A seemingly endless controversy that has already lasted over a decade surrounds Yaroslav Sergeyev's proposed theory of infinity.  He has [over twenty publications](https://mathscinet.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=sergeyev&co4=AND&pg5=TI&s5=&co5=AND&pg6=JOUR&s6=&co6=AND&pg7=ALLF&s7=infinit*&co7=AND&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&review_format=html&Submit=Search) on this topic.  Critics maintain that his theory is inconsistent, with whatever valid material actually derived from earlier rigorous sources.  He also has some backers including [one that posted an *answer* at MO](https://mathoverflow.net/a/226910/28128).

Complete disclosure: I am one of the authors of [this 2017 publication in *Foundations of Science*](http://dx.doi.org/10.1007/s10699-016-9485-8) critical of Sergeyev's production.

Some other possibilities are mentioned in [this *question*](https://mathoverflow.net/q/303325/28128).