Mathematical question

[GrowingSmaller]

IIRC Godel's incompleteness theorem says something like in a formal system of maths there are some statements which cannot be proven to be true or false within that system. What I want to know is could Goldbach's conjecture be one such statement.

Every even integer greater than 2 is a Goldbach number, a number that can be expressed as the sum of two primes.[1]

 

[Wiccan_Child]

I think Godel's theorms basically say any language capable of doing basic arithmetic cannot be both complete (everything is independantly defined) and consistent (everything tallies up, no contradictions). Since consistency is better than completeness, modern mathematics is consistent (you can't prove 1+1=2 and 3) but not complete (pretty much all mathematical concepts boils down to the cardinality of a set).

I don't know if Goldbach's conjecture is unprovable, just unproven.

 

[GrowingSmaller]

But if it is unprovable, could that be a result of incompleteness, or would it be a different type of sentence that is unprovable?

Or, is there anything which is unprovable (yet could be true), which unprovability is not an example of incompleteness?

 

[Wiccan_Child]

But if it is unprovable, could that be a result of incompleteness, or would it be a different type of sentence that is unprovable?

I don't think so, because we can prove things using our incomplete mathematics. It's inconsistency that would make things unprovable.

Or, is there anything which is unprovable (yet could be true), which unprovability is not an example of incompleteness?

Wikipedia has a nice list.

 

[JonF]

IIRC Godel's incompleteness theorem says something like in a formal system of maths there are some statements which cannot be proven to be true or false within that system. What I want to know is could Goldbach's conjecture be one such statement.

No. Godel’s incompleteness theorems by no means even suggest that everything can’t be proven. Just that there exist somethings that can’t be proven. Formally it says systems that contain something similar to first order logic and can formulate the natural numbers contain true statements about the natural numbers that can’t be proven within the system, thus the system is incomplete.

(pretty much all mathematical concepts boils down to the cardinality of a set).

This is not true at all, and I died a little inside reading it.

But if it is unprovable, could that be a result of incompleteness, or would it be a different type of sentence that is unprovable?

Or, is there anything which is unprovable (yet could be true), which unprovability is not an example of incompleteness?

you can think of incomplete as meaning Unprovable (it’s actually a lot more complicated, but you’d need a fairly strong math background to go into detail). Undecidable is something else. There are statements that can’t be proven true or false in a given system and aren’t true or false, and we 100% know this about them. We call these things axiom independent. For example the continuum hypothesis is independent of ZFC. The well ordering theorem is independent of ZF.

 

[Wiccan_Child]

This is not true at all, and I died a little inside reading it.

With as rigorous a disproof as that, how can I possibly attempt a rebuttal...

 

[JonF]

Okay, explain how topology, algebra, analysis, functional analysis, are all questions of cardinality.

 

[Rilke's Granddaughter]

Okay, explain how topology, algebra, analysis, functional analysis, are all questions of cardinality.

"God made the integers; all the rest is the work of man."

 

[Rilke's Granddaughter]

Originally Posted by Wiccan_Child: "(pretty much all mathematical concepts boils down to the cardinality of a set)."

This is not true at all, and I died a little inside reading it.

I once was given a problem: begin with the null set & five astronomical observations and derive the Kepler wave equations.

It was fun.

 

[JonF]

"God made the integers; all the rest is the work of man."

God made ZFC and first order logic, the rest is a lot of work of a lot of men.

 

[Rilke's Granddaughter]

God made ZFC and first order logic, the rest is a lot of work of a lot of men.

That's what he said.

 

[Wiccan_Child]

Okay, explain how topology, algebra, analysis, functional analysis, are all questions of cardinality.

Key qualifier: 'pretty much'. That said, algebra is abstraction of the complex numbers, is an abstraction of the real numbers, is an abstraction of the integers, is an abstraction of the natural numbers, is an abstraction of 1 and 0, where 0 = |{}|, and 1 = |{0}|. Addition, and all subsequently more abstract operations, are manipulations of sets and the elements therein. Algebra is most obviously derived from sets, while topology takes a detour through Euclid's Elements. But it pretty much all boils down to the cardinality of sets.

 

[GrowingSmaller]

So do we know in advance, or can we completely derive a priori, which statements are provable and which aren't? Ie. given the axioms or definitions, then it follows that x, y, x and only x, y, z is provable and a, b, c and only a, b, c is not?

I suppose not, because if it were true there would not be unsolved theorems like Fermat's etc.

 

[Wiccan_Child]

So do we know in advance, or can we completely derive a priori, which statements are provable and which aren't? Ie. given the axioms or definitions, then it follows that x, y, x and only x, y, z is provable and a, b, c and only a, b, c is not?

Nothing about the real world, I think. A priori arguments can only prove things which must exist, and there's not a whole lot of that...

 

[JonF]

Key qualifier: 'pretty much'. That said, algebra is abstraction of the complex numbers, is an abstraction of the real numbers, is an abstraction of the integers, is an abstraction of the natural numbers, is an abstraction of 1 and 0, where 0 = |{}|, and 1 = |{0}|. Addition, and all subsequently more abstract operations, are manipulations of sets and the elements therein. Algebra is most obviously derived from sets, while topology takes a detour through Euclid's Elements. But it pretty much all boils down to the cardinality of sets.

Look i'm not trying to be rude, but you have no idea what you're talking about do you? Have you ever taken any courses in any of the subjects I mentioned?

Also being "derived from sets" doesn't mean it's a question of cardinality, it means sets exist. You described relations on sets, not cardinality. Math is not just a question of cardinality - very few things in math are actually questions of cardinality. I know, I've spent extensive amounts of time doing work in the field (math pun). If you want I can go grab a book in any of the subjects I gave you and start pulling out hundreds of theorems that aren't questions of cardinality, and then show the one or two that are.

 

[JonF]

So do we know in advance, or can we completely derive a priori, which statements are provable and which aren't? Ie. given the axioms or definitions, then it follows that x, y, x and only x, y, z is provable and a, b, c and only a, b, c is not?

I suppose not, because if it were true there would not be unsolved theorems like Fermat's etc.

Most often we can't tell what isn't provable, unless it's axiom independent. Even then sometimes we can't show it. Also you should note, only a priori arguments are accepted in math.

Also, Fermat's Last Theorem has a solution now.

 

[Steve Petersen]

Any of you familiar with this guy?

He posits that mathematics IS reality.

 

[JonF]

Is it Pythagoras? That guy is most likely a crack pot, math seems to track gobs and gobs of them.

 

[Wiccan_Child]

Look i'm not trying to be rude, but you have no idea what you're talking about do you? Have you ever taken any courses in any of the subjects I mentioned?

Also being "derived from sets" doesn't mean it's a question of cardinality, it means sets exist. You described relations on sets, not cardinality. Math is not just a question of cardinality - very few things in math are actually questions of cardinality. I know, I've spent extensive amounts of time doing work in the field (math pun). If you want I can go grab a book in any of the subjects I gave you and start pulling out hundreds of theorems that aren't questions of cardinality, and then show the one or two that are.

Yea, I'm not gonna rise to urinating contest. If you want to talk maths, sort your attitude out and PM me.

 

[JonF]

It's not trying to get in a urinating contest, what you are saying is blatantly wrong. When I called you on it you insisted you were right and tried to defend it with non-sense. I asked you to show me how the body of work in several of the major subjects in math are questions of cardinality, you couldn't. If you want I can start grabbing specific theorems, and you can try to show me how it's a cardinality question? I take it from what you said you don't have a formal math background. I do. Math isn't just a question of cardinality. So if you wanna drop this that's fine, but like everything else in math opinion doesn't matter and i'm right on this.