ZF+CH), and this would work; or, you could just as easily postulate an alternative system in which CH is false.

This situation is a bit like the discovery of non-Euclidean geometries, in which the parallel postulate is not accepted as axiomatically true.

In fact, Godel’s incompleteness theorems have far greater consequences than that related to set-theoretic claims like CH. His work showed that in ANY axiomatic system of mathematics, it is possible to formulate a logical statement within that system that cannot be proved or disproved using those axioms.

Something for us nerds to ponder, if we haven’t already. Interesting that he didn’t go into linear infinities or dimensional infinities or quantum infinities or the other infinities that I can conceptualize but cannot think of a title for them…. I suppose there may be an infinity number of types of infinities

@atomic:
Quite so. Perhaps you could also include ordinal numbers, cardinal numbers, and perhaps even V=L đ

Seriously, how many others who follow TR have the slightest clue about what we are talking about?

I happen to think that this video, which is embedded in the URL I gave in my earlier post (well worth a read – and follow up with the Mother Jones article in Sources & Resouces), is an excellent, short introduction to two major points about mathematics for its target audience, people with minimal mathematical sophistication. (1) multiple infinities (2) some problems cannot be solved within mathematics. In addition to the fundamentalist issue of more than one infinity a/k/a God, I’m sure that the ‘holes’ in mathematics are also a problem to a group which believes that everything has a neat, tidy unique solution – theirs.

@ATOMIC: Well that’s it, then! Marry me. OK? I mean, I’d love to have a man who could whisper sweet axioms in my ear while he’s parallel-postulating me.

Here’s what you said: “His work showed that in ANY axiomatic system of mathematics, it is possible to formulate a logical statement within that system that cannot be proved or disproved using those axioms.”

Instead, what he showed was that in any FINITE axiomatic system of mathematics *POWERFUL*ENOUGH*TO*MODEL*SIMPLE*ARITHMETIC*, then there will be statements that can be constructed within the system that cannot be proven by the system.

Not all axiomatic systems of mathematics are finite and not all finite systems are powerful enough to model simple arithmetic. For example, Presburger arithmetic is finite (only 5 axioms) and is both complete and consistent, but it only defines addition and thus is not powerful enough to model simple arithmetic.

Tarski’s axioms of elementary Euclidean geometry are also consistent, but it isn’t powerful enough to model simple arithmetic and thus there is no contradiction of Godel.

As for why he didn’t go into other aspects of real analysis (and by “real” I mean the mathematical definition of “real”), he only had about 7 minutes and was trying to explain something complex to people who don’t know much math. There simply wasn’t time. After all, the way in which Godel’s proof regarding the continuum is usually stated as: “Assuming the Continuum Hypothesis is true does not lead to a contradiction.” Cohen’s work is stated similarly, “Assuming the Continuum Hypothesis is false does not lead to a contradiction.”

There’s a subtle distinction there that goes to how mathematicians conduct proof. It isn’t that they did a direct proof that no such proof exists. Instead, they did an indirect proof that showed there was no way to contradict the claim that it is true. One would think that would be enough since not being able to prove it false seemingly indicates that it must be true…but Cohen showed that just because you can’t prove it to be false doesn’t mean it actually isn’t.

But to get into all of that takes time that you just don’t have in the format of a TED Talk. It’s supposed to be brief and accessible.

B says

I LOVE MATH

Diogenes Arktos says

It’s this notion of infinity that makes fundamentalists condemn set theory as un-Biblical. They would like to believe that there is only one infinity: God.

http://boingboing.net/2012/08/07/what-do-christian-fundamentali.html

atomic says

Good presentation but somewhat oversimplified regarding Godel and Cohen’s statement about the Continuum hypothesis (CH). It’s not that one cannot prove or disprove CH at all, but rather, that CH is an example of a logical mathematical statement that can be formulated using only the language of the Zermelo-Frankel axioms (ZF), but cannot be proved or disproved within the mathematical structure those axioms define.

In other words, ZF can be used to state the claim of CH, but is insufficient to prove or disprove that claim. You could postulate a system of mathematics in which CH is true (e.g., ZF+CH), and this would work; or, you could just as easily postulate an alternative system in which CH is false.

This situation is a bit like the discovery of non-Euclidean geometries, in which the parallel postulate is not accepted as axiomatically true.

In fact, Godel’s incompleteness theorems have far greater consequences than that related to set-theoretic claims like CH. His work showed that in ANY axiomatic system of mathematics, it is possible to formulate a logical statement within that system that cannot be proved or disproved using those axioms.

atomic says

Good presentation but somewhat oversimplified regarding Godel and Cohen’s statement about the Continuum hypothesis (CH). It’s not that one cannot prove or disprove CH at all, but rather, that CH is an example of a logical mathematical statement that can be formulated using only the language of the Zermelo-Frankel axioms (ZF), but cannot be proved or disproved within the mathematical structure those axioms define.

In other words, ZF can be used to state the claim of CH, but is insufficient to prove or disprove that claim. You could postulate a system of mathematics in which CH is true (e.g., ZF+CH), and this would work; or, you could just as easily postulate an alternative system in which CH is false.

This situation is a bit like the discovery of non-Euclidean geometries, in which the parallel postulate is not accepted as axiomatically true.

In fact, Godel’s incompleteness theorems have far greater consequences than that related to set-theoretic claims like CH. His work showed that in ANY axiomatic system of mathematics, it is possible to formulate a logical statement within that system that cannot be proved or disproved using those axioms.

TC says

Um, what Atomic sez.

murraygoround says

Something for us nerds to ponder, if we haven’t already. Interesting that he didn’t go into linear infinities or dimensional infinities or quantum infinities or the other infinities that I can conceptualize but cannot think of a title for them…. I suppose there may be an infinity number of types of infinities

anon says

Did they include waiting at the DMV?

RJP says

My head just exploded.

Stan says

I love this type of stuff. NERDS ROCK!!!

Diogenes Arktos says

@atomic:

Quite so. Perhaps you could also include ordinal numbers, cardinal numbers, and perhaps even V=L đ

Seriously, how many others who follow TR have the slightest clue about what we are talking about?

I happen to think that this video, which is embedded in the URL I gave in my earlier post (well worth a read – and follow up with the Mother Jones article in Sources & Resouces), is an excellent, short introduction to two major points about mathematics for its target audience, people with minimal mathematical sophistication. (1) multiple infinities (2) some problems cannot be solved within mathematics. In addition to the fundamentalist issue of more than one infinity a/k/a God, I’m sure that the ‘holes’ in mathematics are also a problem to a group which believes that everything has a neat, tidy unique solution – theirs.

jamal49 says

@ATOMIC: Well that’s it, then! Marry me. OK? I mean, I’d love to have a man who could whisper sweet axioms in my ear while he’s parallel-postulating me.

Rrhain says

@ATOMIC: Close, but not quite.

Here’s what you said: “His work showed that in ANY axiomatic system of mathematics, it is possible to formulate a logical statement within that system that cannot be proved or disproved using those axioms.”

Instead, what he showed was that in any FINITE axiomatic system of mathematics *POWERFUL*ENOUGH*TO*MODEL*SIMPLE*ARITHMETIC*, then there will be statements that can be constructed within the system that cannot be proven by the system.

Not all axiomatic systems of mathematics are finite and not all finite systems are powerful enough to model simple arithmetic. For example, Presburger arithmetic is finite (only 5 axioms) and is both complete and consistent, but it only defines addition and thus is not powerful enough to model simple arithmetic.

Tarski’s axioms of elementary Euclidean geometry are also consistent, but it isn’t powerful enough to model simple arithmetic and thus there is no contradiction of Godel.

As for why he didn’t go into other aspects of real analysis (and by “real” I mean the mathematical definition of “real”), he only had about 7 minutes and was trying to explain something complex to people who don’t know much math. There simply wasn’t time. After all, the way in which Godel’s proof regarding the continuum is usually stated as: “Assuming the Continuum Hypothesis is true does not lead to a contradiction.” Cohen’s work is stated similarly, “Assuming the Continuum Hypothesis is false does not lead to a contradiction.”

There’s a subtle distinction there that goes to how mathematicians conduct proof. It isn’t that they did a direct proof that no such proof exists. Instead, they did an indirect proof that showed there was no way to contradict the claim that it is true. One would think that would be enough since not being able to prove it false seemingly indicates that it must be true…but Cohen showed that just because you can’t prove it to be false doesn’t mean it actually isn’t.

But to get into all of that takes time that you just don’t have in the format of a TED Talk. It’s supposed to be brief and accessible.

Chris McCoy says

A good introduction to set theory and the concept of Infinity is the 1982 book ‘Infinity and the Mind’ by Rudy Rucker.