But it’s not possible for a theory to be true without evidence, because theories are formed from evidence. A theory isn’t “an idea”, it’s something for which there’s so much evidence that it would be foolish to not accept it.

A corollary is a statement that follows naturally from some other statement that has either been proven or is generally accepted as true. A corollary may be undeniably true if the concept or theory it’s based on is true. For example, the sum of the interior angles of any triangle is always 180 degrees.

A Corollary could be described as a “post-proof.” A corollary is something that follows almost obviously from a theorem you’ve proved. You work to prove something, and when you’re all done, you realize, “Oh my goodness! If this is true, than [another proposition] must also be true!”

Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”). Proposition — a proved and often interesting result, but generally less important than a theorem. Axiom/Postulate — a statement that is assumed to be true without proof.

An axiom or postulate is a fundamental assumption regarding the object of study, that is accepted without proof.

According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent). These results have had a great impact on the philosophy of mathematics and logic.

For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

However, interestingly, even this “Church-sentence” is decidable by humans: in fact it is pre-decided through its contruction by Church. Church defined it so we can know: this statement is actually true.

A formal system (deductive system, deductive theory, . . .) S is said to be inconsistent if there is a formula A of S such that A and its negation, lA, are both theorems of this system. Hence, employing such a category of logics, the inconsistent systems do not present any proper logico-mathematical interest.

Gödel numbers are integers, and integers only factor into primes in a single way. So the only prime factorization of 243,000,000 is 26 × 35 × 56, meaning there’s only one possible way to decode the Gödel number: the formula 0 = 0. Gödel then went one step further.

A numbering of the set of computable functions can then be represented by a stream of Gödel numbers (also called effective numbers). Rogers’ equivalence theorem states criteria for which those numberings of the set of computable functions are Gödel numberings.

An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) ‘that which is thought worthy or fit’ or ‘that which commends itself as evident.

Consistency proofs for ZFC are essentially proofs by reflection, meaning that we note, in some way or another, that since the axioms of ZFC are true, they are consistent.

Most sets commonly encountered are not members of themselves. For example, consider the set of all squares in the plane. This set is not itself a square in the plane, thus it is not a member of itself.

extensionality

Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent.

As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature. Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity.

Here are a few best practices:

1. Isolate one goal. Developing consistency goes against human nature.
2. Focus on incremental improvement. You’re not going to develop a positive, worthwhile habit overnight.
3. Fight your emotions. The brain is a taxing organ.
4. Forgive your failures.

A proof is sufficient evidence or a sufficient argument for the truth of a proposition.

In mathematics, logic, and computer science, a type system is a formal system in which every term has a “type” which defines its meaning and the operations that may be performed on it. Type theory was created to avoid paradoxes in previous foundations such as naive set theory, formal logics and rewrite systems.

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference.