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It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, these interpretations are important because they establish the basic coherence of the other theory—they give us the relative consistency result, which gives us the confidence that the other theory has its own basic integrity, at least as much as the standard alternative.

Similarly, it is not circular when we construct a model of ZF+$\neg$AC by forcing over a model of ZFC, or a model of ZFC+$\neg$CH by forcing over a model of ZFC+CH. Rather these arguments show the basic coherence and relative consistency of the other theory. The theories ZF+AC, ZF+$\neg$AC, ZFC+CH, ZFC+$\neg$CH are all equiconsistent with each other, equally safe from a consistency point of view.

It seems to be the same situation in your case. At issue historically was whether the infinitesimal approach to calculus was even coherent. Robinson's nonstandard analysis and the approach you mention show various different senses in which it is.

It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, these interpretations are important because they establish the basic coherence of the other theory—they give us the relative consistency result, which gives us the confidence that the other theory has its own basic integrity, at least as much as the standard alternative.

Similarly, it is not circular when we construct a model of ZF+$\neg$AC by forcing over a model of ZFC, or a model of ZFC+$\neg$CH by forcing over a model of ZFC+CH. Rather these arguments show the basic coherence and relative consistency of the other theory. The theories ZF+AC, ZF+$\neg$AC, ZFC+CH, ZFC+$\neg$CH are all equiconsistent with each other, equally safe from a consistency point of view.

It seems to be the same situation in your case. At issue historically was the worry whether the infinitesimal approach to calculus might simply be incoherent. Robinson's nonstandard analysis and the approach you mention show various (different) senses in which it is coherent.

It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, these interpretations are important because they establish the basic coherence of the other theory—they give us the relative consistency result.

Similarly, it is not circular when we construct a model of ZF+$\neg$AC by forcing over a model of ZFC, or a model of ZFC+$\neg$CH by forcing over a model of ZFC+CH. Rather these arguments show the basic coherence and relative consistency of the other theory. The theory ZF is equiconsistent with ZFC, with ZFC+CH, and with ZFC+$\neg$CH.

It seems to be the same situation in your case. At issue historically was whether the infinitesimal approach to calculus was even coherent. Robinson's nonstandard analysis and the approach you mention show various different senses in which it is.

It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, these interpretations are important because they establish the basic coherence of the other theory—they give us the relative consistency result, which gives us the confidence that the other theory has its own basic integrity, at least as much as the standard alternative.

Similarly, it is not circular when we construct a model of ZF+$\neg$AC by forcing over a model of ZFC, or a model of ZFC+$\neg$CH by forcing over a model of ZFC+CH. Rather these arguments show the basic coherence and relative consistency of the other theory. The theories ZF+AC, ZF+$\neg$AC, ZFC+CH, ZFC+$\neg$CH are all equiconsistent with each other, equally safe from a consistency point of view.

It seems to be the same situation in your case. At issue historically was whether the infinitesimal approach to calculus was even coherent. Robinson's nonstandard analysis and the approach you mention show various different senses in which it is.

It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, we are establishing the basic coherence of the other theory—it is a relative consistency result.

Similarly, it is not circular when we construct a model of ZF+$\neg$AC by forcing over a model of ZFC, or a model of ZFC+$\neg$CH by forcing over a model of ZFC+CH. Rather these arguments show the basic coherence and relative consistency of the other theory.

It seems to be the same situation in your case. At issue historically was whether the infinitesimal approach to calculus was even coherent. Robinson's nonstandard analysis and the approach you mention show various senses in which it is.

It is not circular for us to prove the consistency of noneuclidean geometry by providing an interpretation of noneuclidean geometry within euclidean geometry, such as with the Poincaré disk model of hyperbolic geometry. Rather, these interpretations are important because they establish the basic coherence of the other theory—they give us the relative consistency result.

Similarly, it is not circular when we construct a model of ZF+$\neg$AC by forcing over a model of ZFC, or a model of ZFC+$\neg$CH by forcing over a model of ZFC+CH. Rather these arguments show the basic coherence and relative consistency of the other theory. The theory ZF is equiconsistent with ZFC, with ZFC+CH, and with ZFC+$\neg$CH.

It seems to be the same situation in your case. At issue historically was whether the infinitesimal approach to calculus was even coherent. Robinson's nonstandard analysis and the approach you mention show various different senses in which it is.