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When I took, last year, my university's course on set theory, I remember my teacher introduced the topic by saying that ZFC Set Theory is an axiomatic system over which math is founded, but it's not the only one since there are many other of these, that, just like ZFC, are built upon first order logic. But then he said one can go further and find other axiomatic systems that serve as a foundation, built upon other logics or theories; He cited topoi as the axiomatic system built upon category theory that serves as a foundation for math. Then he mentioned there are analogous axiom systems built upon fuzzy logic, but said there are not such systems built upon modal logic.

I would like to know

  • Is it true that there are not axiomatic systems built upon Modal Logic that serve as a foundation of mathematics? If so, why is that? Where can I find more info. about this?
  • Where can I find more information on all the axiomatic systems there are, and the logics and theories they are based upon? What is this area of math called?
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    $\begingroup$ I can't speak for your teacher, but it is the case that none of the main systems that have been proposed as possible foundations for mathematics involve modal connectives. I suspect this is down to philosophical considerations: what modalities would be appropriate for a subject that is felt to be timeless and absolute? ,,, $\endgroup$
    – Rob Arthan
    Commented Jan 21, 2020 at 23:33
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    $\begingroup$ ... I should add that intuitionistic systems in some sense embrace the notion that mathematical knowledge grows over time, but they achieve this without having a modal connective to represent the state of mathematical knowledge. Instead that is implicit in the intuitionistic interpretation of implication. $\endgroup$
    – Rob Arthan
    Commented Jan 21, 2020 at 23:41
  • $\begingroup$ Good answer; however, do you think that the fact that there exist fuzzy mathematics based on fuzzy logic, is a sign that it is possible to create a modal foundation for math? Also, Do you know where can a I find those "main systems that have been proposed as possible foundations for mathematics" you mention? and more info. in general, about this topic? $\endgroup$
    – Daniel Bonilla Jaramillo Haase
    Commented Jan 21, 2020 at 23:56
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    $\begingroup$ We could give modal truth values to undecidable statements, perhaps. It would be interesting. $\endgroup$
    – Mr. Fox
    Commented Jan 23, 2020 at 1:05
  • $\begingroup$ Let me add two comments: (i) Clearly, since any formula of first order logic is also a formula in first order modal logic, it should be no problem to carry over the axiomatization of ZFC to first order modal logic. So, there is an axiomatic system in modal logic for ZFC. However, such an axiomatization makes no use of the modalities. (ii) Another foundation of set theory (including non-well-founded sets) can be found in Peter Aczel's book "Non-well-founded sets", where he shows that any set can uniquely be identified with an accessible pointed directed graph. $\endgroup$
    – Willem Hagemann
    Commented Feb 12, 2020 at 16:09

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