Waiting for Gödel (2016)
&gt; There are true statements that are unprovable within the system.<p>I really don&#x27;t like this, and it verges on being flat-out incorrect: the first incompleteness theorem does <i>not</i> say this at all. It says that there are sentences (and indeed it constructs one, the so-called <i>Gödel Sentence</i>) that <i>cannot be proven or refuted</i> within the system.<p>For first-order theories it follows from the contraposition of Gödel&#x27;s <i>completeness</i> theorem that there exist (classical) models of the theory where this sentence holds and models where it doesn&#x27;t: the existence of models where the sentence does not hold means that this must be a very different meaning of &quot;true&quot; to the one used in common parlance.<p>In higher-order logics, which do not have a completeness theorem, it makes sense to talk about true statements which are unprovable: there can exist tautologies, sentences which hold <i>in all models</i> of a theory, which cannot be proven from the axioms of a higher-order theory (and you don&#x27;t need the incompleteness theorem for this); on the other hand, in a first-order theory all tautologies are provable.<p>Sentences which cannot be proven or refuted within a theory are said to be <i>logically independent</i> of the theory. Famously, the Axiom of Choice is independent of the axioms of Zermelo–Fraenkel set theory: it&#x27;s up to mathematicians to decide whether they accept the Axiom of Choice. If they do, they can work in ZF+C; if not, they can work in ZF. Neither system is &quot;more true&quot; from a purely logical perspective, so I really don&#x27;t like describing logically independent sentences as &quot;true but unprovable&quot;: it almost certainly doesn&#x27;t mean what people think it means.<p>The first incompleteness theorem could perhaps be stated better for a lay audience as:<p><pre><code>    No recursive set of axioms can capture our notion of arithmetic it its entirety.
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This is a limitation on how we can use axiom systems to represent mathematical objects: even more informally, we might say:<p><pre><code>    Truth is subjective in sufficiently complex systems.</code></pre>