The problem, in this context, with informally formulated set theories, not derived from ( and implying ) any particular axiomatic theory, is that there may be several widely differing formalized versions, that have both different sets and different rules for how new sets may be formed, that all conform to the original informal definition.
22.
He gave a particularly noteworthy lecture on probability theory ( NL Hausdorff : Capsule 21 : Fasz 64 ) in the summer semester 1923, in which he grounded this theory in measure-theoretic axiomatic theory, and this occurred ten years before A . N . Kolmogorov's " Basic concepts of probability theory " ( reprinted in full in the collected works, Volume V ).
23.
If we equip this with the topology generated by first order formulas, then it is known from A . Gregorczyk, A . Mostowski, C . Ryll-Nardzewski, " Definability of sets of models of axiomatic theories ", Bulletin of the Polish Academy of Sciences ( series Mathematics, Astronomy, Physics ), vol . 9 ( 1961 ), pp . 163 7 that the resulting space is Polish.
24.
Following an idea of Grassmann's father, "'A1 "'also defined the exterior product, also called " combinatorial product " ( in German : " �u�eres Produkt " or " kombinatorisches Produkt " ), the key operation of an algebra now called exterior algebra . ( One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined . ) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule " e p e p " = 0 by the rule " e p e p " = 1 . ( For quaternions, we have the rule " i " 2 = " j " 2 = " k " 2 = " 1 . ) For more details, see exterior algebra.
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