Starts to geometric proofs

Usage examples of axioms.

No logical system rich enough to contain the axioms of simple arithmetic can ever be made complete.

It is always possible to construct statements which can be neither disproved nor proved by deduction from the axioms.

But how does it feel, to be a sentient structure with bones of axioms, sinews of logic?

A finite sequence of logical steps linked the axioms of number theorywhich included some simple generalities about all numbersto Fermat's own sweeping assertion.

Everything implied by a given set of axioms is logically connected to them, timelessly, eternally .

Omega-inconsistent number theories were non-standard versions of arithmetic, based on axioms that "almost" contradicted each othertheir saving grace being that the contradictions could only show up in "infinitely long proofs" (which were formally disallowed, quite apart from being physically impossible).

What you're talking about is taking ordinary arithmeticno weird counter-intuitive axioms, just the stuff every ten-year-old knows is trueand proving that it's inconsistent, in a finite number of steps?

In spite of everything, I still wasn't ready to put a bullet in anyone's brain for the sake of defending the axioms of number theory.

Seduced by the success of the axioms in developing a system of geometry, the Greeks came to think of the axioms as "absolute truths" and to suppose that other branches of knowledge could be developed from similar "absolute truths.

Thus in astronomy they eventually took as self-evident axioms the notions that (I ) the earth was motionless and the center of the universe, and (2) whereas the earth was corrupt and imperfect, the heavens were eternal, changeless, and perfect.

Similarly, Aristotle worked up fanciful theories of motion from "self-evident" axioms, such as the proposition that the speed of an ob'ect's fall was proportional to its weight.

After the Greeks had worked out all the implications of the axioms, further important discoveries in mathematics or astronomy seemed out of the question.

Through the sudden manufacture of an ultra-democratic constitution, through a convocation of the primary assemblies, and a ratification of its work by the people in these assemblies, through the summoning of delegates to Paris, through the assent of these converted, fascinated, or constrained delegates, it exonerates and justifies itself, and thus deprives the Girondins of the grievances to which they had given currency, of the axioms they had displayed on their standards, and of the popularity they thought they had acquired.

Nouns, adjectives, adverbs—all parts of speech—were losing meaning for her as these clowns endlessly debated the basic axioms of ontology and epistemolo-gy.

This Utopian proposition—which the sane reader will regard with acute skepticism—has been seen to be sound by a rigorously logical demonstration, based on the axioms of economics, in The Cause of Business Depressions by Hugo Bilgrim and Edward Levy.