Spahn continuum hypothesis (Rev #3, changes)

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Definition

The continuum hypothesis asserts that there is no strict inequality of cardinal numbers?

$|\mathbb{N}|\lt \alpha\lt |\mathbf{2}^\mathbb{N}|$

For context see the topos Set?.

Definition

The continuum hypothesis asserts that there is no strict inequality of cardinal numbers?

|\mathbb{N}|\lt \alpha\lt |\Omega^\mathbb{N}|

where the leftest symbol dnotes the cardinality of the natural-numbers object? $\mathbb{N}$ in Set? and the rightest symbol denotes its power object?.

Theorem

There exists a boolean topos in which the axiom of choice holds and the continuum hypothesis fails.

Definition

(Cohen topos?)

References

~~André Joyal, Ieke Moerdijk, sheaves in geometry and logic, VI.2, VI.3~~

André Joyal, Ieke Moerdijk, sheaves in geometry and logic, VI.2, VI.3
M.C. Fitting, “Intuitionistic logic, model theory and forcing” , North-Holland (1969)

Revision on June 19, 2012 at 14:24:38 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.