Definitions
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 noun The
quality of beinguncountable .
Etymologies
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Examples

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

Given this, the LÃ¶wenheimSkolem 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 settheoretic notions like countability and uncountability come out relative on the algebraic conception.

Given this algebraic conception of axiomatization, then, Skolem appeals to the LÃ¶wenheimSkolem 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.

Thus, as long the basic set theoretic notions are characterized simply by looking at the model theory of firstorder 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. [

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

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 settheoretic notions ” including the notions of countability and uncountability ” will turn out to be relative.
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