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It is not the case that Arguing for the consistency of a set of axioms by observing that the intended structure itself satisfies those axioms begs the question
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Reasons For
2 perspectives
Reason for 1 of 2
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1.
Gödel's own practice shows that semantic consistency arguments via intended models have independent epistemic value even absent formal proof.
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2.
The charge of circularity conflates ontological presupposition with logical circularity: assuming M exists is not the same inference-step as concluding θ_M is consistent.
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3.
Hilbert's distinction between contentual and formal reasoning permits intended-model arguments as pre-formal evidence that guides, rather than replaces, rigorous consistency proofs.
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Reason for 2 of 2
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1.
Parsons and Shapiro argue that the iterative conception of sets provides independent, non-circular grounds for accepting the cumulative hierarchy V before any axiomatization is formulated.
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2.
If the intended structure M is apprehended through a conceptually prior informal understanding, invoking M is not question-begging but rather an appeal to the evidential basis that motivated the axioms in the first place.
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Reasons Against
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Reason against
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1.
The trivial approach to showing θ_M is consistent is to note that the structure M satisfies θ_M
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2.
This is circular because the existence of M is precisely what is at issue
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