A few questions on nonstandard analysis I know that nonstandard analysis is analysis plus the existence of infinitesimal numbers. Does it mean that nonstandard analysis is the same theory as $ZF+\exists$infinitesimal numbers? 

From what I read about it on Wikipedia there seem to be a few different approaches to it. There is "Robinson's method" which is "based on studying models". There is "Nelson's method" which is "Internal set theory" which is an extension of ZF. Are they the same? Or are there theorems that are true in one but not in the other?

If IST is an extension of ZF how much stronger than ZF is the "theory of nonstandard analysis"?
$\mathsf{IST}$ is a conservative extension of $\mathsf{ZFC}$: any internal formula that can be proved within internal set theory can be proved in $\mathsf{ZFC}$. This is proved in the Appendix to Edward Nelson’s original $1977$ paper in the Bulletin of the American Mathematical Society, ‘Internal Set Theory: A New Approach to Nonstandard Analysis’.
To understand the theory nonconstrutivly of Robinson, it's sufficient to use the model theory, in this link: https://hal.inria.fr/hal-01248379/ , we can find a new approach to nonstandard analysis without using t
he ultrafiltries.