Mathematical question

[Wiccan_Child]

Wiccan, I think the problem has to do with the word "cardinality". It has a specific meaning mathematics. If you said, it all has to do with the properties and behavior of sets, you might get less push-back.

Somehow I doubt that very much I've only ever said it pretty much all boils down to sets, and the cardinality thereof. It should go without saying that, if we're talking about sets, then the full theory regarding sets is assumed, properties (such as cardinality) and behaviour included. If I say pretty much all matter boils down to up and down quarks, I am implicitly (if not explicitly) presuming the Standard Model. But JonF seems to enjoy semantics. Go figure.

 

[Rilke's Granddaughter]

Somehow I doubt that very much I've only ever said it pretty much all boils down to sets, and the cardinality thereof. It should go without saying that, if we're talking about sets, then the full theory regarding sets is assumed, properties (such as cardinality) and behaviour included. If I say pretty much all matter boils down to up and down quarks, I am implicitly (if not explicitly) presuming the Standard Model. But JonF seems to enjoy semantics. Go figure.

But you didn't say sets and all their properties like cardinality. You said cardinality.

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). (emphasis added)

Even I find that confusing.

 

[Wiccan_Child]

But you didn't say sets and all their properties like cardinality. You said cardinality.

I said pretty much cardinality. The major concepts of mathematics, such as integers, can be, and by convention are, defined in terms of the cardinality of sets. 0 = |{}|, 1 = |{{}}| and so on. Cardinality isn't the only key concept, and talking about the cardinality of sets doesn't mean I'm explicitly excluding all other aspects of Set Theory. Pointing out that most matter is made of protons doesn't mean I'm explicitly denying the existence of neutrons. It doesn't even mean that neutrons are of lesser importance to protons.

Even I find that confusing.

The point of my original statement was that mathematics is incomplete: you cannot completely define everything without either a) leaving some things blank, or b) going round in circles, or c) having your definitions lead to inconsistent results.

 

[Rilke's Granddaughter]

I said pretty much cardinality. The major concepts of mathematics, such as integers, can be, and by convention are, defined in terms of the cardinality of sets. 0 = |{}|, 1 = |{{}}| and so on. Cardinality isn't the only key concept, and talking about the cardinality of sets doesn't mean I'm explicitly excluding all other aspects of Set Theory. Pointing out that most matter is made of protons doesn't mean I'm explicitly denying the existence of neutrons. It doesn't even mean that neutrons are of lesser importance to protons.


The point of my original statement was that mathematics is incomplete: you cannot completely define everything without either a) leaving some things blank, or b) going round in circles, or c) having your definitions lead to inconsistent results.

So now that we've wasted three pages on semantics... what was the OP about again?

 

[JonF]

If you said, it all has to do with the properties and behavior of sets, you might get less push-back.

I would agree with this, since it's true.

And no wiccan you didn't say pretty much cardinality, you said pretty much all math. There is a difference. Words matter. I'm not the one playing semantic games, taking well defined terms and changing their meaning. Also again, you are redefining completeness, completeness with regard to set theory isn't about definitions, it's about provability. The proof of GICCT actually assumes there is a set of maximally definable terms. Words have a very specific meaning in math. There is no ambiguity in them, and if you want to change their meaning you're not doing math.

This is the problem. You are taking a math term, saying it means "about the same thing" as something else and then using the results of the math term for "something else". That doesn't work in math. If you really had the formal math background you claim you would know how exact mathematical language is, and how important that is. I'm not focusing on semantics, you are by altering the meaning of key terms, like "complete" or "cardinality".

 

[GrowingSmaller]

So now that we've wasted three pages on semantics... what was the OP about again?

I asked whether the Goldbach conjecture might be unprovable due to the incompleteness of the math system it is formulated in.

 

[JonF]

It might be, it might even be neither true nor false.

 

[GrowingSmaller]

It might be, it might even be neither true nor false.

What about the Law of the Excluded Middle? Can that be abandoned? I have heard of 'paraconsistemt logic' - is that anything to do with it?

 

[JonF]

Excluded Middle doesn’t conflict with things that are neither true nor false in the math sense (called axiom independent). You can affirm, for some proposition P: P or ~P while P itself doesn't have a definite truth value.

Here or here is a good example. Here is how math proves something is axiom independent.

 

[GrowingSmaller]

In philosophy it is said that square circles cannot exist, being logivally impossible. But it is possible to transform a circle into a square by changing the "space" on which it is realised from Euclidian to something else?

 

[Wiccan_Child]

In philosophy it is said that square circles cannot exist, being logivally impossible. But it is possible to transform a circle into a square by changing the "space" on which it is realised from Euclidian to something else?

Sure. It's vanishingly improbable in the real world, but you can mathematically warp space to do it. Squaring a circle is how you construct a square from a circle with both having equal area using a compass and straightedge.

 

[Chatter]

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.

So far as we know, yes. The interesting thing about Goldbach's conjecture is that, if it is false, then you could definitely prove that it is false, by exhibiting one of its counterexamples. The result is that formal systems such as arithmetic can prove the following claim:

if Goldbach's conjecture cannot be disproven (in this system), then Goldbach's conjecture is true.

So proving that Goldbach's conjecture cannot be proven false (in formal theories such as arithmetic) amounts to proving it true.

 

[GrowingSmaller]

Could it be proven in one system, and unproven in another?

 

[Chatter]

Could it be proven in one system, and unproven in another?

So far as we know, sure.

 

[JonF]

[FONT=&quot]Any statement that Godel’s incompleteness theorem shows is true but not provable has an extension of the system where it’s provably true. This extended system then will have other unprovable statements of course.[/FONT]

 

[Chatter]

[FONT=&quot]Any statement that Godel’s incompleteness theorem shows is true but not provable has an extension of the system where it’s provably true. This extended system then will have other unprovable statements of course.[/FONT]

But we're talking about the Goldbach conjecture, of which the Incompleteness Theorems are silent.

The question in the OP isn't really a mathematical question, and the significance of the Incompleteness Theorems outside of mathematics is disputable to say the least. If we interpret the OP question in a really pedantic way, it falls apart at the seams:

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.

Godel's Theorems don't apply to arbitrary formal systems, only to consistent formal systems which embed a sufficiently powerful arithmetic, and the statement which is undecidable varies with the system. There are, of course, systems in which Goldbach's conjecture is provable. For instance, the theory in arithmetic where Goldbach's conjecture is the only axiom is a theory which trivially proves Goldbach's conjecture.

This isn't, I feel, in the spirit of the OP's question. More informally, GrowingSmaller is perhaps asking whether, whatever methods we take to be the sound methods of mathematics, it might be that we are never able to prove Goldbach's conjecture. The answer is: as far as we know, it might be. However, as I hinted, if we ever manage to prove that the Goldbach conjecture is irrefutable from a system that embeds basic arithmetic, then we will have a proof of Goldbach's conjecture.

 

[GrowingSmaller]

. For instance, the theory in arithmetic where Goldbach's conjecture is the only axiom is a theory which trivially proves Goldbach's conjecture.

This isn't, I feel, in the spirit of the OP's question.

Who cares?

 

[Chatter]

Who cares?

If you don't, then I guess no one.

If you'd be willing to say a bit about what motivates your question, that would be good. If you want to hear a few of us talk about straight-up mathematical logic and the theory behind Godel's Theorems and formal systems, and go from there, that's cool. I'm sure JonF and I could give it a decent crack.

 

[JonF]

But we're talking about the Goldbach conjecture, of which the Incompleteness Theorems are silent.

This is not true. For all we know the Goldbach's conjecture is one of the statements described by Godel's incompleteness theorems. If that is the case, it may not be provable in ZFC, but it will be provable in some extension of ZFC. Which is directly related to the question GrowingSmaller asked a few post ago and his question in the OP. Most mathematicians would probably "guess" this isn't the case, because Goldbach's Conjecture is so different than the type of unprovable statement constructed in Godel's Incompleteness Theorems proof. If it's not provable, it's probably axiom independent - which is a completely different equally complicated issue.

 

[JonF]

Godel's Theorems don't apply to arbitrary formal systems, only to consistent formal systems which embed a sufficiently powerful arithmetic, and the statement which is undecidable varies with the system. There are, of course, systems in which Goldbach's conjecture is provable. For instance, the theory in arithmetic where Goldbach's conjecture is the only axiom is a theory which trivially proves Goldbach's conjecture.

This is misleading. You can't really have just Goldbach's conjecture as your only axiom. To even formulate Goldbach's conjecture you'd need something like a first (or higher) order logic and ZFC. These necessitate the standard construction of N. So it's possible that Goldbach's conjecture is false in any system it can be stated in. Also Godel's theorem's do apply to inconsistent systems, they actually say a lot about them! Granted, the inconsistent parts of the theorems tend to be largely ignored by non-mathematicians.