Well it's an interesting idea. Besides, I'm sure anyone who does math is more satisfied with an "effective" proof than a reductio ad absurdum. Then again, I would not reject non constructive math. It does not bother me that much, the constructive proofs are like a bonus.
The developments of set theory in 1960's led to an era of independence in which many of the central questions were shown to be unresolvable on the basis of the standard system of mathematics, ZFC. This is true of statements from areas as diverse as analysis (“Are all projective sets Lebesgue measurable?”), cardinal arithmetic (“Does Cantor's Continuum Hypothesis (CH) hold?”), combinatorics (“Does Suslin's Hypotheses hold?”), and group theory (“Is there a Whitehead group?”).
These developments gave rise to two conflicting positions. The first position—which we shall call pluralism—maintains that the independence results largely undermine the enterprise of set theory as an objective enterprise. On this view, although there are practical reasons that one might give in favour of one set of axioms over another—say, that it is more useful for a given task—, there are no theoretical reasons that can be given; and, moreover, this either implies or is a consequence of the fact—depending on the variant of the view, in particular, whether it places realism before reason, or conversely—that there is no objective mathematical realm at this level. The second position—which we shall call non-pluralism—maintains that the independence results merely indicate the paucity of our standard resources for justifying mathematical statements. On this view, theoretical reasons can be given for new axioms and—again, depending on the variant of the view—this either implies or is a consequence of the fact that there is an objective mathematical realm at this level.
>>7780367 But I don't mind Banach-Tarski. Besides, if you were to remove Zorn's lemma there would be whole lot of stuff in functional analysis, algebra and topology you would not be able to do anymore (sure, applications are usually done on seperable vector spaces or noetherian rings so, in most cases you don't need its full strength but it can happen)
>>7780379 >get a load of this science pleb. The empirical sciences care about falsifiability and other crap we don't care about. Constructive math is a special case of constructive math (basically constructive math + axiom of choice). So we can work in a greater generalization of mathematics where we can assume anti-classical axioms.
We've finally transcended the need for boxing gloves, anon.
>>7780286 Hell yea anon. Doing math in intuitionistic logic is the future.
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