Proofs 1 The axiomatic Method The standard procedure for establishing truth in mathematics was invented by Euclid,a mathematician working in Alexadria, Egypt around 300 BC. His idea was to begin with five assumptions about geometry, which seemed undeniable based on direct experience (For example, There is a straight line segment between every pair of points. )Proposi tions like these that are simply accepted as true are called axioms Starting from these axioms, Euclid established the truth of many additional proposi tions by providing proofs". A proof is a sequence of logical deductions from axioms and previously-proved statements that concludes with the proposition in question. You probably wrote many proofs in high school geometry class, and you'll see a lot more in this course There are several common terms for a proposition that has been proved. The different terms hint at the role of the proposition within a larger body of work Important propositions are called theorems A lemma is a preliminary proposition useful for proving later propositions A corollary is an afterthought, a proposition that follows in just a few logical steps from a theorem The definitions are not precise. In fact, sometimes a good lemma turns out to be far more important than the theorem it was originally used to prove Euclids axiom-and-proof approach, now called the axiomatic method, is the founda- tion for mathematics today. Amazingly, essentially all mathematics can be derived from just a handful of axioms called ZFC together with a few logical principles. This does not completely settle the question of truth in mathematics, but it greatly focuses the issue You can still deny a mathematical theorem, but only if you reject one of the elementary ZFC axioms or find a logical error in the proof 1.1 Our Axioms For our purposes, the ZFC axioms are too primitive- by one reckoning, proving that 2+2=4 requires more than 20,000 steps! So instead of starting with ZFC, were going to take a huge set of axioms as our foundation: we'll accept all familiar facts from high school math This will give us a quick launch, but you will find this imprecise specification of the axioms troubling at times. For example, in the midst of a proof, you may find yourself wondering, " Must I prove this little fact or can I take it as an axiom? Feel free to ask for guidance, but really there is no absolute answer. Just be upfront about what you're assuming, and don' t try to evade homework and exam problems by declaring everything an axiom.Proofs 3 1 The Axiomatic Method The standard procedure for establishing truth in mathematics was invented by Euclid, a mathematician working in Alexadria, Egypt around 300 BC. His idea was to begin with five assumptions about geometry, which seemed undeniable based on direct experience. (For example, “There is a straight line segment between every pair of points.) Proposi￾tions like these that are simply accepted as true are called axioms. Starting from these axioms, Euclid established the truth of many additional proposi￾tions by providing “proofs”. A proof is a sequence of logical deductions from axioms and previously­proved statements that concludes with the proposition in question. You probably wrote many proofs in high school geometry class, and you’ll see a lot more in this course. There are several common terms for a proposition that has been proved. The different terms hint at the role of the proposition within a larger body of work. • Important propositions are called theorems. • A lemma is a preliminary proposition useful for proving later propositions. • A corollary is an afterthought, a proposition that follows in just a few logical steps from a theorem. The definitions are not precise. In fact, sometimes a good lemma turns out to be far more important than the theorem it was originally used to prove. Euclid’s axiom­and­proof approach, now called the axiomatic method, is the founda￾tion for mathematics today. Amazingly, essentially all mathematics can be derived from just a handful of axioms called ZFC together with a few logical principles. This does not completely settle the question of truth in mathematics, but it greatly focuses the issue. You can still deny a mathematical theorem, but only if you reject one of the elementary ZFC axioms or find a logical error in the proof. 1.1 Our Axioms For our purposes, the ZFC axioms are too primitive— by one reckoning, proving that 2 + 2 = 4 requires more than 20,000 steps! So instead of starting with ZFC, we’re going to take a huge set of axioms as our foundation: we’ll accept all familiar facts from high school math! This will give us a quick launch, but you will find this imprecise specification of the axioms troubling at times . For example, in the midst of a proof, you may find yourself wondering, “Must I prove this little fact or can I take it as an axiom?” Feel free to ask for guidance, but really there is no absolute answer. Just be upfront about what you’re assuming, and don’t try to evade homework and exam problems by declaring everything an axiom!