This is an argument I thought up after a class on combinatrical properties of the model [itex]\textbf{L}[/itex]. Our course is about set theory, not logic, so this paradox desn't seem relevant in its context. Can you help me figure out where I got it wrong?(adsbygoogle = window.adsbygoogle || []).push({});

The constructible heirarchy of sets is a series [itex]L_{\alpha}[/itex] that is defined for all ordinal numbers [itex]\alpha[/itex]. The important properties for my argument are:

- [itex]L_{\alpha}[/itex] is transitive for every [itex]\alpha[/itex]
- If [itex]\alpha < \beta[/itex], then [itex]L_{\alpha}\subset L_{\beta}[/itex]
- The transitive collapse (aka Montowski collapse) of every elementary submodel [itex]M \prec L_{\alpha}[/itex] is [itex]L_{\beta}[/itex] for some [itex]\beta[/itex]
- [itex]L_{\omega_{1}}[/itex] satisfies "every set is countable" and [itex]L_{\omega_{2}}[/itex] does not
- [itex]L_{\alpha}[/itex] is coutable iff [itex]\alpha[/itex] is countable

So, we take an countable elementary submodel (CESM) [itex]M_{1} \prec L_{\omega_{1}}[/itex], and look at its transitive collapse, [itex]L_{\alpha_{1}}[/itex] for some countable [itex]\alpha_{1}[/itex]. We then take an CESM [itex]M_{2} \prec L_{\omega_{2}}[/itex] that contains [itex]L_{\alpha_{2}}[/itex], and collapse it to get [itex]L_{\alpha_{2}}[/itex] with countable [itex]\alpha_{2}[/itex]. Then the same procedure yields a model [itex]L_{\alpha_{3}} \supset L_{\alpha_{2}}[/itex] that has an elementary embedding into [itex]L_{\omega_{1}}[/itex]. We generate an infinite series, switching between modelling [itex]L_{\omega_{1}}[/itex] and [itex]L_{\omega_{2}}[/itex].

The limit [tex]L_{\alpha}=L_{\lim_{n<\omega}\alpha_{n}}=\bigcup_{n<\omega}L_{\alpha_{n}}[/tex] is then the union of both subseries [itex]\{L_{\alpha_{n}}\}_{n=1,3,\ldots}[/itex] and [itex]\{L_{\alpha_{n}}\}_{n=2,4,\ldots}[/itex]. But a union of a series of elementary submodels is itself an elementary submodel, since it is a direct limit. In particular [itex]L_{\alpha}[/itex] should be elementary equivalent to both [itex]L_{\omega_{1}}[/itex] and [itex]L_{\omega_{2}}[/itex]. This is impossible because of property (4), namely there is a statement true in one and not in another.

Where did I go wrong in my reasoning? All kinds of tips are appreciated...

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# Paradox with elementary submodels of the constructible tower

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