Timeline for Is a paraconsistent and provably non-trivial foundation for math possible?
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13 events
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Jan 23, 2022 at 18:16 | comment | added | user21820 | @JohannesHahn: I can't make any sense of your comment. To the last sentence, how can a proof that assumes axioms of ZFC lend any credence to consistency of ZFC? To the first two sentences, how can reflection support consistency of ZFC, since it would be meaningless if ZFC is inconsistent, and since it is trivial to extend any theory that interprets PA− to one that proves every finite fragment consistent? | |
Nov 20, 2016 at 13:01 | comment | added | Johannes Hahn | Another argument for ZFC's consistency are the reflection principle(s). In my view they prove that ZFC is "as consistent as it possibly can be without violating Gödel's theorem". I would even go so far as to say that it is a nontrivial mental exercise to precisely find out why the reflection principle doesn't actually prove consistency. Or put in another way: To someone who isn't familiar with the finer points of logic the reflection principle seems just as good as a true proof of consistency. | |
Oct 13, 2016 at 12:41 | comment | added | arsmath | That said, a paraconsistent foundation for some fragment of arithmetic is a perfectly worthy research goal. | |
Oct 13, 2016 at 12:40 | comment | added | arsmath | @ThomasBenjamin Sure, but I cannot express the level of certainty I have that Nelson is wrong. I am more certain that PA is consistent than I am that a person named Ed Nelson existed, or that a place called Princeton exists, or even that you exist. I would put my certainty about PA below my certainty that I exist, but only by a little bit. | |
Oct 13, 2016 at 12:28 | comment | added | Thomas Benjamin | @arsmath: Are you aware that the late Ed Nelson of Princeton believed that $PA$ was inconsistent (and $PA$ is equiconsistent to $ZFC$ $-$ $Infinity$, as you know)? What if it were true? Would a paraconsistent logic be needed then? | |
Oct 2, 2016 at 0:21 | comment | added | Nik Weaver | I like the argument from large to small better, I think this is your strongest case. (Still would disagree about the significance of answering questions in real analysis --- I think the questions you're referring to are very peripheral to mainstream analysis.) Anyway, thank you for taking the time to answer my comment. | |
Oct 2, 2016 at 0:20 | comment | added | Nik Weaver | for the argument from small to large, whether the intuition extends to infinite sets is something you really have to think about separately for each axiom. I find the power set axiom very dubious for infinite sets --- in particular, I do not know any informal argument in its favor that would not also support full comprehension. | |
Oct 1, 2016 at 20:20 | comment | added | arsmath | The argument from large to small. Set theorists have added many additional axioms to ZFC. If ZFC was inconsistent, then these theories are also inconsistent, plus it should be easier to prove consistency. But set theorists have largely been unable to prove them inconsistent. Not only that, they seem to form a coherent picture and even answer questions in real analysis that are independent of ZFC (and give the nicest possible answer). | |
Oct 1, 2016 at 20:10 | comment | added | arsmath | I will give two arguments. First, the argument from small to large. All of the operations in ZFC correspond to natural constructions with sets. Bounded Zermelo set theory is pretty much the list of natural constructions with sets. Axiom of replacement is a more sophisticated but natural idea. They are all consistent for finite sets, so it's then just a question of whether the same intuition extends to infinite sets. I feel that it does. | |
Sep 30, 2016 at 19:49 | comment | added | Nik Weaver | "There's basically no reason to think that ZFC is inconsistent." --- this is not a terribly strong argument. Are there positive reasons for thinking it's consistent? The fact that no inconsistency has been discovered is some evidence, although not as impressive as it seems, given that the vast bulk of modern mathematics can be formalized in much weaker systems. Does the system have a compelling philosophical coherence? Some would say yes, but I don't see it --- to me ZFC looks like a bit of a hodgepodge, an attempt to skirt the paradoxes without really understanding their source. | |
Sep 30, 2016 at 17:50 | vote | accept | Myren | ||
Sep 30, 2016 at 16:22 | vote | accept | Myren | ||
Sep 30, 2016 at 17:18 | |||||
Sep 30, 2016 at 15:55 | history | answered | arsmath | CC BY-SA 3.0 |