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The attached screenshot, from Goldblatt's Topoi, shows at a glance the distinction between the constructive concept of inhabited set versus that of the classical nonempty set.

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These definitions seem only equivalent if the axiom of excluded middle is assumed. However, in the absence of this axiom, the distinction seems fundamental (Why isn't this taught in the math curriculum?)

By analogy to Herrlich's Axiom of Choice which lists "disasters without choice", "disasters with choice" and "disasters either way", is there a published list of consequences of this difference?

In other words, which major results that rest on provably nonempty sets in classical logic cannot be proven to be inhabited sets without excluded middle?

By the way, at the end of Khamsi and Kirk's Introduction to metric spaces and fixed point theory the authors point out three variations on constructive mathematics developed by different mathematicians such as Brouwer and A Markov (Jr). It would be interesting then to also compare what effect these variations have on the emergent set theory.

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    It isn’t taught in the usual math curriculum because the law of the excluded middle is (properly, in my view) taken for granted in most mathematical practice, and the odd exceptional areas aren’t part of the common curriculum.2012-11-11
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    Isn't that a bias, like only teaching linear systems theory or Euclidean geometry or only consequences of ZFC? What's the point of Herrlich's book for example? It's that you cannot have Choice and 3-dimensional geometry as the same time.2012-11-11
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    @alancalvitti, what does $\forall a$ mean there? (what is being quantified over)2012-11-11
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    @spernerslemma, that's a question for Goldblatt. Are you suggesting that the quantification is invalid because (I presume) of the absence of a universal set to quantify over?2012-11-11
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    @spernerslemma, I've emailed Dr. Goldblatt with a link to this question. Perhaps he'll answer directly or if he replies to me I'll post the content.2012-11-11
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    I was fairly certain that the axiom of choice and geometry were fine with one another. I don't see why the axiom of choice means that you can't have geometry in three dimensions.2012-11-11
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    As far as I’m concerned, the real point of Herrlich’s book is that AC is necessary for doing a great deal of lovely mathematics. It’s interesting for its own sake to know just what depends on it and to what extent, but otherwise I don’t care: I don’t consider AC any more problematic than the axiom of infinity. The question of what depends on EM also has some intrinsic interest, but as far as I’m concerned not much more than the question of what depends on modus ponens.2012-11-11
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    @AsafKaragila, according to the material assembled by Herrlich, because in ZFC - but not, for example in ZF+AD - paradoxical Banach-Tarski decompositions arise in 3-dimensions. (On the other hand, in ZF+AD, algebra can fail badly, for example, no free ultrafilters on the natural numbers, and - if I remember - the reals don't have a vector space basis over the rationals. Please correct me if I'm off base.2012-11-11
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    Of course it’s a bias. Unlike teaching only Euclidean geometry, however, it’s a desirable bias.2012-11-11
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    @BrianM.Scott, all mathematical results are contingent on their assumptions by Modus Ponens. This is without exception, correct? Therefore, on what basis can one say what is and is not desirable? The best that can be done is to study one set of assumptions versus another, with no claim to "inside track" to reality.2012-11-11
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    I don't remember Euclid, Archimedes, Hilbert or Gauss failing to do any three-dimensional geometry because of the Banach-Tarski paradox.2012-11-11
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    @AsafKaragila, by a similar token, I'm not aware of any mathematician other than Cohen getting a Fields medal for work in foundations. Does that mean foundations are moot then?2012-11-11
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    A bias in teaching mathematics that accords with actual mathematical practice is obviously desirable. As for my modus ponens comment, one can certainly investigate deductive systems that don’t include MP, but in my view this hasn’t much less importance for the rest of mathematics than investigating what happens when you don’t assume EM.2012-11-11
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    @alancalvitti: You tried that argument with me before. I think that you have a very strong bias against set theory, and it has come up before in your comments. I don't wish to repeat my argument with you because frankly I find such arguments childish. I also think that prizes are not everything in mathematics, but apparently you seem to disagree.2012-11-11
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    @AsafKaragila, this is not a personal argument, just seeking to find out what follows from what assumptions. Euclid, Archimedes and the others you mention didn't have axiomatic set theory to base their geometric results on to begin with, correct? If so, then geometry is independent of set theory to that degree.2012-11-11
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    @alan: While it's not personal against me or against you, it *does* feel that you have ill feelings towards the foundational research. You keep pointing out that if no one got a Fields medal, then it's a worthless field. As if mathematics revolves around the Fields medal, or any other prize. You should sometime read about Shelah. He is by far the best mathematician I have heard of. He sees things more clearly than anyone and he does things in an immensely complicated way that it's inhuman. For some reason... he is working on foundational research. Shame, wouldn't you say?2012-11-11
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    @AsafKaragila, I don't know anything about Shelah (I'm an engineer) other than he's published 1000's of papers so it seems construction work on foundations never ends (so how can it be called foundational?). Could you please address my previous question?2012-11-11
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    Tarski gave a very good axiomatization of geometry. Sure it uses logic and set theory, but it's not *very* dependent on them (what is a line if not a set of points? you have *some* naive need for sets there). The question is how would you like to formalize things, and what you are trying to work on. Regardless to that, calling a whole field "moot" is inappropriate. I would also find a topic whose research "ends" quite boring, I'm also sure I am not the only one (even if you count non-logicians).2012-11-11
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    If you are an engineer it would be quite appropriate for you to learn to respect people that have earned that respect. In mathematics things are not measured in prizes, money, or fame. You should learn about the society you wish to dis before dissing it, especially when the majority of members of that society are likely to be as intelligent as you if not more intelligent.2012-11-11
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    I'm not "dissing" anyone. I'm asking the question, because I want answers that I don't know. I used the word "moot" in a question after all.2012-11-11
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    @spernerslemma, Dr. Goldblatt replied that the quantifier ranges over the universe of sets. In return, I forwarded this link: http://logic.harvard.edu/EFI_Woodin_TheTransfiniteUniverse.pdf, quoting: Skeptic's Attack: The Continuum Hypothesis is neither true nor false because the entire conception of the universe of sets is a complete fiction. Further, all the theorems of Set Theory are merely finitistic truths, are a reflection of the mathematician and not of any genuine mathematical \reality".2012-11-14
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    @AsafKaragila, "one can certainly investigate deductive systems that don’t include MP" [Modus Ponens]- I've read somewhere that MP is the one common denominator to all constructive maths. If it is removed, what does inference look like?2012-11-14
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    I haven't tried to remove MP from my process, so I can't tell you. To your comment above that one, I should probably tell you that $\frac12$ is a fiction as well, not to mention $10^{2000}$ or $\sqrt2$. Let alone an infinite object like the real numbers. Also $\mathbb R^{42}$ is a serious fiction and certainly *doesn't exist*. For some reason, though, it's easier to pick on set theory...2012-11-14
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    @AsafKaragila, it's Woodin's comment, not mine. Perhaps the distinction between "the" universe of sets versus $\frac12$, $\sqrt 2$, $10^{2000}$ and $\mathbb R^{42}$ is that the former is not *uniquely* defined, while the rest are uniquely defined (regardless of physical existence). For example, who disagrees on the definition of a vector space (particularly finite dim)?2012-11-14
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    Woodin's point of view is not the only point of view in set theory, and I am not even sure that his point of view is the common one. Furthermore if one accepts the fact the theory of the real numbers can be developed within SFC, the. The fact there is no canonical model of ZFC means there is no canonical model of the real numbers either.2012-11-14
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    That there are multiple points of view in set theory is precisely the problem I was alluding to. On the other hand, results and computations based on the properties of the real line, complex plane, and finite-dim vector spaces are built into real world control systems in communication, avionics, medical imaging, spectroscopy, mechatronics, etc. This is where math meets nature and these properties do not vary. The structures are canonical in that sense.2012-11-14

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