MIT6_042JF10_lec02

MIT6_042JF10_lec02 - "mcs-ftl 2010/9/8 0:40 page 23#29 2...

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2 “mcs-ftl” — 2010/9/8 — 0:40 — page 23 — #29 Patterns of Proof 2.1 The Axiomatic Method The standard procedure for establishing truth in mathematics was invented by Eu- clid, a mathematician working in Alexandria, Egypt around 300 BC. His idea was to begin with five assumptions about geometry, which seemed undeniable based on direct experience. For example, one of the assumptions was “There is a straight line segment between every pair of points.” Propositions like these that are simply accepted as true are called axioms . Starting from these axioms, Euclid established the truth of many additional propo- sitions 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. ±² Important propositions are called theorems . ±² A lemma is a preliminary proposition useful for proving later propositions. ±² A corollary is a proposition that follows in just a few logical steps from a lemma or 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 axiom-and-proof approach, now called the axiomatic method , is the foundation for mathematics today. In fact, just a handful of axioms, collectively called Zermelo-Frankel Set Theory with Choice ( ZFC ), together with a few logical deduction rules, appear to be sufficient to derive essentially all of mathematics. 2.1.1 Our Axioms The ZFC axioms are important in studying and justifying the foundations of math- ematics, but for practical purposes, they are much too primitive. Proving theorems in ZFC is a little like writing programs in byte code instead of a full-ﬂedged pro- gramming language—by one reckoning, a formal proof in ZFC that 2 C 2 D 4 requires more than 20,000 steps! So instead of starting with ZFC, we’re going to

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24 “mcs-ftl” — 2010/9/8 — 0:40 — page 24 — #30 Chapter 2 Patterns of Proof 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 may 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 up front about what you’re assuming, and don’t try to evade homework and exam problems by declaring everything an axiom!
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