|May26-12, 05:07 PM||#1|
Is this a good way to explain Skolem's Paradox?
"The paradox: Let T be a standard first-order formulation of ZFC. Assume T has a model. By Skolem's Theorem, T has a countable model M. Since T ⊢ ∃A(A is uncountable), M ⊨ ∃A(A is uncountable). But how can M—i.e. a model that “sees” only countably many things in the universe—“say” some sets contain uncountably many elements? How can M account for all the “extra” members of A? It can’t. According to M, A can be at most countable as there are only countably many “things” available (in the domain of M) to be in A. So A paradoxically looks countable and uncountable."
Is there anything WRONG? UNECESSARY? MISSING?
Or, is there a simpler way to put it, so that a 10 year old could understand it?
|May26-12, 05:12 PM||#2|
I like to use colors. Countable is a property of sets in the set-theoretic universe in which we've formulated logic. It means there is a function that provides a bijection from the set to the natural numbers.
Countable is a property of sets in the theory T. It means there is a function that provides a bijection from the set to the natural numbers.
Countable is a property of sets in a model of T. It means there is a function that provides a bijection from the set to the natural numbers.
We can assume the model is regular, so that every set is a set. Countable is, of course, the interpretation of countable in the model.
Continuing with the assumption, every function between sets is also a function. But the reverse might not be true.
So, a set can be countable without being countable.
|Jun11-12, 04:44 PM||#3|
After this explanation by Hurkyl (that is rigorously correct), we may ask further questions : how is it possible that a function between given sets in the model, may exist outside the model but not inside it ? The theory gives a name to the set of all functions between given sets (say the set of functions from E to F is named FE), but this name may have different interpretations between models.
In each model this name means the set of all functions from E to F that exist inside this model, so that they are in this set whenever they are in this model; but it cannot exclude the existence of such functions ouside the model (that do not coincide with any function inside).
This sort of incompleteness is a specific character of the powerset, that does not happen for some other constructions of sets (union, image of a function, subset defined by formulas with bounded quantifiers).
I have explained this difference and other paradoxical aspects of the foundations of mathematics in my web site.
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