Re: Gödel's system P, Principia Mathematica,

On Jan 28, 10:34 pm, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
you say
Aatu did not make any comment about whether the axiom of reducibility is
"justified
your baise is clouding what your eyes see

Aatu says clearly

It was rather based on the simple observation that there is ***no apparent
way of justifying*** ... the axiom of reducibility as logical truths

cant you read


Here is what Aatu is trying to tell you.

Russell tried to prove that mathematics can be reduced to logic. He
decided that he failed, because he needed the axiom of reducibility,
and he didn't think that was a logical truth.

That's what Aatu's trying to tell you. He's not expressing any
particular view himself. He's telling you what Russell thought. And
the issue of whether the axiom of reducibility is a logical truth is
different to the issue of whether it is "justified". Most people agree
now that we need some non-logical axioms for mathematics, we haven't
come to the conclusion that we aren't justified in using those axioms.

Perhaps Aatu also wished to imply that Russell thought that you're
only justified in using an axiom in mathematics if it's a logical
truth, and therefore we should regard a mathematical theorem as
doubtful if it relies on the axiom of reducibility. Perhaps that's
where he's coming from with his choice of wording. I doubt it though.
I think he's just saying, Russell conceded that the logicist project
failed, but you don't need to vindicate logicism to justify
mathematics. Aatu can speak for himself about this.

Aatu certainly expressed no view himself about whether the axiom of
reducibility is justified. For all I know he may not think this
question is meaningful, he may be a mathematical antirealist. In any
case, as I have told you time and time again, this issue is completely
irrelevant. Goedel's theorem can be proved in a metatheory which is
acceptable to people of all foundational stripes. And it applies to an
enormous range of object theories, some of which use the axiom of
reducibility, some of which don't. Your position on the axiom of
reducibility simply doesn't have any bearing on the correctness of
Goedel's reasoning or the interest of his work.
.

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