# Mathematicians are in deep trouble for 2 reasons

pam69ur

Sun 18 May, 2008 09:04 pm
@pam69ur,
we have been talking about self referencing in godels work and how texts books on logic tels us such statements are invalid - and thus according to dean godels theorem is invalid

this takes us full circle back to the thread in regard to ZFC being inconsistent due to the skolem paradox

well

the australian philospher colin leslie dean points out a source in ZF thus
ZFC for its inconsistency ie the skolem paradox

poincare and russell argued that impredicative statements led to paradox
in mathematics

now
the seperation axiom of ZFC is impredicative thus we would expect that ZFC would end in paradox and it does due to the skolem paradox

solomon feferman page 12

http://math.stanford.edu/~feferman/papers/predicativity.pdf

Quote:
"in ZF the fundamental source of impredicativity is the seperation axiom which asserts that for each well formed function p(x)of the language ZFthe existence of the set x : x } a ^ p(x) for any set a Since the formular p may contain quantifiers ranging over the supposed "totality" of all the sets this is impredicativity according to the VCP this impredicativity is given teeth by the axiom of infinity "
now Adding to ZF either the axiom of choice (AC) or a statement equivalent
thereto, yields ZFC.

thus WE HAVE A SOURCE FOR THE INCONSISTENCY IN ZF thus ZFC IE SKOLEM PARADOX

that the seperation axiom is impredicative is doubly interesting as
zermelo introduced it to avoid the russell paradox which showed naive set theory is inconsistent

Zermelo""Fraenkel set theory - Wikipedia, the free encyclopedia

Quote:
3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): If z is a set, and is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox and its variants.

now impredicative statements are considered invalid by text books on logic poicare russell et al
therefore the separation axiom is invalid
thus
russells paradox stands and and set theory is inconsistent

Aedes

Mon 19 May, 2008 06:06 am
@pam69ur,
Wait a second --

Godel does not actually use the axiom of reducibility as you like to point out. He makes a contention ABOUT the axiom of reducibility, but this contention does not require that it be valid or invalid. I can make an argument about the proposition of the earth being flat, but the fact that it's round does not affect my argument.

pam69ur wrote:
simple colin leslie dean has not presented it to the academic world
no journals
no nothing
he is a no body who preferes to step out side the grove
he believes that the academic world has and should not have a monopoly on truth
Ok, so rather than the academic world you'd rather present it to a bunch of amateurish dilettante hobbyists (like myself) on an online forum? How does that advance your career or recognition? If you're interested in your conclusions actually resonating in the math and philosophy worlds, you need to present your arguments to them, not to us. Most of us are not professional philosophers, and of those here who are very few are logicians. So you're devoting effort to a cause that doesn't really matter, because we're not the audience you care about.

I found this discussion elsewhere on the web:

Thought is not in language or images or concepts or anything else - BrainMeta.com Forum

You assert here that all views are meaningless and that language contains no thought. If so, then what are we to make of your non-mathematical assertions about Godel? If your arguments are solely that he used AR and that he doesn't define truth, those are LOGICAL arguments that can only be presented through this language that you find so bereft of thought.

Incidentally, we ALL know that you are Colin Leslie Dean. I ask you this out of respect for you: speak in the first person, and take both credit and responsibility for your thoughts.

Arjen

Mon 19 May, 2008 11:04 am
@pam69ur,
Say pam69ur, do you mean meaninglesness as in falso?

pam69ur

Mon 19 May, 2008 06:21 pm
@Arjen,
Quote:
Say pam69ur, do you mean meaninglesness as in falso?

i see you are a perceptive reader

NO

meaninglessness just means self contradictory

things can be contradictory and still "true"

read colin leslie deans book

http://gamahucherpress.yellowgum.com/books/philosophy/irrationality.pdf

Quote:
Absurdities or meaninglessness or irrationality is no hindrance [sic] to something being 'true' rationality, or, Freedom from contradiction or paradox is not a necessary an/or sufficient condition for 'truth': mathematics and science examples

the Dean theorem says

Quote:
EXAMPLES FROM MATHEMATICS AND SCIENCE SHOW THE
DEAN THEOREM

CONTRADICTION, OR INCONSISTENCY WITHIN A VIEW AS WELL AS MUTUAL CONTRADICTION, OR INCOMMENSURABLITY BETWEEN VIEWS DOES NOT PRECLUDE THE VIEW OR BOTH VIEWS FROM BEING 'TRUE'

pam69ur

Mon 19 May, 2008 06:23 pm
@pam69ur,
Quote:
How does that advance your career or recognition?

i think colin leslie dean could not care less about a career or recognition

pam69ur

Mon 19 May, 2008 06:48 pm
@pam69ur,
Quote:
Godel does not actually use the axiom of reducibility as you like to point out.

he uese it
he even tells you he uses it

Quote:
Proposition VI: To every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r).

Quote:
In the proof of Proposition VI the only properties of the system P employed were the following:
1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of Proposition V).

Quote:
P is essentially the system obtained by superimposing on the Peano axioms the logic of PM

the axioms of system P are
Quote:
IV. Every formula derived from the schema
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free. This axiom represents the axiom of reducibility (the axiom of comprehension of set theory).
V. Every formula derived from the following by type-lift (and this formula itself):
1. x1 ∀ (x2(x1) ≡ y2(x1)) ∨ x2 = y2.
This axiom states that a class is completely determined by its elements.

A formula c is called an immediate consequence of a and b, if a is the formula (~(b)) ∨ (c), and an immediate consequenee of a, if c is the formula v ∀ (a), where v denotes any given variable. The class of provable formulae is defined as the smallest class of formulae which contains the axioms and is closed with respect to the relation "immediate consequence of".24
The basic signs of the system P are now ordered in one-to-one correspondence with natural numbers,

forumular 40 is used in the proof
f
Quote:
40. R-Ax(x) ≡ (∃u,v,y,n)[u, v, y, n <= x & n Var v & (n+1) Var u & u Fr y & Form(y) & x = u ∃x {v Gen [[R(u)*E(R(v))] Aeq y]}]:>
x is a formula derived from the axiom-schema IV [ie axiom of reducibility], 1 by substitution

Quote:
"A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), andthe axioms of reducibility and of choice (for all types)" ((K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.5). NOTE HE SAYS HE IS USING 2ND ED PM- WHICH RUSSELL ABANDONED REJECTED GAVE UP DROPPED THE AXIOM OF REDUCIBILITY.

Aedes

Tue 20 May, 2008 05:44 am
@pam69ur,
pam69ur wrote:
he uese it
he even tells you he uses it
He makes examples of it and other axioms. His proof doesn't rely on its soundness. He's making a point about axioms in general.

Strange, for all your citations, you seem not to have carefully read his proof. He CLEARLY states that he is commenting on the nature of axioms and not constructing a proof out of the axiom itself, in fact he's critiquing the axiom.

From:
Godel's first Incompleteness Proof at MROB at MROB

Quote:
The most comprehensive formal systems yet set up are, on the one hand, the system of Principia Mathematica (PM)2 and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later extended by J. v. Neumann).3 These two systems are so extensive that all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference. It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case...

pam69ur

Tue 20 May, 2008 05:53 am
@Aedes,
Quote:
He makes examples of it and other axioms. His proof doesn't rely on its soundness. He's making a point about axioms in general.

it is very clear
Quote:
In the proof of Proposition VI the only properties of the system P employed were the following:
1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of Proposition V)

he says he employs ie uses the axioms of system P
which are

note
IV. Every formula derived from the schema
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free. This axiom represents the axiom of reducibility (the axiom of comprehension of set theory).

Quote:
IV. Every formula derived from the schema
1. (∃u)(v ∀ (u(v) ≡ a))
on substituting for v or u any variables of types n or n + 1 respectively, and for a a formula which does not contain u free. This axiom represents the axiom of reducibility (the axiom of comprehension of set theory).
V. Every formula derived from the following by type-lift (and this formula itself):
1. x1 ∀ (x2(x1) ≡ y2(x1)) ∨ x2 = y2.
This axiom states that a class is completely determined by its elements.

Aedes

Tue 20 May, 2008 05:59 am
@pam69ur,
Yes, I read what you wrote the first time you pasted it in. And you apparently didn't read Godel's own introduction in which he sets out to disprove the necessary internal coherence of the axioms that follow.

pam69ur

Tue 20 May, 2008 06:22 am
@Aedes,
Quote:
Yes, I read what you wrote the first time you pasted it in. And you apparently didn't read Godel's own introduction in which he sets out to disprove the necessary internal coherence of the axioms that follow.
but

Quote:
In the proof of Proposition VI the only properties of the system P employed were the following:
1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of Proposition V)
if you bothered to read deans books you would see the paradox godel falls into when he states

Quote:
It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case...

quoting dean

http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

Quote:
Godel makes the claim that there are undecidable propositions in a constructed system [PM and ZF] that dont depend upon the special nature of the constructed system [PM and ZF]
Quote

As he states
"It is reasonable therefore to make the conjecture that these axioms and rules of inference are also sufficent to decide all mathematical questions which can be formally expressed in the given systems. In what follows it will be shown that this is not the case but rather that in both systems cited [PM and ZF] there exist relatively simple problems of ordinary whole numbers [undecidability] which cannot be decided on the basis of the axioms. [NOTE IT IS CLEAR]This situation [ undecidability which cannot be decided on the basis of the axioms]. does not depend upon the special nature of the constructed systems [PM and ZF] but rather holds for a very wide class of formal systems among which are included in particular all those which arise from the given systems [PM and ZF] by addition of finitely many axioms" (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.6).( K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.6)

Thus Godel says he is going to show that undecidability is not dependent on the
axioms of a system or the speacial nature of PM and ZF
Also
Godels refers to PM and ZF AS FORMAL SYSTEMS

"the most extensive formal systems constructed .. are PM ZF" ibid, p.5
so when he states that
"This situation does not depend upon the special nature of the constructed
systems but rather holds for a very wide class of formal systems"
he must be refering to PM and ZF as belonging to these class of formal systems- further down you will see this is true as well

thus he is saying
the undecidability claim is independent of the axioms of the formal system but PM is a formal system

Godel says he is going to show undecidabilitys by using the system of PM (ibid)
he then sets out to show that there are undecidable propositions in PM (ibid. p.8)

where Godel states
"the precise analysis of this remarkable circumstance leads to surprising results concerning consistence proofs of formal systems which will be treated in more detail in section 4 (theorem X1) ibid p. 9 note this theorem comes out of his system P
he then sets out to show that there are undecidable propositions in his system P -which uses the axioms of PM and Peano axioms.
at the end of this proof he states
"we have limited ourselves in this paper essentially to the system P and have only indicated the applications to other systems" (ibid p. 38)

now
it is based upon his proof of undecidable propositions in P that he draws out broader conclusions for a very wide class of formal systems
After outlining theorem V1 in his P proof - where he uses the axiom of choice- he states
"in the proof of theorem V1 no properties of the system P were used other than the following
1) the class of axioms and the riles of inference- note these axioms include reducibility
2) every recursive relation is definable with in the system of P
hence in every formal system which satisfies assumptions 1 and 2 [ which uses system PM] and is w - consistent there exist undecidable propositions ". (ibid, p.28)

CLEARLY GODEL IS MAKING SWEEPING CLAIMS JUST BASED UPON HIS P PROOF . Note from above the version of PM he is using AR was abandoned rejected given up DROPPED So system P is completely artificial and invalid as it uses the invalid axiom of reducibility. Thus his theorem has no value outside this invalid artificial system P

Godel has said that undecidability is not dependent on the
axioms of a system or the special nature of PM and ZF

There is a paradox here
He says every formal system which satisfies assumption 1 and 2 ie
based upon axioms - but he has said undecidablity is independent of axioms

Arjen

Tue 20 May, 2008 09:37 am
@pam69ur,

pam69ur

Tue 20 May, 2008 10:04 am
@Arjen,
Quote:

you say
Quote:

right

dean attacks godel for many reasons
one being
his incompleteness theorem is invalid-regardless of what it says

ie
Quote:

because
he uses in his proof of that theorem an invalid axiom ie AR

godels theorem could say pigs fly -that is irrelavent
the point is he uses in his proof of that theorem an invalid axiom ie AR

Arjen

Tue 20 May, 2008 10:21 am
@pam69ur,
AR meaning in this case?

Aedes

Tue 20 May, 2008 11:22 am
@Arjen,
Arjen wrote:
Dean doesn't even do that. He completely misses the point in his fervor to discredit Godel. He assumes that Godel is using the AR (axiom of reducibility) as an integral part of a proof. But this is patently false, as Godel uses it and other axioms as an example, not as part of a proof.

Arjen

Tue 20 May, 2008 11:31 am
@Aedes,
And there I was thinking of Acces Routers..

Aedes

Tue 20 May, 2008 12:41 pm
@Arjen,
I thought it was either Aortic Regurgitation or Allergic Rhinitis at first.

Aedes

Tue 20 May, 2008 12:43 pm
@pam69ur,
pam69ur wrote:
dean attacks godel for many reasons
one being
his incompleteness theorem is invalid-regardless of what it says he uses in his proof of that theorem an invalid axiom ie AR
Except for the fact that it's not part of his proof in the way you describe.

Furthermore, parts of the AR are apparently valid as used by Godel (according to the entry in the Stanford Encyclopedia):

Quote:
Interestingly, the basic idea underlying Russell's ramified hierarchy of types is a crucial ingredient for Godel's later consistency proof of the continuum hypothesis via its inner model L of constructible sets. Also, as already observed in Godel (1944), a form of AR becomes true in L in the sense that, roughly, an arbitrary propositional function of natural numbers is extensionally equivalent to some function of order alpha, for some countable ordinal alpha.

pam69ur

Tue 20 May, 2008 06:04 pm
@Aedes,
Quote:
Except for the fact that it's not part of his proof in the way you describe.

Furthermore, parts of the AR are apparently valid as used by Godel (according to the entry in the Stanford Encyclopedia):
what he does not use AR
but
you says AS USED BY GODEL

seems you are in contradiction to

go read his words

Quote:
In the proof of Proposition VI the only properties of the system P EMPLOYED were the following:
1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).
2. Every recursive relation is definable in the system P (in the sense of Proposition V)

have a look at formular 40 -which is used in the proof

from the uni of california maths department
Quote:

"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).

and read the paradox godel falls into as dean shows you

you quote standford so read this

Quote:
the standford encyclopdeia of philosophy says of AR

Principia Mathematica (Stanford Encyclopedia of Philosophy)

"many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically

Aedes

Tue 20 May, 2008 06:50 pm
@pam69ur,
My quote was from that very article you cited. It names a special example in which AR was valid as used by Godel.

pam69ur

Tue 20 May, 2008 07:10 pm
@Aedes,
Quote:
My quote was from that very article you cited. It names a special example in which AR was valid as used by Godel.

well quote the parts that show you are wrong

Quote:
"many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically

from editors of godels complete works
Quote:
From Kurt Godels collected works vol 3 p.119

Collected Works - Google Book Search

"the axiom of reducibility is generally regarded as the grossest philosophical expediency