It looks to me like this has some interesting ideas, but none of the theorems that would be required for them to have any validity. The form of this work is "wouldn't it be cool if there was a structure, the extended real numbers, with some of the properties of the real numbers, but including infinite numbers? And wouldn't it be cool if there was a map from sets of numbers to the extended real numbers, that represents the 'size' of the set, that has the following properties, as well as some natural properties we would like a size function to have?". To which my answer is "Yes, it would be cool. But it's not clear that a system and a map with the properties you're hoping for exists". All the results are of the form "If such a map exists, and the extended reals and the map have the following properties, then the "size" of the following set of numbers is the following extended real number". But the set of properties of the extended real numbers and the map are never specified. Instead, whatever properties seem natural when deriving a certain result are used, with no complete specification of these properties provided. I think it's entirely possible that there is some interesting mathematics here. But it's also entirely possible that a contradiction could arise, in which case all the individual results are useless, and all that remains is a single theorem that no "size" function with all the desired properties exists. Andy On Tue, May 29, 2018 at 9:32 AM, James Propp <jamespropp@gmail.com> wrote:
Can any of you who know more about the theory of distributions (and other generalizations of classical analysis) comment on the work of Ilya Chernykh?
See
http://extended.wikia.com/wiki/Extended_Wiki#Some_extended_numbers
for instance.
Chernykh is operating in a mode that is not in fashion these days (though we forgive such freewheeling manipulations of divergent expressions in the work of Euler and Ramanujan).
I cannot assess how novel this work is, nor can I assess whether there's a consistent theory lurking beneath Chernykh's manipulations.
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