uncountability love

Definitions

from Wiktionary, Creative Commons Attribution/Share-Alike License

  • n. The quality of being uncountable.

Etymologies

from Wiktionary, Creative Commons Attribution/Share-Alike License

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

Examples

  • 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

  • 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

  • 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

  • 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

  • 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. [

    Skolem's Paradox

  • 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

  • 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

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