Cardinality in a paraconsistent and paracomplete set theory
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911576286560256 |
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| author | Oddsson, Hrafn Valtýr |
| author_facet | Oddsson, Hrafn Valtýr |
| contents | This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that ``$x\in A$'' is neither true nor false for some $x$). We carefully analyze what it means for two potentially incomplete or inconsistent sets to have ``the same size'', construct the corresponding cardinal numbers, and develop the basic theory of cardinal arithmetic. A surprising result is that the cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, our cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_07094 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cardinality in a paraconsistent and paracomplete set theory Oddsson, Hrafn Valtýr Logic 03E70 (Primary) 03B50, 03B60, 03E10 (Secondary) This paper develops a rich theory of cardinality in the paraconsistent and paracomplete set theory $\mathrm{BZFC}$, where sets can be inconsistent ($A$ such that ``$x\in A$'' is both true and false for some $x$) or incomplete ($A$ such that ``$x\in A$'' is neither true nor false for some $x$). We carefully analyze what it means for two potentially incomplete or inconsistent sets to have ``the same size'', construct the corresponding cardinal numbers, and develop the basic theory of cardinal arithmetic. A surprising result is that the cardinality of any set can be expressed as a linear combination of three fundamental cardinal numbers with classical cardinals as coefficients. In that sense, our cardinal numbers form a three-dimensional space over the usual cardinals, much like how the complex numbers form a two-dimensional space over the reals. |
| title | Cardinality in a paraconsistent and paracomplete set theory |
| topic | Logic 03E70 (Primary) 03B50, 03B60, 03E10 (Secondary) |
| url | https://arxiv.org/abs/2604.07094 |