ece190_rec4 - Induction: Equivalence between forms of...

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Induction: Equivalence between forms of induction Periklis A. Papakonstantinou ? University of Toronto In this handout we show equivalence between the Well-Ordering Principle of natural numbers and the Principle of Induction. Admitting the obvious to be true Most people believe that mathematical induction is a proof technique. This is true. What people call as a “proof technique”? A proof technique could be something as simple as a logical rule of inference; e.g. modus ponens. It could also be the case that we have a general enough theorem that creates a “framework” into which we can easily work to get results (=proofs of other theorems). In the context we discuss it here, induction is none of these things. Taking a deeper look into our system of reasoning we realize that induction is an axiom 1 . In formal Logic most of the things we encounter in our reasoning system get a precise (mathematical/technical) meaning. For example, we can formalize the notion of the rea- soning system itself and e±ectively the notion of a proof. Unfortunately, we cannot get into this right now and we will stick only with the intuitive facts. It may seem that this way we introduce a bit of ambiguity but it really isn’t that bad. Note that before the 19th century people could still understand what an axiom and a proof is, even though they were not aware about the logic systems formulations (in the form we have it today). Intuitively, a system of reasoning is a very small set of logical rules (e.g. modus ponens) and a set of axioms. The axioms 2 are roughly a bunch of statements that we cannot prove to be true but rather we accept them as true statements. Those who study discrete mathematics they are in constant use of such a reasoning system all the time. The main theme in this system is the natural numbers and in particular finite sets of natural numbers. So one question that may pop-up is what is a natural number. And when we say a natural numbers we also refer to their basic properties. By basic properties we mean the elementary/basic concepts that we use in order to infer more complicated theorems. Agreeing upon a small number of axioms enable us to describe what kind of creatures are the natural numbers. In 1889 G. Peano gave such a satisfactory description by introducing five simple and axioms that formalize the intuition about natural numbers. All but the fifth one is straightforwardly ? Please submit any typos, remarks, suggestions to . 1 Read the following and then promptly forget it: technically, induction is not just one axiom but rather an infinite set of axioms, one for each formula expressing P ( · ). 2 The axioms we mention here are known as “non-logical axioms”.
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2 Periklis Papakonstantinou (ECE 190, Fall 2006) accepted by almost everybody. Roughly speaking the last one is the principle of induction 3 . Let
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This note was uploaded on 04/19/2008 for the course ECE 190 taught by Professor Carter during the Fall '06 term at University of Toronto- Toronto.

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ece190_rec4 - Induction: Equivalence between forms of...

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