2012/12/22 by Garcia, Max
#03C10 #03C20 #03C50 #03H05 #12L10 #26A03 #26A06 #26E35 #30G06 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1212.5740
We study free filters and their maximal extensions on the set of natural numbers. We characterize the limit of a sequence of real numbers in terms of the Frechet filter, which involves only one quantifier as opposed to the three non-commuting quantifiers in the usual definition. We construct the field of real non-standard numbers and study their properties. We characterize the limit of a sequence of real numbers in terms of non-standard numbers which only requires a single quantifier as well. We are trying to make the point that the involvement of filters and/or non-standard numbers leads to a reduction in the number of quantifiers and hence, simplification, compared to the more traditional epsilon, delta-definition of limits in real analysis.