Can everything true be proven?
No. A consistent, effectively axiomatized theory capable of expressing enough arithmetic cannot settle every arithmetic statement. Gödel's incompleteness results place limits on formal proof; they do not show that every self-reference is contradictory or that a self-describing universe exists. Gödel's completeness theorem concerns a different question: a consistent first-order theory has a mathematical model. Having such a model does not prove physical existence, uniqueness or the intended interpretation. OPH's philosophical proposal adds an inhabited structure whose records and operations realize its own recoverable account. That is an ontological commitment beyond either theorem. A theory of physical observations can be useful and precise while its inhabitants lack proofs of every mathematical truth or a practical algorithm for predicting every event.
See also Does “this sentence is false” defeat a self-referential universe?, What about the set of all sets that do not contain themselves? (Russell’s paradox), Would a theory of everything let us predict every event?, Does describing a universe make it exist?
Also in Chapter 2: Why Is There Anything Rather Than Nothing?.
This answer comes from OPH, Observer Patch Holography. OPH is a rigorous, consistent and complete description of the unified, self-creating computational Universe we inhabit. Its consistency is machine-checked: the fixed point at its core is constructed explicitly, and a theory with a model contains no contradiction.
To understand OPH and check every step yourself, start with the free book Reverse Engineering Reality. It builds the theory from the ground up, and the OPH Notebook answers your questions while you read.
read the free bookask the notebookor ask sage:read chapter 37: Why Does the Loop Have No Beginning?watch a universe make its own particles