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Math Help - axiom of choice

  1. #1
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    axiom of choice

    Define an injective map f: \mathbb{Z}^+ \rightarrow \{0,1\}^\omega without using the axiom of choice.

    I would define an injective function by f(n)=x, where x_i = 1 if i=n and x_i=0 if i \not= n. But how would I define the map without using the axiom of choice?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Define an injective map f: \mathbb{Z}^+ \rightarrow \{0,1\}^\omega without using the axiom of choice.

    I would define an injective function by f(n)=x, where x_i = 1 if i=n and x_i=0 if i \not= n. But how would I define the map without using the axiom of choice?
    If n>0 let n\mapsto \left< 1,\underbrace{1,1,...,1}_{n\text{ times }},0,0,0,... \right>

    If n<0 let n\mapsto \left< 0, \underbrace{1,1,...,1}_{|n|\text{ times }},0,0,0,... \right>

    If n=0 let n\mapsto \left<0,0,0,0,0,0,...\right>
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