En las páginas que siguen pretendemos dar una visión panorámica y esquemática de la evolución del programa formalista que resulta de los estudios recientes de notas de curso hasta hace poco inéditas. Analizaremos primeramente ciertos elementos del programa (la preferencia por el método axiomático, el estructuralismo y el logicismo). En segundo lugar observaremos cómo, una vez el programa establecido en 1920 (aunque con cierta vaguedad), diversos finitismos con una base común fueron ensayados por Hilbert y Bernays hasta 1931, en una tentativa por definir con precisión su programa y llevarlo a buen término. El resultado es el de un complejo programa de investigación en continua evolución.

We construct a countable (hence, sigma-compact) monothetic topological group G which is not compactly generated, thus answering in the negative a question posed by Fujita and Shakhmatov. In addition, our group G is precompact and sequentially complete.

The following problem is considered: If a topological group G is the union of an increasing chain of subspaces and certain cardinal invariants of the subspaces are known, what can be said about G? We prove that if G is locally compact and every subspace in the chain has countable pseudocharacter or tightness, then G is metrizable. We also prove a similar assertion for σ-compact and totally bounded groups represented as the union of first countable subspaces, when the length of the chain is a regular cardinal greater than ω1. Finally, we show that these results are not valid in general, not even for compact spaces.

We consider the following problem: If a topological group G is the union of an increasing chain of subgroups and certain cardinal invariants of the subgroups in the chain are known, what can be said about G? We prove that if the index of boundedness of each subgroup is strictly less than λ for some infinite cardinal λ, then the index of boundedness of G is at most λ. We also prove that if both the index of boundedness and the pseudocharacter of each subgroup in the chain are at most λ and G is countably compact, then │G│≤2 λ. Finally, we show that the last assertion is not valid in general, not even for pseudocompact groups.