We first prove a version of Tietze-Urysohn's theorem for proper functions taking values in non-negative real numbers defined on $\sigma$-compact locally compact Hausdorff spaces. As its application, we an extension metrics, which states that if $X$ is space, $A$ closed subset $X$, and $d$ metric generates the same topology $A$, then there exists such $D$ $D|_{A^{2}}=d$. Moreover, retraction, ca...

?1. Historical introduction. The modern problem of infinity was first raised by Aristotle who held (at least on the popular interpretation2) that infinite sets exist potentially (i.e. one more number can always be counted, one more division can always be made in a line segment) but not actually (i.e. the numbers or divisions cannot all exist at one time). In fact, Aristotle not only held that c...

Proper consistency is defined by the properties that each player takes all opponent strategies into account (is cautious) and deems one opponent strategy to be infinitely more likely than another if the opponent prefers the one to the other (respects preferences). When there is common certain belief of proper consistency, a most preferred strategy is properly rationalizable. Any strategy used w...