Definition 2.1.1: write in words what each of these statements means.
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"By property 3, ℕ itself has a minimum. We will call it 1." Explain why the minimum of ℕ must be equal to the element of ℕ that does not have a predecessor.
Section 2.2
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Equation (2.6): why do they say that we did not prove any of these statements "by themselves?"
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Equation (2.9): Say in words what this statement means, i.e. write a sentence that starts with "A is the set of ..."
Also Equation (2.9): what does the vertical bar (between "k ∈ ℕ" and "P(k) is false") mean?
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Discussion below Equation (2.10): explain why this argument holds only if we assume that "true" and "false" are mutually exclusive, i.e. that a statement is either true or false, but not both.
Where else in the discussion that follows, is it assumed that "true" and "false" are mutually exclusive?
"The subtraction property of ℕ tells us that ..." Is there a way to justify the existence of s using only the basic properties of ℕ, i.e. without invoking subtraction? Explain.
Section 2.3
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Equation (2.12): why do the authors define S this way, and why do they say "we want S to have a minimum?"
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Case 1: What is the point of introducing q since we say it is equal to p?
"Now p is still an element of S because we made this observation before we began case 1." Why does it matter that this observation was made before the discussion of case 1?
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Definition 2.3.3: can the statement "n = 2 ⋅ k − 1" be understood without invoking subtraction? If so, how?
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"Since n = 2 ⋅ k − 1, we know that n is odd." Why does this prove that 1 is odd? Why did they introduce this number k?
Case 2 of Step 1: what is m in this discussion?
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Why is the statement 2 k' = 2 k + 1 a direct consequence of 2 k = 2 k' - 1?