Posted by: Alexandre Borovik | June 25, 2008

The zero and the empty set

A discussion on the Dialogue on Infinity blog on an exciting topic: ” Is zero a anatural number?” led me to these comments about zeroes and empty sets:

Zero is not natural, even less so is the empty set.

A proof that any two empty sets are equal usually involves the use of the outrageously infinitistic Axiom of Extensionality. To many students in my Foundation Year course (and they represent countries from Afghanistan to Zambia) a proof of the uniquiness of THE empty set is a revelation — not so much mathematical as linguistic, because for many of them their native tongue has no definite article, or usage of definite articles is different from that in English. In some languages, definite articles can be used only to point a finger to a concrete object and could be best translated not as “the” but “that”. Try to say “take that empty set”.

A few words for an incidental Russian reader of this blog. A colleague of mine once explained to me that in Russian regional dialects one can still detect the remnants of definite articles of the past:

Тарелку-ту на стол-тот поставь.

In the mainstream usage, however, we have only remnants of remnants of definite articles which are used not for emphasising the logical structure of a sentence, but for adding some specific emotional colour which I cannot even define:

Желание-то у меня есть…

Ты ружье-то положь.

Они глядь – а в кассе-то пусто.



  1. It just occured to me that Bulgarian has definite “suffixed” articles,
    e.g. Wikipedia gives an example:
    indefinite: човек, “person”; definite: човекът, “the person”
    that’s probably shows a common root with Russian…

  2. Most likely so, I agree with you.

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