How recent work in mathematical logic relates to the foundations of mathematics (Q1085151)

This paper is related to an article of \textit{S. MacLane} [ibid. 5, No. 4, 53--55 (1983; Zbl 1366.00035)]. The aim of the author is to explain the connections between recent work in mathematical logic and foundations, contrary to what was argued by MacLane. In order to clarify this connection, the paper contains a survey of some results of mathematical logic (from weak arithmetic to large cardinals) from the point of view of the partial ordering of consistency strengths.