Given the set of standard axioms [of some mathematical theory], do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.

In addition to all the interesting discussion about Godel's and Goodstein's Theorems, I want to suggest also another "thread" of discussion, regarding epistemology of mathematical knowledge.

During the '60s and '70s, the philosophy of science debate was concerned with the distinction between :

• the Context of Discovery and the Context of Justification.

Roughly speaking, the context distinction regards : how science (e.g.physics) dicover a new fact or law; the second is  : how science explain it (ref.Paul Hoyningen-Huene, On the Varieties of the Distinction between the Context of Discovery and the Context of Justification, 2002).

Applied to mathematics, this points to the difference between :

the discovery of a new math idea or concept vs the proof of a theorem.

As far as I know, very few philosophers of mathematics are concerned with this kind of issue ; the only book I've read about something similar was Lakatos' Proofs and Refutations, (1976).

The connection I see is this :

when we don't have a proof of a mathematical "fact" , what are the gorund for asserting or believing it ?

Here some comments about comments in the above debate :

a) "not all true statements are theorems, that is the content of Gödel's incompleteness theorem"

They are not theorems of the formal arithmetic in question (i.e. first-order PA) but THEY ARE proved via Godel's "construction" provided by G's Theorem itself (i.e.proved in the meta-theory): isn't it ?

b) "Goldbach's conjecture is not proven, but it is almost surely true due to statistical evidence"

Are there research about "inductive" grounds for unproven mathematical facts ?

A single contradiction con destroy a theory (Russell's Paradox in front of Frege's system) but how many years (?) of absence of contradiction can support our sound belief in a theory (e.g.ZFC) ?

Given the set of standard axioms [of some mathematical theory], do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.

In addition to all the interesting discussion about Godel's and Goodstein's Theorems, I want to suggest also another "thread" of discussion, regarding epistemology of mathematical knowledge.

During the '60s and '70s, the philosophy of science debate was concerned with the distinction between :

• the Context of Discovery and the Context of Justification.

Roughly speaking, the context distinction regards : how science (e.g. physics) discover a new fact or law; the second is: how science explain it (ref.Paul Hoyningen-Huene, On the Varieties of the Distinction between the Context of Discovery and the Context of Justification, 2002).

Applied to mathematics, this points to the difference between :

the discovery of a new math idea or concept vs the proof of a theorem.

As far as I know, very few philosophers of mathematics are concerned with this kind of issue ; the only book I've read about something similar was Lakatos' Proofs and Refutations, (1976).

The connection I see is this :

when we don't have a proof of a mathematical "fact" , what are the gorund for asserting or believing it ?

Here some comments about comments in the above debate :

a) "not all true statements are theorems, that is the content of Gödel's incompleteness theorem"

They are not theorems of the formal arithmetic in question (i.e. first-order PA) but THEY ARE proved via Godel's "construction" provided by G's Theorem itself (i.e.proved in the meta-theory): isn't it ?

b) "Goldbach's conjecture is not proven, but it is almost surely true due to statistical evidence"

Are there research about "inductive" grounds for unproven mathematical facts ?

A single contradiction con destroy a theory (Russell's Paradox in front of Frege's system) but how many years (?) of absence of contradiction can support our sound belief in a theory (e.g.ZFC) ?