uncountability love

# uncountability

## Definitions

• noun The quality of being uncountable.

## Etymologies

Un- not + count + -ability, nounal form of -able.

## Examples

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

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

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

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

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

Set Theory

• 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. [