# [FOM] constructible sets

Max Weiss 30f0fn at gmail.com
Thu Feb 25 17:56:55 EST 2010

[[[Dear Moderator: here is a slight improvement of the previous
message, replacing Tc(x) with Tc({x}).]]]

Assume that every subset of every constructible set is constructible.

Let x be a set and let y be the set of nonconstructible elements of
the transitive closure of {x}.  Suppose y is nonempty, and let z be an
element of y that is of least rank.  Since the transitive closure of z
contains no nonconstructible sets, therefore z is a subset of L_\alpha
for some \alpha.  But L_\alpha is a constructible set, so by the
hypothesis it follows that z is constructible.  This is a
contradiction, and so the transitive closure of {x} contains no
nonconstructible elements.  Hence x is constructible.

= = = = = =
Max M Weiss
Doctoral student
Dep't of philosophy
U. British Columbia

On 25 February 2010 10:16, <jbell at uwo.ca> wrote:
>
> Can someone answer the following question: does V = L follow from the
> assertion that every subset of every constructible set is constructible?
> Maybe I'm missing something obvious!
>
> John Bell
>
>
>
> Professor John L. Bell
> Department of Philosophy
> University of Western Ontario
> London, Ontario N6A 3K7
>
> http://publish.uwo.ca/%7Ejbell/
>
>
>
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