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Topic: My set theory
Replies: 6   Last Post: Apr 7, 2014 7:32 AM

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Aatu Koskensilta

Posts: 2,639
Registered: 6/28/08
Re: My set theory
Posted: Apr 7, 2014 7:32 AM
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Virgil <> writes:

> In article <>,
> Aatu Koskensilta <> wrote:

>> Peter Percival <> writes:

>> > I think that in a theory of sets without the axiom of choice there can
>> > be non-comparable cardinalities.

>> Sure. But even in such theories the inequality |N| < |R| will hold,
>> witnessed by the injection taking the natural n to the real n.

> That would certainly justify |N| <= |R|, but why would it necessitate
> strict inequality?

By the uncountability of the reals. In any case, this more or less
trivial injection -- the identity map, morally speaking -- suffices to
establish comparability.

Aatu Koskensilta (

"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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