# Russell's Paradox/Historical Note

## Historical Note on Russell's Paradox

**Russell's Paradox** was devised by Bertrand Russell in $1901$.

He communicated it in a letter to Gottlob Frege on $16$ June $1902$, and apparently also to Giuseppe Peano some time earlier.

It was first published in 1903: Bertrand Russell: *The Principles of Mathematics*.

This paradox is one of a series of antinomies arising from the inherent contradiction in allowing unrestricted use of the comprehension principle. In this case, it is being used to obtain a set which contains itself if and only if it does not contain itself.

A desire to avoid such antinomies was a motivation towards the development of various systems of axiomatic set theory.

The best-known system of axiomatic set theory, **ZFC**, includes only (relatively) restrictive methods of generating new sets by using properties. In particular:

- The axiom of specification allows us to use properties to define only subsets of sets which have already been demonstrated as being allowed to exist.
- The axiom of replacement uses only those properties which define functions.

These restrictions both make the above argument invalid in ZFC, since the justification for the existence of the set $R$ is removed.

Some authors, for example 1965: J.A. Green: *Sets and Groups*, sidestep this issue:

*This logical impasse can be avoided by restricting the notion of set, so that 'very large' collections ... are not counted as sets. However this is done at some cost in simplicity, and in this book we shall do no more than keep to sets which appear to be harmless, and hope that paradoxes will not appear.*

Again, from 1971: Allan Clark: *Elements of Abstract Algebra*:

*It is well known that an informal point of view in the theory of sets leads to contradictions. These difficulties all arise in operations with very large sets. We shall never need to deal with any sets large enough to cause trouble in this way, and, consequently, we may put aside all such worries.*

1975: T.S. Blyth: *Set Theory and Abstract Algebra* puts it as:

*This is not a serious problem since fortunately most sets encountered in mathematics are of a harmless nature ... It is very fortunate that most of the properties dealt with in mathematics (and indeed all the properties we will deal with) are "set forming" in the sense that there is a set whose elements satisfy the property in question.*

1982: P.M. Cohn: *Algebra Volume 1* (2nd ed.) remarks:

*The paradox is resolved by restricting the ways in which sets can be formed, so that it becomes inadmissible to consider 'the set of all those sets that are not members of themselves'. There are several ways of doing this, but they need not concern us here; they will not play a role in the rather simple set-theoretical arguments we shall meet.*

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.8$. Sets of sets - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Set Theory - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 2$. Sets of sets - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Introduction - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $6$: Theorem $1.2$