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Post Closed as "Duplicate" by Nate Eldredge, Asaf Karagila♦, Johannes Hahn, Joonas Ilmavirta, Alex Degtyarev
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Logica
Logica
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We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).

I was wondering, WITHOUT AC, whether or not there exists any finitely additive measure defined over $\mathcal{P}(\mathbb{N})$? Is there a result showing that ZF + {nonexistence of finitely additive measure over $\mathcal{P}(\mathbb{N})$} is consistent? Thanks!

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).

I was wondering, WITHOUT AC, whether or not there exists any finitely additive measure defined over $\mathcal{P}(\mathbb{N})$? Is there a result showing that ZF + {nonexistence of finitely additive measure over $\mathcal{P}(\mathbb{N})$} is consistent?

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).

I was wondering, WITHOUT AC, whether or not there exists any finitely additive measure defined over $\mathcal{P}(\mathbb{N})$? Is there a result showing that ZF + {nonexistence of finitely additive measure over $\mathcal{P}(\mathbb{N})$} is consistent? Thanks!

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Logica
Logica
  • 19
  • 2

Finitely additive measure over integers

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).

I was wondering, WITHOUT AC, whether or not there exists any finitely additive measure defined over $\mathcal{P}(\mathbb{N})$? Is there a result showing that ZF + {nonexistence of finitely additive measure over $\mathcal{P}(\mathbb{N})$} is consistent?