As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms of what actual set theorists do I have no idea (my idea of forcing is people somehow shoving axioms into $\sf ZFC$).

I've heard many, many people talk about how $\sf ZFC$ is somehow wrong. Some examples:

  • I've read Wildberger's paper where he states the axiom of infinity as "There exists an infinite set" and proceeds to bash how informal it is. I've seen the MSE thread where this is discussed and seen arguments for why it is indeed not a strawman, but they didn't convince me. (He doesn't appear to have a problem with choice specifically, but rather with infinity to start with [and I guess without infinity choice isn't even a thing anyway.])

  • I've seen the last section of the paper on dividing by three (it's available online) where they talk about how choice is probably not true, without giving any mathematical support for this hypothesis. They even claim that there is work being done to show ZFC is inconsistent - is this still happening today?

  • I've also seen the Banach-Tarski paradox used as an argument against choice. Basically, they say that since choice on the continuum implies such an astounding result which is so obviously not true (why, exactly?) that it must be wrong. I find this argument similar to one in quantum physics - when Heisenberg proved the uncertainty principle no one panicked and seriously started talking about how quantum states must not form a Hilbert space after all.

This is about it. Are there any arguments with mathematical content against $\sf ZFC$ whether it be refuting infinity, choice, or something else?

  • $\begingroup$ The argument that you cannot extract programs from classical proofs? (This is an issue with the underlying logic, of course -- I don't know if this is what you are looking for.) $\endgroup$ – darij grinberg Jan 1 '15 at 13:31
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    $\begingroup$ No, there aren't especially convincing arguments in that sense. There are arguments that begin with other philosophical positions (e.g. Wildberger), but these are only convincing if you accept the philosophical position of the author. The stronger arguments, instead of arguing that ZFC is inconsistent, argue that it's not the right theory to use. These arguments propose other foundational systems that are claimed to be a better fit for mathematical practice. One such alternative is called "homotopy type theory". $\endgroup$ – Carl Mummert Jan 1 '15 at 13:36
  • $\begingroup$ Voevodsky's arguments seem very relevant to me (see e.g. this lecture). They are not aimed, however, at ZFC in particular - he is concerned with formal systems with arithmetic in general (and the implicit assumption that these have to be necessarily consistent in order to be suited for mathematics). $\endgroup$ – Pavel Čoupek Jan 1 '15 at 13:43
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    $\begingroup$ @Carl: The arguments are not against set theory either, but rather against set theory as a foundational theory. I'm sure that nobody from the HTT has an issue with set theorists studying implications of large cardinals and whatnot. They just feel that mathematics will be better served using type theory as a foundation. Whether or not they are right is a personal philosophical question, and at best time will tell whether or not there will be a "migration" movement, or something like that. $\endgroup$ – Asaf Karagila Jan 1 '15 at 18:33

The question is what do you consider a "logical argument".

All the arguments that you brought up are philosophical and based on the idea that mathematics should somehow describe our physical reality, and must therefore be compatible with our physical intuition. Personally I find this argument to be lacking, since our physical intuition changes all the time anyway, and we have such feeble minds when it comes to grasping very small or very large objects anyway. So what do we even know about physical reality?

Once you remove the argument "this is incompatible with how I perceive reality", and agree that mathematics whether formalistic games on paper, or describing an ideal mathematical universe, but not our physical universe as we understand it, any argument is really against mathematics as a whole, rather than just specifically $\sf ZFC$.

These could be about using the law of excluded middle, or other non-constructive methods, but that's not specific against $\sf ZFC$. Many people use the law of excluded middle, all across mathematics. Or about how set theorists are actively exploiting the incompleteness phenomenon which can confuse even experienced mathematicians (true and provable are not the same thing in set theory, and in any sufficiently strong theory; this can be baffling, but this is reduced to "this is incompatible with how I see mathematics" rather than be a mathematical argument).

What do set theorists do? They study the structure of different models of $\sf ZFC$ or $\sf ZF$. Much like other mathematicians might be studying schemes or modules, or monoids, or simplicial objects in a low-dimensional space. We study the structure of models of set theory, and we ask what you can prove or disprove. If someone wants to disprove the claim "Every Banach space has a basis with such and such properties", they will work hard and construct this space. If someone wants to disprove the claim "Every model of set theory has such and such properties", they will work hard and construct a model where that fails. Forcing is one of the tools in the utility belt of a set theorist.


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