Talk:Skolem's paradox

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[edit] Error

The article states:

In mathematics, specifically model theory, Skolem's paradox is a direct result of the Löwenheim-Skolem Theorem, which states that every infinite model has an elementarily equivalent countably infinite submodel.

which is not quite right: the theorem only states this if the model is of a language with no more than countably many constants and functions. I'm not going to correct this all that soon. --- Charles Stewart 21:27, 14 Jun 2005 (UTC)

[edit] Paradox? section--poss copyvio, pov problems

The "Is it a paradox?" section copies significant amounts of text from http://uk.geocities.com/frege@btinternet.com/cantor/skolem_moore.htm . Is this asserted to be fair use?

Another problem with this section is the following passage:

But this in turn is not possible unless we endorse the error that there is a set containing all the sets we mean to talk about.

It needs to be clarified whether this is a description of Moore's (or Skolem's) position, or whether the article itself is asserting that this is an error. In the last case it would be POV and would have to be removed (or recast as someone's position). --Trovatore 20:35, 3 October 2005 (UTC)

[edit] Could P(w) be countable?

It says:

If Skolem's explanation is true, ideas such as countability and uncountability are inherently relative. Our belief that the power set of the natural numbers, P(w), as uncountable, is correct, but must be understood relative to our own current "viewpoint". From another viewpoint this set may in fact be countable.

I'm not sure if this is true. Doesn't the uncountability of P(w) follow from the axioms of ZF? -- Baarslag 21:58, 15 November 2005 (UTC)

Yes, it does. What is probably intended here is that what we think of as P(omega), though uncountable in the model we're considering, may be countable in some larger model. To be sure, the point isn't worded very clearly, and IMHO that's because it's kind of a muddled idea to start with. But even though I don't agree with it, there's no question that some otherwise sensible people subscribe to the idea, so it should probably be represented somehow. --Trovatore 22:00, 15 November 2005 (UTC)

Sorry, still not with you; if the uncountability of P(w) follows from the axioms of ZF, then this would hold in any model of ZF, so how could it be countable in some larger model then? -- Baarslag 21:57, 22 November 2005 (UTC)

The larger model would think the smaller model's version of P(omega) is countable. It would not believe its own version of P(omega) to be countable. --Trovatore 22:03, 22 November 2005 (UTC)