Spahn continuum hypothesis (Rev #4, changes)

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For context see the topos Set?.

Definition

The Letcontinuum hypothesisEE asserts be that an there is no strict inequality of cardinal elementary numbers topos? with subobject classifier? Ω\Omega and natural-numbers object? nn.

||<α<|Ω ||\mathbb{N}|\lt \alpha\lt |\Omega^\mathbb{N}|

The continuum hypothesis asserts that there is no sequence of monomorphisms

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

nbΩ nn \hookrightarrow b\hookrightarrow \Omega^n

which are not isomorphisms.

In the classical case this statement reads: The continuum hypothesis asserts that there is no strict inequality of cardinal numbers?

||<α<|Ω ||\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?)

The topos in which the theorem holds is called Cohen topos; it is the topos of sheaves with respect to the dense topology? (also called ¬¬\neg\neg-toology) on the Cohen poset. In this topos will exist a monomorphism B2 B\hookrightarrow 2^\mathbb{N}

The cohen topos will be constructed from the topos Set? of sets. for this recall that the subobject classifier of SetSet is 2:={0,1}2:=\{0,1\}.

Definition

(Cohen topos?)

Let \mathbb{N} be the set of natural numbers; i.e. the natural-numbers object in SetSet. Let BB be a set with strictly larger cardinality |B|>|||B|\gt |\mathbb{N}|; e.g. B:=2 2 B:=2^{2^\mathbb{N}} will do because of the ‘’diagonal argument’’.

Then the Cohen poset PP is defined to be the set of morphisms

p:F p2p:F_p\to 2

where F pB×F_p\subseteq B\times \mathbb{N} is a subset the order relation on PP is defined by

qpiffF qF pandq| F p=pq\le p\; iff\; F_q\supseteq F_p\;and\;q|_{F_p}=p

where the rightest condition means that qq restricted to F pF_p shall coincide with pp.

We think of PP as a sequence of approximations to the function F:B×F:B\times\mathbb{N} being the transpose? of the putative monomorphism

f:B2 f:B\to 2^\mathbb{N}

and the smaller elements considered as the better approximations. The very rough intuition is that pqp\to q\to \dots (if ppp\ge p\ge \dots) forms a codircted diagram? of monomorphism with domains of increasing size whose colimit is ff and that by free cocompletion? (here: forming (pre)sheaves) we obtain a topos in which this colimit exists.

Lemma

The dense? Lawvere-Tierney topology? on Psh(P)Psh(P) is subcanonical. In other words: For any pPp\in P we have y(p)=hom(,p)Sh(p,¬¬)y(p)=hom(-,p)\in\Sh(p,\neg\neg)

Lemma

Let k B×:{PSet ptoB×k_{B\times\mathbb{N}}:\begin{cases}P\to Set\\p\toB\times\mathbb{N}\end{cases} denote the functor constant on B×B\times\mathbb{N}. Let

A:{PSet p{(b,n)|p(b,n)=0}B×A:\begin{cases} P\to Set \\ p\mapsto \{(b,n)|p(b,n)=0\}\subseteq B\times \mathbb{N} \end{cases}

Then we have ¬¬A=A\neg\neg A=A in Sub(k B×)Sub(k_{B\times\mathbb{N}}); i.e. AA is a closed subobject with respect to the dense topology ¬¬\neg\neg in the algebra of subobjects? of k B×k_{B\times\mathbb{N}}.

Lemma

Let Ω\Omega denote the subobject classifier? of Psh(P)Psh(P). Let Ω ¬¬\Omega_{\neg\neg} denote the subobject classifier of Sh(P,¬¬)Sh(P,\neg\neg). Recall that Ω ¬¬\Omega_{\neg\neg} is the equalizer Ω ¬¬=eq(id Ω,¬¬)\Omega_{\neg\neg}=eq(id_\Omega,\neg\neg).

The characteristic morphism? χ a\chi_a of the subobject a:Ak B×=k B×k a:A\hookrightarrow k_{B\times\mathbb{N}}=k_B\times\k_\mathbb{N} factors through some f:k B×Ω ¬¬f:k_{B\times\mathbb{N}}\to \Omega_{\neg\neg}.

Then the adjoint g:k BΩ ¬¬ k g:k_B\to \Omega_{\neg\neg}^{k_{\mathbb{N}}} of ff is a monomorphism.

Corollary

The associated-sheaf functor sends gg to a monomorphism in the Cohen topos.

References

  • 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 17:13:39 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.