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1. Suppose we want to argue, by using the definitions and axioms from our lecture on Peano arithmetic, that if n + 1 = 2 for some n € No, then n = 1. Give a justification for each of steps in the proof: each justification is either an axiom or a definition. We start by assuming that n + 1 = 2, where n € No. (a) Therefore n + s(0) = s(s(0)), because ... (b) Therefore s(n+0) = s(s(0)), because... (c) Therefore s(n) = s(s(0)), because... (d) Therefore n = s(0), because ... (e) Therefore n = 1, because...

1. Suppose we want to argue, by using the definitions and axioms from our lecture on Peano arithmetic, that if n + 1 = 2 for some n € No, then n = 1. Give a justification for each of steps in the proof: each justification is either an axiom or a definition. We start by assuming that n + 1 = 2, where n € No. (a) Therefore n + s(0) = s(s(0)), because ... (b) Therefore s(n+0) = s(s(0)), because... (c) Therefore s(n) = s(s(0)), because... (d) Therefore n = s(0), because ... (e) Therefore n = 1, because...

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Suppose we want to argue, by using the definitions and axioms from our lecture on Peano arithmetic, that if \( n + 1 = 2 \) for some \( n \in \mathbb{N}_0 \), then \( n = 1 \). Give a justification for each of steps in the proof: each justification is either an axiom or a definition. We start by assuming that \( n + 1 = 2 \), where \( n \in \mathbb{N}_0 \). (a) Therefore \( n + s(0) = s(s(0)) \), because . . . (b) Therefore \( s(n + 0) = s(s(0)) \), because . . . (c) Therefore \( s(n) = s(s(0)) \), because . . . (d) Therefore \( n = s(0) \), because . . . (e) Therefore \( n = 1 \), because . . .
Transcribed Image Text:1. Suppose we want to argue, by using the definitions and axioms from our lecture on Peano arithmetic, that if \( n + 1 = 2 \) for some \( n \in \mathbb{N}_0 \), then \( n = 1 \). Give a justification for each of steps in the proof: each justification is either an axiom or a definition. We start by assuming that \( n + 1 = 2 \), where \( n \in \mathbb{N}_0 \). (a) Therefore \( n + s(0) = s(s(0)) \), because . . . (b) Therefore \( s(n + 0) = s(s(0)) \), because . . . (c) Therefore \( s(n) = s(s(0)) \), because . . . (d) Therefore \( n = s(0) \), because . . . (e) Therefore \( n = 1 \), because . . .
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