On first-order arithmetic truth
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arXiv
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| Format: | Preprint |
| Published: |
2011
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| _version_ | 1866910209940652032 |
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| author | Boyce, Stephen |
| author_facet | Boyce, Stephen |
| contents | The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class of PA theorems is semantically defined by a class sign of PA itself, (E x_2) Pf(x_2, x_1), in the following sense: with b' the PA numeral for the number b, (E x_2) Pf(x_2, b') is true under the standard interpretation if and only if b is the Godel number of a PA theorem. From this however it is easily established, by a modification of Godel's proof, that the class of PA theorems, and hence the standard interpretation of PA itself, is not well defined after all. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1105_0321 |
| institution | arXiv |
| publishDate | 2011 |
| record_format | arxiv |
| spellingShingle | On first-order arithmetic truth Boyce, Stephen General Mathematics 03C62, 03B25, 03F40, 03F03 The standard interpretation of first-order number theory (PA), according to the generally accepted view, associates well-defined set-theoretic entities with each and every well-formed formula of this system. But this implies that the class of PA theorems is semantically defined by a class sign of PA itself, (E x_2) Pf(x_2, x_1), in the following sense: with b' the PA numeral for the number b, (E x_2) Pf(x_2, b') is true under the standard interpretation if and only if b is the Godel number of a PA theorem. From this however it is easily established, by a modification of Godel's proof, that the class of PA theorems, and hence the standard interpretation of PA itself, is not well defined after all. |
| title | On first-order arithmetic truth |
| topic | General Mathematics 03C62, 03B25, 03F40, 03F03 |
| url | https://arxiv.org/abs/1105.0321 |