contradiction involved in postulating a new , non-finite , number which is to be the first after all the finite numbers .” 137
On the other hand , Cantor , as a philosopher , argues for an actual intelligible existence of the transfinite numbers on account of their perfectly determined place in our mind , their clear distinction from other concepts , and their definite relations that manifest their properties . The transfinite numbers “ thus modify , in a definite way , the substance of our mind .” 138 Cantor distinguishes between three types of an infinite : the absolute infinite – reserved for God alone , the transfinite as existing “ in the abstract ,” ( the transfinite numbers ), and the transfinite as existing in the real world . 139 “ Cantor is one of very few people – perhaps the only one – to have asserted the reality of the actual infinite in all three of these forms .” 140 The potential infinite is not listed among Cantor ’ s types of an infinite because “ Cantor holds that the potential infinite is not infinite at all , since it is always finite at any given stage ; for this reason he sometimes calls it the ‘ false infinite .’” 141
Beside mathematical and philosophical defense of the transfinite numbers , Cantor provided also theological arguments for his theory , which was inspired by his correspondence with Constantine Gutberlet . For Gutberlet , the existence of an actual infinite can be established by arguments from God ’ s omniscience . For an example , God knows all decimal places of π ; hence , in the mind of God , the complete decimal extension of π exists as an actual infinity . For God there is only an actual infinite . Claiming that God knows a potential infinite leads to a
137
Cantor , Contributions , 54 .
138
Ibid , 67 .
139
Ibid , 409 .
140
Ibid .
141
Ibid , 408 . Page 56 of 62