13 problems

A universal proposition is a statement asserting a property for every integer; a general proof is a proof establishing that proposition for all integers. Gödel's conjecture. One ma…

Let a set be fractally countable relative to a base system when it is obtained as a union of countable sets definable in a sequence of extensions of that system, as in … Let $HYP$…

Let von Neumann's function theory be the formal theory referred to in the surrounding discussion, and let $\mathrm{NBG}$ denote von Neumann–Bernays–Gödel set theory. Two theories a…

Gödel–Hilbert programme conjecture. Gödel may have intended to argue that, for every system of the specified kind, the sentence $G$ is axiomatically undecidable but is decidable vi…

The paper distinguishes Church's metalinguistic definition of $j$-equivalence from an object-linguistic recursive definition. Church's metalinguistic recursion conjecture. Church r…

Herbrand considers intuitionistic proofs and their formalizability in Russell's system. Herbrand's conjecture. It is impossible to prove that every intuitionistic proof is formaliz…

Critical-period conjecture. The critical period of set theory will be realized with the advent of second-order logic and the revival of Hilbert's program.

Constituent-structure conjecture for the integers. Like the natural numbers, each integer has a constituent structure that specifies all of its properties and nothing else. When th…

Unbounded proof-length conjecture. For most “complex”, “deep” theorems $T$, and in particular for Fermat's Last Theorem $FLT$, it is impossible to provide and justify any upper bou…

Classical mathematics is founded on set theory, but ordinary sets do not have symmetries. The proposal is to enlarge the notion of set so that sets may carry symmetries. Symmetry-e…

A sequence of continuous functions has a limit function, and the conjecture concerns whether that limit is continuous. Continuity conjecture. The limit function of a convergent seq…

The paper discusses human beings' motivation for sustained cogitation and the possibility that the human mind has some appreciation of its own consistency. Conjecture on quasi-auto…

Human advantage conjecture. A computer will never give an answer to questions of this type as quickly as a human because it does not know the meaning of the instructions to pick…