In the standard model of cosmology (ΛCDM) the first stars, star clusters, and galaxies are expected to have formed in short bursts of star formation in low-mass dark matter halos at high redshifts (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z%5C,%20%5Csim%20%5C,6-10" data-classname="equation_inline" />). Up to this point, attempts to predict the properties and abundances of these luminous objects have made use of numerically expensive cosmological simulations. On top of being numerically expensive, these simulations often lack the required sub-parsec resolution needed to resolve the formation of compact star clusters and/or neglect possibly dominant stellar feedback processes. Motivated by this, I introduce Anaxagoras, as far as I know the most detailed analytical ab initio model of starbursts in low-mass halos to date. The model incorporates sub-models for gas cooling (including a new determination of the H2-cooling threshold in minihalos), central gas accretion and disk formation (using a new selfsimilarsolution), stellar feedback from radiation pressure (direct stellar radiation, Lyman-<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" data-classname="equation_inline" data-title="" /> scattering in H I, and multiple scattering of IR photons by dust), stellar winds, expanding HII regions, and (crudely) supernovae. The resulting star formation efficiency is used to predict the fraction of stars that remain gravitationally bound in a cluster following gas expulsion, andwhat fraction escape the central region of the halo, yet remain bound by the dark matter halo. I apply Anaxagoras to study star formation at <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z%5C,%20%3E%20%5C,6" data-classname="equation_inline" /> in satellite halos of the Milky Way using a halo merger tree code, as well as Population III (Pop III) star formation in minihalos. For the Milky Way setup, hundreds of galaxies are predicted to form with luminosities (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?L_%7B%5Crm%20V%7D%20%5C,%3C%5C,%20%5Crm%7Bfew%7D%5C,%20%5Ctimes%20%5C,%2010%5E4%20%5C:%20%5Crm%7BL%7D_%7B%5Codot%7D" data-classname="equation_inline" />), half-mass radii (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csim%2010-200%5C:%5Crm%20pc" data-classname="equation_inline" />), mass-to-light ratios (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M/L_%7B%5Crm%20V%7D%20%5Csim%20100%20-%20%5Crm%7Bfew%7D%20%5C,%5Ctimes%5C,%2010%5E3%20%5C:%5Crm%7BM%7D_%7B%5Codot%7D/%5Crm%7BL%7D_%7B%5Codot%7D" data-classname="equation" />), and ages (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?13.18%5E%7B+0.29%7D_%7B-0.31%7D%5C:%5Crm%20Gyrs" data-classname="equation_inline" data-title="" />) in good agreement with the observed local population of Ultra-Faint Dwarfs. This shows that ΛCDM is able to explain the properties ofthe faintest dwarf galaxies without fine-tuning. Furthermore, at least ~ 40 compact (initial half-mass radii <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csim%200.1-5%5C;%5Crm%7Bpc%7D" data-classname="equation_inline" />), old (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?13.27%5E%7B+0.21%7D_%7B-0.39%7D%5C:%5Crm%20Gyrs" data-classname="equation_inline" />) globular cluster (GC) candidates with initial stellar masses <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?10%5E5%20-%2010%5E6%20%5C:%20%5Crm%7BM%7D_%7B%5Codot%7D" data-classname="equation_inline" />  are predicted to form at the center of low-mass halos, and could survive to the present-day and explain at least a fraction of the observed metal-poor GCs. Their properties are consistent with recent candidates for GCs residing in dark matter halos. Thus, Anaxagoras lends support to the viability of the scenario of GC formation in minihalos. Finally, the formation of Population III (Pop III) stars in minihalos is studied, with the conclusion that if Pop III stars are not overly massive (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?25%5C:%5Crm%7BM%7D_%7B%5Codot%7D" data-classname="equation_inline" />) between ~ 1 − 30 stars could form per minihalo at <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z%5C,%20%3E%20%5C,20" data-classname="equation_inline" />, with the number increasing to ~ 10 − 500 stars per minihalo at <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?z%5C,%3C%5C,15" data-classname="equation_inline" /> as Lyman-Werner feedback delay star formation until halos reach larger masses. In the case where Pop III stars are more massive (<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?140%5C:%5Crm%7BM%7D_%7B%5Codot%7D" data-classname="equation_inline" data-title="" />) most minihalos form just a single star. Due to self-shielding of H2 in minihalos, I find that the cosmological Lyman-Werner background is insufficient to produce Pop III galaxies in atomic-cooling halos, with the implication that the number of massive Pop III galaxies/star clusters in the early Universe has been greatly overestimated in the literature that ignores self-shielding.