Russell's Paradox/Proof 1

Theorem

The Axiom of Abstraction leads to a contradiction.


Proof

Sets have elements.

Some of those elements may themselves be sets.

So, given two sets $S$ and $T$, we can ask the question:

Is $S$ an element of $T$?

The answer will either be yes or no.

In particular, given any set $S$, we can ask the question:

Is $S$ an element of $S$?

Again, the answer will either be yes or no.

Recall the definitions for a set to be:

ordinary if and only if it is not an element of itself
extraordinary if and only if it is an element of itself.


Thus, $\map P S = S \in S$ is a property on which we can use the Axiom of Abstraction to build the set $T$ of all extraordinary:

$T = \set {S: S \in S}$

which is the set of all sets which contain themselves.


Or we can apply the Axiom of Abstraction to build the set $T$ of all ordinary sets:

$R = \set {S: S \notin S}$

($R$ for Russell, of course.)


We ask the question:

Is $R$ itself an element of $R$?

There are two possible answers: yes or no.

If $R \in R$, then $R$ must satisfy the property that $R \notin R$.

So from that contradiction we know that $R \in R$ does not hold.

So the only other answer, $R \notin R$, must hold instead.

But now we see that $R$ satisfies the conditions of the property that $R \in R$.

So we can see that $R \notin R$ does not hold either.

Thus we have generated a contradiction from the Axiom of Abstraction.

$\blacksquare$


Sources

  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 2$: The Axiom of Specification
  • 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next). Sets of sets: $\S 1.8$
  • 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
  • 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Paradox
  • 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Introduction
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Russell's paradox
  • 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $6$: Theorem $1.2$
  • 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Remark $1.2.1$
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Russell's paradox
  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 8$ Russell's paradox
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Russell's paradox
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