This proof system does not contain all of PA. Other proofs have been constructed. If Zermelo-Fraenkel set theory is consistent, then number theory is consistent, because ZF provides a model for the natural numbers. (An inconsistent set of axioms would have no model.)

By the age of 25 Kurt Gödel had produced his famous “Incompleteness Theorems.” His fundamental results showed that in any consistent axiomatic mathematical system there are propositions that cannot be proved or disproved within the system and that the consistency of the axioms themselves cannot be proved.

Gödel’s incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.

1 : a statement accepted as true as the basis for argument or inference : postulate sense 1 one of the axioms of the theory of evolution. 2 : an established rule or principle or a self-evident truth cites the axiom “no one gives what he does not have”

AxiomsEdit. An axiom is something that is assumed, or believed to be true. It is where mathematical proof starts; you cannot prove the axioms, you merely believe them and use them to prove other things. There are different sets of axioms, the most current and widely-used being Zermelo–Fraenkel set theory.

A mathematical statement that we know is true and which has a proof is a theorem. So if a statement is always true and doesn’t need proof, it is an axiom. If it needs a proof, it is a conjecture. A statement that has been proven by logical arguments based on axioms, is a theorem.

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!”

A penny saved is a penny earned

SYNONYMS FOR maxim 1 aphorism, saying, adage, apothegm.

1 : a general truth, fundamental principle, or rule of conduct Mother’s favorite maxim was “Don’t count your chickens before they hatch.” 2 : a proverbial saying advised her daughter with the maxim “marry in haste, repent at leisure”

The moral status of an action must be determined by evaluating its maxim. A maxim is a rule that connects an action to the reasons for the action, i.e., a motivation/goal/context. So, when you formulate a maxim you must name the action and give the reason.

“The bigger the better” / “Good things come in small packages.” “What’s good for the goose is good for the gander.” / “One man’s meat is another man’s poison.” “Birds of a feather flock together.”/ “Opposites attract.” “Actions speak louder than words.” / “The pen is mightier than the sword.”