Table of contents |

2 Burali Forti paradox 3 Endnotes 4 References 5 See also 6 External links |

- A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a sequence. Now I envisage the system of all numbers and denote it
*Omega*. The system*Omega*in its natural ordering according to magnitude is a "sequence". Now let us adjoin 0 as an additional element to this sequence, and certainly if we set this 0 in the first position then*Omega** is still a sequence ... of which one can readily convince oneself that every number occurring in it is the [ordinal number] of the sequence of all its preceding elements. Now*Omega** (and therefore also*Omega*) cannot be a consistent multiplicity. For if*Omega** were consistent, then as a well-ordered set, a number D [delta] would belong to it which would be greater than all numbers of the system*Omega*; the number D, however, also belongs to the system*Omega*, because it comprises all numbers. Thus D would be greater than D, which is a contradiction. Thus the system*Omega*of all ordinal numbers is an inconsistent,**absolutely infinite**multiplicity."

This seems paradoxical, and is closely related Cesar Burali Forti's "paradox" that there can be no greatest ordinal number. There is a quick fix in Zermelo's system by his Axiom of Separation, which stipulates that sets cannot be independently defined by any arbitrary logically definable notion, but must be separated as subsets of sets already "given". This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers".

But it is a philosophical problem. It is a problem for the view that a set of individuals must exist, so long as the individuals exist. Moreover, Zermelo's fix commits us to rather mysterious objects called "proper classes". The expression "x is a set" is the name of such a class, what sort of object is it? So is the object named by "x is a thing". Is it a thing or not?

As A.W. Moore notes, there can be no end to the process of set formation, and thus no such thing as the *totality of all sets*, or the *set hierarchy*. Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.

- Ivor Grattan-Guinness has shown that this "letter" is really an amalgam by Cantor's editor Ernst Zermelo of several letters written at different times (I. Grattan-Guiness, "The rediscovery of the Cantor-Dedekind Correspondence",
*Jahresbericht der deutschen Mathematik-Vereinigung*76, 104-139

- [1] Rudy Rucker,
*Infinity and the Mind*, Princeton University Press, 1995. - [2] Ruckerbook
*Mind Tools* - [3] Heijenoort 1967
- [4] Moore, A.W.
*The Infinite*, New York, Routledge, 1990 - [5] Moore, A.W. "Set Theory, Skolem's Paradox and the Tractatus",
*Analysis*1985, 45 - [6] G. Cantor, 1932.
*Gesammelte Abhandlungen mathematischen und philosophischen Inhalts.*E. Zermelo, Ed. Berlin: Springer; reprinted Hildesheim: Olms, 1962; Berlin/Heidelberg/New York: Springer, 1980.

- Gödel's ontological proof
- Infinity
- Reflection principle
- The Ultimate
- Paradoxes
- Cantor's paradox
- Hilbert's paradox

- Class (set theory)

- Theism and Mathematical Realism by John Byl
- See Hilbert's Paradox for a detailed technical and historical discussion (note: in Adobe PDF format)