In this note we list some relationships between forms which are provable in \(ZF^0\), but whose derivations are not readily available via the method of chained implications and non-implications:
| Number | Hypothesis | Relationship | Conclusion | Reference(s) |
|---|---|---|---|---|
| 1 | \(\) 8 + 163 + 0 | \(\leftrightarrow\) | 1 + 0 | [Brunner-1982a] |
| 2 | \(\) 67 + 164 + 0 | \(\leftrightarrow\) | 1 + 0 | [Brunner-1982a] |
| 3 | \(\) 49 + 51 + 0 | \(\leftrightarrow\) | 1 + 0 | [Morris-1969] |
| 4 | \(\) 64 + 133 + 0 | \(\leftrightarrow\) | 1 + 0 | clear |
| 5 | \(\) 88 + 133 + 0 | \(\leftrightarrow\) | 1 + 0 | [Brunner-1982a] |
| 6 | \(\) 15 + 91 + 0 | \(\leftrightarrow\) | 1 + 0 | [Brunner-1982a] |
| 7 | \(\) 8 + 191 + 193 | \(\leftrightarrow\) | 1 + 0 | [Blass-1979] |
| 8 | \(\) 191 + 192 + 0 | \(\leftrightarrow\) | 1 + 0 | [Blass-1979] |
| 9 | \(\) 30 + 126 + 0 | \(\to\) | 8 + 0 | [Keremedis-2000a] |
| 10 | \(\) 106 + 154 + 0 | \(\leftrightarrow\) | 43 + 0 | [Brunner-1984b] [Brunner-1983c] |
| 11 | \(\) 14 + 297 + 0 | \(\leftrightarrow\) | 50 + 0 | [Bell-1988] [Banaschewski-1988] |
| 12 | \(\) 14 + 299 + 0 | \(\leftrightarrow\) | 50 + 0 | [Bell-1988] |
| 13 | \(\) 14 + 300 + 0 | \(\leftrightarrow\) | 50 + 0 | [Bell-1988] |
| 14 | \(\) 14 + 301 + 0 | \(\leftrightarrow\) | 50 + 0 | [Bell-1988] |
| 15 | \(\) 14 + 302 + 0 | \(\leftrightarrow\) | 50 + 0 | [Bell-1988] |
| 16 | \(\) 30 + 118 + 0 | \(\to\) | 15 + 0 | [Krom-1986] |
| 17 | \(\) 127 + 133 + 0 | \(\to\) | 132 + 0 | [Brunner-1984b] |
| 18 | \(\) 106 + 127 + 0 | \(\to\) | 64 + 0 | [Brunner-1984b] |
| 19 | \(\) 17 + 133 + 0 | \(\to\) | 134 + 0 | [Brunner-1984b] |
| 20 | \(\) 9 + 70 + 0 | \(\to\) | 63 + 0 | clear |
| 21 | \(\) 39 + 41 + 0 | \(\to\) | 135 + 0 | [van Douwen-1980-1] |
| 22 | \(\) 128 + 130 + 0 | \(\to\) | 18 + 0 | [Brunner-1984f] |
| 23 | \(\) 10 + 163 + 0 | \(\to\) | 231 + 0 | Note 123 |
| 24 | \(\neg\) 206 + 0 + 0 | \(\to\) | 253 + 0 | [Howard-1975] |
| 25 | \(\) 431 + 131 + 0 | \(\to\) | 8 + 0 | [Keremedis-2000a] Note 132 |
| 26 | \(\) 52 + 286 + 0 | \(\leftrightarrow\) | 1 + 0 | [Rubin-Rubin-1985-4] |
| 27 | \(\) 30 + 296 + 0 | \(\to\) | 295 + 0 | [Pincus-1997] |
| 28 | \(\) 30 + 144 + 294 | \(\leftrightarrow\) | 1 + 0 | clear |
| 29 | \(\) 62 + 99 + 0 | \(\leftrightarrow\) | 14 + 0 | Note 22 |
| 30 | \(\) 163 + 10 + 0 | \(\to\) | 165 + 0 | [Brunner-1982a] |
| 31 | \(\) 62 + 67 + 0 | \(\leftrightarrow\) | 1 + 0 | clear |
| 32 | \(\) 122 + 328 + 0 | \(\leftrightarrow\) | 40 + 0 | clear |
| 33 | \(\) 327 + 328 + 0 | \(\to\) | 322 + 0 | clear |
| 34 | \(\) 122 + 330 + 0 | \(\leftrightarrow\) | 165 + 0 | clear |
| 35 | \(\) 327 + 330 + 0 | \(\to\) | 324 + 0 | clear |
| 36 | \(\) 62 + 329 + 0 | \(\leftrightarrow\) | 60 + 0 | clear |
| 37 | \(\) 333 + 334 + 0 | \(\leftrightarrow\) | 1 + 0 | [Keremedis-1996a] |
| 38 | \(\) 8 + 335 + 0 | \(\leftrightarrow\) | 1 + 0 | [Keremedis-1996b] |
| 39 | \(\) 61 + 67 + 0 | \(\leftrightarrow\) | 218 + 0 | [Bleicher-1965] |
| 40 | \(\) 336 + 0 + 0 | \(\to\) | 342 + 0 | Note 111 |
| 41 | \(\) 67 + 122 + 0 | \(\leftrightarrow\) | 1 + 0 | It's clear that [67 B] + Form 122 implies Form 1. See Levy [1962] |
| 42 | \(\) 14 + 43 + 0 | \(\leftrightarrow\) | 407 + 0 | [Bacsich-1972b] |
| 43 | \(\) 20 + 97 + 0 | \(\to\) | 359 + 0 | [Higasikawa-1995] |
| 44 | \(\) 20 + 3 + 0 | \(\to\) | 359 + 0 | [Higasikawa-1995] |
| 45 | \(\) 14 + 253 + 0 | \(\leftrightarrow\) | 1 + 0 | [Howard-1975] |
| 46 | \(\) 40 + 152 + 0 | \(\leftrightarrow\) | 1 + 0 | clear |
| 47 | \(\) 151 + 152 + 0 | \(\leftrightarrow\) | 1 + 0 | clear |
| 48 | \(\) 214 + 329 + 0 | \(\to\) | 67 + 0 | clear |
| 49 | \(\) 32 + 131 + 0 | \(\leftrightarrow\) | 8 + 0 | Note 132 |
| 50 | \(\) 432 + 131 + 0 | \(\leftrightarrow\) | 8 + 0 | [Keremedis-2000a] Note 132 |
| 51 | \(\) 433 + 131 + 0 | \(\leftrightarrow\) | 8 + 0 | [Keremedis-2000a] Note 132 |
| 52 | \(\) 434 + 131 + 0 | \(\leftrightarrow\) | 8 + 0 | [Keremedis-2000a] Note 132 |
| 53 | \(\) 131 + 350 + 0 | \(\leftrightarrow\) | 126 + 0 | [Keremedis-2000a] Note 132 |
| 54 | \(\) 435 + 131 + 0 | \(\leftrightarrow\) | 126 + 0 | [Keremedis-2000a] Note 132 |
| 55 | \(\) 10 + 106 + 0 | \(\leftrightarrow\) | 43 + 0 | [Blass-1979] |
| 56 | \(\) 70 + 344 + 0 | \(\leftrightarrow\) | 123 + 0 | [Morillon-1988] |
| 57 | \(\) 20 + 75 + 0 | \(\to\) | 3 + 0 | [Higasikawa-1995] |
| 58 | \(\) 20 + 75 + 0 | \(\to\) | 359 + 0 | [Higasikawa-1995] |
| 59 | \(\) 8 + 385 + 0 | \(\leftrightarrow\) | 1 + 0 | [Herrlich-Steprans-1997] |
| 60 | \(\) 10 + 389 + 0 | \(\leftrightarrow\) | 43 + 0 | [Keremedis-2001a] |
| 61 | \(\) 30 + 393 + 0 | \(\leftrightarrow\) | 1 + 0 | clear |
| 62 | \(\) 30 + 396 + 0 | \(\leftrightarrow\) | 67 + 0 | clear |
| 63 | \(\) 30 + 401 + 0 | \(\leftrightarrow\) | 15 + 0 | clear |
| 64 | \(\) 8 + 70 + 0 | \(\leftrightarrow\) | 8 + 406 | [Alas-1994-1] [Solovay-1971-3] |
| 65 | \(\) 8 + 386 + 0 | \(\leftrightarrow\) | 8 + 406 | [Alas-1994-1] |
| 66 | \(\) 14 + 65 + 0 | \(\to\) | 1 + 0 | [Rubin-Rubin-1985] |
| 67 | \(\) 200 + 291 + 0 | \(\to\) | 290 + 0 | clear |
| 68 | \(\) 200 + 290 + 0 | \(\to\) | 291 + 0 | clear |
| 69 | \(\) 291 + 290 + 0 | \(\to\) | 200 + 0 | clear |
| 70 | \(\) 9 + 371 + 0 | \(\to\) | 206 + 0 | [Morillon-1993-1] |
| 71 | \(\) 10 + 163 + 0 | \(\to\) | 231 + 0 | Note 123 |
| 72 | \(\) 14 + 8 + 0 | \(\to\) | 113 + 0 | [Howard-Rubin-1996] |
| 73 | \(\) 170 + 6 + 0 | \(\to\) | 34 + 0 | [Howard-Rubin-Stanley-Tachtsis-Keremedis-2001] |
| 74 | \(\neg\) 64 + 0 + 0 | \(\to\) | 371 + 0 | [Morillon-1993-1] |
| 75 | \(\) 14 + 43 + 0 | \(\leftrightarrow\) | 345 + 0 | [Morillon-Haddad-1990] |
| 76 | \(\) 62 + 57 + 0 | \(\to\) | 9 + 0 | [Sageev-1981] Note 44 |
| 77 | \(\) 14 + 106 + 0 | \(\leftrightarrow\) | 345 + 0 | [Morillon-1988] |