Editing Absolute infinite (section)

== Cantor's view ==
Cantor said:

{{Blockquote|The actual infinite was distinguished by three relations: first, as it is realized in the supreme [[perfection]], in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or order type. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it ''[[Transfinite number|Transfinitum]]'' and strongly contrast it with the absolute.{{#tag:ref|https://www.uni-siegen.de/fb6/phima/lehre/phima10/quellentexte/handout-phima-teil4b.pdf<br/>Translated quote from German: {{Blockquote|Es wurde das Aktual-Unendliche (A-U.) nach drei Beziehungen unterschieden: erstens, sofern es in der höchsten Vollkommenheit, im völlig unabhängigen außerweltlichen Sein, in Deo realisiert ist, wo ich es Absolut Unendliches oder kurzweg Absolutes nenne; zweitens, sofern es in der abhängigen, kreatürlichen Welt vertreten ist; drittens, sofern es als mathematische Größe, Zahl oder Ordnungstypus vom Denken in abstracto aufgefaßt werden kann. In den beiden letzten Beziehungen, wo es offenbar als beschränktes, noch weiterer Vermehrung fähiges und insofern dem Endlichen verwandtes A.-U. sich darstellt, nenne ich es ''Transfinitum'' und setze es dem Absoluten strengstens entgegen.}}[Ca-a,<ref name="Cantor.1932"/> p. 378].}}}}

While using the [[Latin language|Latin]] expression ''in Deo'' (in God), Cantor identifies [[absolute (philosophy)|absolute]] infinity with [[God]] (GA 175–176, 376, 378, 386, 399). According to Cantor, Absolute Infinity is beyond [[transcendence (philosophy)|mathematical comprehension]] and shall be interpreted in terms of [[negative theology]].<ref>{{Cite journal|author1=Gutschmidt, Rico|author2=Carl, Merlin|url=https://link.springer.com/article/10.1007/s11153-023-09897-8#citeas|title=The negative theology of absolute infinity: Cantor, mathematics, and humility|journal= International Journal for Philosophy of Religion|volume=95|pages=233-256|year=2024|publisher=[[Springer Publishing|Springer]]|doi=10.1007/s11153-023-09897-8|access-date=January 18, 2025|ISSN=0020-7047|OCLC=10146601115}} (peer-reviewed, [[Open Access]]). Also available on [https://kops.uni-konstanz.de/entities/publication/7b6361a2-2a1b-431d-b38b-9393e4439eac KOPS Universität Konstanz] website.</ref>

Cantor also mentioned the idea in his letters to [[Richard Dedekind]] (text in square brackets not present in original):{{refn|''Gesammelte Abhandlungen'',<ref name="Cantor.1932"/> Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931'', ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as [[Ivor Grattan-Guinness]] has discovered,<ref>[https://eudml.org/doc/146637 The Rediscovery of the Cantor-Dedekind Correspondence], I. Grattan-Guinness, ''Jahresbericht der Deutschen Mathematiker-Vereinigung'' '''76''' (1974/75), pp. 104–139, at p. 126 ff.</ref> this is in fact an amalgamation by Cantor's editor, [[Ernst Zermelo]], of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.|name=GesammelteAbhandlungen}}

{{quotation|
A multiplicity [he appears to mean what we now call a [[set (mathematics)|set]]] is called [[well-ordered]] if it fulfills the condition that every sub-multiplicity has a first [[element (mathematics)|element]]; such a multiplicity I call for short a "sequence".<br />
...<br />
Now I envisage the system of all [ordinal] numbers and denote it ''Ω''.<br />
...<br />
The system ''Ω'' in its natural ordering according to magnitude is a "sequence".<br />
Now let us adjoin 0 as an additional element to this sequence, and place it, obviously, in the first position; then we obtain a sequence ''{{prime|Ω}}'':<br />
0, 1, 2, 3, ... ω<sub>0</sub>, ω<sub>0</sub>+1, ..., γ, ...
<br />
of which one can readily convince oneself that every number γ occurring in it is the type [i.e., order-type] of the sequence of all its preceding elements (including 0). (The sequence ''Ω'' has this property first for ω<sub>0</sub>+1. [ω<sub>0</sub>+1 should be ω<sub>0</sub>.])<br />
<br />
Now ''{{prime|Ω}}'' (and therefore also ''Ω'') cannot be a consistent multiplicity. For if ''{{prime|Ω}}'' were consistent, then as a well-ordered set, a number ''δ'' would correspond to it which would be greater than all numbers of the system ''Ω''; the number ''δ'', however, also belongs to the system ''Ω'', because it comprises all numbers. Thus ''δ'' would be greater than ''δ'', which is a contradiction. Therefore:<br />
<br />
''The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity.''}}