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I have two questions:

  • How do you read the following expression in words: (here is $A\subseteq \mathbb{N}$)

$\forall k\in \mathbb{N}(k\in A\implies k+1\in A)$?

What I would translate it as is: for all $k\in \mathbb{N}$, we have $k+1\in A$ whenever $k\in A$. Not sure if it makes sense. Are there alternate translations?

  • How do you prove a statement using the expression above?

What I would do is: Let $k\in \mathbb{N}$ be given. Assume $k\in A$. Then, I use this assumption to prove that $k+1\in A$. Is this correct? I feel like if first two sentences "Let $k\in \mathbb{N}$ be given. Assume $k\in A$." are mentioned in the same line, it seems a bit superfluous. Can I say: "Let $k\in \mathbb{N}$ be given with $k\in A$", or simply "Let $k\in A$ be given", where I omit saying "let $k\in \mathbb{N}$ be given" in the beginning?

Mr.MathDoctor
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1 Answers1

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Your interpretation of this part of the induction axiom is true. But note that this is not the whole induction axiom. The induction axiom says that if the mentioned statement is true for some set A then A is equal to the set of natural numbers. Thus if you have a set A containing zero and you can say that if k is in A then k+1 is also in A then A is equal to N.

amir homayoun Nejah
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  • I know that I haven't included the whole part of induction of axiom. What I am asking for is: how to read the particular point - that is written in propositional logic - into words, and how to prove using that point. What you are suggesting is omitting "$\forall k\in \mathbb{N}$" in the second point. Is it because that they mean the same thing, since $A\subseteq \mathbb{N}$? – Mr.MathDoctor Dec 26 '21 at 05:10
  • I am sorry if I didn't understand your question. The universal quantifier you are asking about just means we are talking about natural numbers. If I have understood your question correctly then yes you can omit it for the exact reason you mentioned. – amir homayoun Nejah Dec 26 '21 at 16:38