Re: Godel uses ad hoc axiom that was not in PM - which is invali

On Mar 3, 7:48 pm, "elsiemelsi" <cyprin...@xxxxxxxxxxxxxxx> wrote:
you say

." I've got the second edition and the axiom of
reducibility still appears in the main text. He wasn't happy with the
axiom of reducibility, that's for sure, but he never found any
satisfactory alternative. The axiom of reducibility is part of PM, there
are no two ways about it.

read the quotes

what does repudiated mean
what does gave up mean
all very clear

“In the Introduction to the second edition of Principia, Russell
repudiated Reducibility as 'clearly not the sort of axiom with which we
can rest content'…Russells own system with out reducibility was rendered
incapable of achieving its own purpose”


Some people have the crazy idea that if you want to do credible
scholarship about "Principia Mathematica", you've got to just once
have a look at the text. In the Introduction to the second edition,
yes, Russell outlined many changes he would make to the system if he
had the energy to re-write the entire book. And, yes, he did say that
the axiom of reducibility was clearly not the sort of axiom with which
we could rest content. But he didn't find a satisfactory alternative.

There is absolutely no way it is correct to say "the axiom of
reducibility was not in PM". Yes, Russell found it an unsatisfactory
axiom. You should just stick to saying that.

If you don't like the axiom of reducibility, that's fine. For reasons
that have been explained many times, this has no bearing on the merits
of Goedel's argument. Anyway, why don't you tell us a theory you do
like, as I keep asking you to?

quote page 14http://www.helsinki.fi/filosofia/gts/ramsay.pdf.

“Russell gave up the Axiom of Reducibility in the second edition of
Principia (1925)”

you say

Goedel proves a bunch of theorems *about* the system P, which as far as he
is concerned might just as well be a game played with symbols,
or a game played with numbers (once you've arithmetized it). That doesn't
mean he *uses* the system P. His results can be proved in a
system much weaker than P.

it is not a matter of his system being proved in weaker systems
his paper is called
On formally undecidable propositions of Principia
Mathematica and related systems

his theorems apply to system P
system P is the system from which he derives his incompleteness theorem

No. System P is the system which his incompleteness theorem is about.
The theorem is also true of many other systems. And a version of it
can be proved in Bounded Arithmetic.

So whether you find the axiom of reducibility acceptable has no
bearing on the merits of Goedel's proof, because

(1) his result is provable in an extremely weak system which does not
use the axiom of reducibility
(2) even if you accepted the utterly silly idea that whether or not
the object theory is "valid" matters to the proof (a belief which
actually indicates complete incompetence), it still doesn't matter,
because Goedel's reasoning applies to many systems that do not use the
axiom of reducibility.

You still clearly do not understand the distinction between the object
theory and the metatheory. You really have no hope of being competent
to say anything about this subject until you understand that
distinction.

For the thousandth time: Goedel is *talking* about the system P. He
does not need to assume it is "valid". It can just be a meaningless
game played with symbols for all he cares. He is using methods of
mathematical reasoning to argue about it which are much weaker than
those which can be formalized in P. He does not need the axiom of
reducibility.

quote from the van Heijenoort translation

[quote]”Theorem XI. Let κ be any recursive consistent63 class of
FORMULAS;
then the SENTENTIAL FORMULA stating that κ is consistent is
notκ-PROVABLE; in particular, the consistency of P is not provable in
P,64
provided P is consistent (in the opposite case, of course, every
proposition is provable [in P])". (Brackets in original added by Gödel
“to help the reader”, translation and typography in van
Heijenoort1967:614)

system P uses peano and the axiom of reducibility


But Theorem XI can be proved in RCA_0, which does not have the axiom
of reducibility. Also, it is true of many systems other than P which
do not have the axiom of reducibility, not that this matters.


"The system P of footnote 48a is Godel’s
streamlined version of Russell’s theory of types built on the natural
numbers as individuals, the system used in [1931]. The last sentence ofthe
footnote allstomindtheotherreferencetosettheoryinthatpaper;
KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing
his approach to set theory, “This axiom plays the role of [Russell’s]
axiom of reducibility (the comprehension axiom of set theory).”

note it says Godel wrote

This axiom plays the role of [Russell’s]
axiom of reducibility (the comprehension axiom of set theory).

you say

. Do you understand the distinction between *using* a theory and talking
*about* a theory?

cleary godel is useing AR and not talking about it
as is clear from formular 40


Quite the reverse. Failure to understand that he is talking about it,
not using it, indicates complete incompetence in the subject.

40. reduAxiom(x) ⇔ ∃u,v,y,n ≤ x.
vtype(n,v) ∧ vtype(n + 1,u) ∧ ¬free(u,y) ∧ isFm(y)∧
x = exists(u,forall(v,equiv(seq(u) ◦ paren(seq(v)),y)))
x is a formula obtained by substitution from the axiom schema IV, 1
, i.e. from the reducibility axiom

CLEARLY USEING AR NOT AND TALKING ABOUT IT

Not at all. He's just defining an arithmetical predicate. He can
define any arithmetcal predicate he likes. Where is he using the axiom
of reducibility?
.

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