# Burali-Forti Paradox

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## Paradox

The ordinal of the set of all ordinals must be larger than any of the ordinals in that set.

Hence this would be an ordinal which is not contained in that set.

That is, it is an ordinal which is not contained in the set of all ordinals.

Hence the set of all ordinals is not an allowable set.

## Refutation

This paradox is an antinomy caused by the incorrect assumption that one can create a set of all ordinals.

This is formally expressed in Existence of Set of Ordinals leads to Contradiction.

$\blacksquare$

## Also known as

Sometimes presented with an apostrophe: **Burali-Forti's Paradox**

## Also see

- Russell's Paradox, another paradox in naive set theory.

## Source of Name

This entry was named for Cesare Burali-Forti.

## Historical Note

The **Burali-Forti Paradox** was first stated by Cesare Burali-Forti in $1897$.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Burali-Forti paradox** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Burali-Forti's paradox** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Burali-Forti's paradox**

*Proof needs to be investigated.*