Le 06-janv.-06, à 02:40, Benjamin Udell a écrit :

I don't know a whole lot about math, and I tend to be fallibilist, so I wonder whether anybody really does "know," like Penrose claims, that those maths are in fact really and truly are consistent, which are consistent _provably_ only on the unprovable assumption of arithmetic's consistency. I think of seeming inconsistencies that get patched up, 0 divided by 0 equals "any number" you want -- so, more carefully define equality to exclude that problem. Denominators seemingly turning to 0 in calculus got remedied. And so on.

I think that all mathematicians (99,999...%) believes correctly in the consistency of Peano Arithmetic (PA). Note that PA is a "simple" example of a lobian theory or machine).

Few mathematician doubt about the consistency of Zermelo-Fraenkel Set Theory (ZF), although George Boolos makes a case that ZF could be inconsistent.

Famous results by Godel and Cohen have shown the relative consistency of many "doubtful" math assertion: precisely it has been shown that IF ZF is consistency THEN ZF + the axiom of choice is consistent. The same for the continuum hypothesis, etc.

Note that (a theorem prover for) ZF is also a Lobian machine.

Much more difficult is the question of the consistency of (rather exotic) set theories like Quine's New Foundation (NF).

There is no sense to ask about the consistency of the whole math, because the whole math cannot be presented in a formal theory or theorem proving machine (if only by Godel incompleteness result). It is a reason to doubt about Penrose' use of the notion of consistency in his "godelian" argument against Mechanism/Comp. All what such types of reasoning show is that: IF I am a sound Lobian Machine THEN I cannot know which machine I am (and then I cannot know which computational history extends me, and that is what I use eventually for solving the OM measure problem).


PS I will answer your long "naming-issue" post, Ben, and some others to, asap.


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