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

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I don't know a whole lot about math, and I tend to be fallibilist, soI wonder whether anybody really does "know," like Penrose claims, thatthose maths are in fact really and truly are consistent, which areconsistent _provably_ only on the unprovable assumption ofarithmetic's consistency. I think of seeming inconsistencies that getpatched up, 0 divided by 0 equals "any number" you want -- so, morecarefully define equality to exclude that problem. Denominatorsseemingly 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).`

Bruno

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

`to, asap.`

http://iridia.ulb.ac.be/~marchal/