TITLE: Can someone explain definition of number 1?
QUESTION [8 upvotes]: I've found the next piece of text:

As an example of the second failing, Poincaré recalled the definition of the number 1 offered by another of the logicists, Burali-Forti:
  $$1 = \imath\,T'\{Ko\cap(u,h)\in(u\in O\,n\,e)\}$$
  This is written in a notation devised by Peano, and indeed in what Poincaré called the "Peanian" language.

And what is the meaning of this equation?

REPLY [2 votes]: The original paper by Cesare Burali-Forti explains this formula as follows:

we say that one is the ordinal type of the ordinal class $(u, h)$ such that $u$ contains only one item.

Here is the Italian original:

Poniamo $$1 = \overline{\imath\,}\,\mathrm{T}`\{\mathrm{Ko}\cap\overline{(u,h)}\mathop{\varepsilon}(u\mathop{\varepsilon}\mathrm{Un})\}$$ cioè diciamo uno il tipo d'ordine delle classi ordinate $(u, h)$ tali che $u$ contiene un solo elemento.

Here the paper uses the older Peano notation $\varepsilon$ for set inclusion that was obsoleted in Russell's Principia. $\textrm{Ko}$ is the class of ordinals, $\mathrm{Un}$ is the class of sets that contain a single element (it can be defined for instance using equality, but this is not discussed in the paper), and $\mathrm{T}`(u,h)$ is the function that maps an equivalence class $(u,h)$ to its ordinal type.
The point of this definition is to link natural numbers and ordinal types, namely stating that natural numbers can be identified with the ordinal types of finite ordinals.
Here is the screenshot of the formula as it appears in the paper called Una Questione sui Numeri Transfiniti that was published in Rendiconti del Circolo Matematico di Palermo in 1897.