uncountability

Given any first-order axiomatization of set theory and any formula Ω (x) which is supposed to capture the notion of uncountability, the Löwenheim-Skolem theorems show that we can find a countable model M which satisfies our axioms.

Skolem's Paradox Bays, Timothy 2009

Given this, the Löwenheim-Skolem theorems show that the notions of countability and uncountability will in fact vary as we move from model to model.

Skolem's Paradox Bays, Timothy 2009

After all, it's still a theorem that set-theoretic notions like countability and uncountability come out relative on the algebraic conception.

Skolem's Paradox Bays, Timothy 2009

Given this algebraic conception of axiomatization, then, Skolem appeals to the Löwenheim-Skolem theorems to argue that the axioms of set theory lack the resources to pin down the notion of uncountability.

Skolem's Paradox Bays, Timothy 2009

The birth of Set Theory dates to 1873 when Georg Cantor proved the uncountability of the real line.

Set Theory Jech, Thomas 2002

Thus, as long the basic set theoretic notions are characterized simply by looking at the model theory of first-order axiomatizations of set theory, then many of these notions ” and, in particular, the notions of countability and uncountability ” will turn out to be unavoidably relative. [

Skolem's Paradox Bays, Timothy 2009

This shows that there is at least one interpretation of Ω (x) under which this formula really does capture ” at least from an extensional perspective ” the ordinary mathematical notion of uncountability.

Skolem's Paradox Bays, Timothy 2009

To summarize, then, the upshot of this discussion is this: if we take a purely algebraic approach to the axioms of set theory, then many basic set-theoretic notions ” including the notions of countability and uncountability ” will turn out to be relative.

Skolem's Paradox Bays, Timothy 2009