In the historical development of set theory, it was mentioned that Russell demonstrated, the following classical definition is not sufficient since it leads to paradoxical constructs. If you are a beginning student of the set theory, the classical definition of Georg Cantor (1845 - 1918) is a good starting point, because it is highly intuitive.
(Original, naive set definition of Georg Cantor (1895))1
A set is a combination of well-distinguishable, mathematical objects. Let \(X\) be a set. * If an object \(x\) belongs to the set \(X\), it is called ist element and written as \(x\in X\). * We write \(x\notin X\), if \(x\) is not an element of the set \(X\). * If $X$ has no elements, we call $X$ empty, and write $X=\emptyset,$ otherwise non-empty and write $X\neq\emptyset.$
Explanations: 1
Axioms: 1 2
Chapters: 3
Corollaries: 4 5 6
Definitions: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99
Examples: 100 101
Explanations: 102 103 104 105 106 107 108 109 110
Lemmas: 111 112 113 114 115 116
Motivations: 117 118
Parts: 119 120 121 122 123 124 125
Problems: 126 127
Proofs: 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164
Propositions: 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192
Theorems: 193 194 195
Nowadays, we use the Zermelo-Fraenkel axioms (ZFA) to define sets. ↩
