• Wayfarer
    22.3k
    The overwhelmingly vast majority of true statements about the natural numbers cannot be expressed in languageTarskian

    Doesn’t two plus two equals four qualify? It’s a true statement about natural numbers isn’t it?
  • Tarskian
    658
    How’s that?
    So they are not well-formed? Something is amiss.
    Banno

    Doesn’t two plus two equals four qualify? It’s a true statement about natural numbers isn’t it?Wayfarer

    You gave an example of a statement about the natural numbers that can be expressed in language. That is an exception and not the rule.

    Take for example {5, 10, 71}. This is a subset of the natural numbers. So, the following statement is a truth about the natural numbers:

    The set {5, 10, 71} is a subset of the natural numbers.

    This statement cannot be expressed in language about the overwhelmingly vast majority of the subsets of the natural numbers, because there are uncountably infinite many such subsets while language itself is countably infinite.

    Hence, the overwhelmingly vast majority of the truths about the natural numbers are ineffable.

    Note: you do not need set-theoretical language to express the notion of "finite subset" in arithmetic, given the bi-interpretability of arithmetic theory (PA) with bounded set theory (ZF-inf). You can express it entirely in the language of arithmetic itself: "On Interpretations of Arithmetic and Set Theory" by Richard Kaye, Tin Lok Wong, https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-48/issue-4/On-Interpretations-of-Arithmetic-and-Set-Theory/10.1305/ndjfl/1193667707.full
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