Logic Problem 2

A not-so-elementary reasoning problem due to Richard Montague. (The solution of this problem requires that you understand the symbolism of modern logic.) The notion of a set or collection is a basic concept in modern mathematics. To avoid paradox, certain axioms are adopted to help us better manage our sets.

Sets may have sets as members, and those sets may again have sets as their members, and so on. An axiom of Zermelo-Fraenkel set theory says that this process does not go on forever; eventually we will hit bottom. We say that all our sets are thus grounded. That is, the process described above halts at a ground level where we can no longer continue picking a set out of a set.

The paradox of grounded sets is this: there is no set which is comprised of all the grounded sets; i.e., the set of all grounded sets does not exist.

Given the uncontroversial premise that any given object is such that there is a set which contains as an element only the given object (think of the singleton set of any given object)

Logid 1

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Logic 2