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Oct 2, 2007, 2:11:10 AM10/2/07

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The Australian philosopher colin leslie dean argues what Gödel proved he

proved this with flawed and invalid axioms- axioms that either lead to

paradox or ended in paradox –thus showing that Godel’s proof is based upon

a misguided system of axioms and that it is invalid as its axioms are

invalid."

proved this with flawed and invalid axioms- axioms that either lead to

paradox or ended in paradox –thus showing that Godel’s proof is based upon

a misguided system of axioms and that it is invalid as its axioms are

invalid."

GÖDEL’S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS

GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS CASE STUDY

IN THE MEANINGLESSNESS OF ALL VIEWS

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

For those who say godel did not use the axiom of reducibility, axiom of

choice, theory of types -or impredicative definitions

Godel states that he is going to use the system of PM -which uses the

axiom of reducibility, axiom of choice, theory of types

quote

“ before we go into details lets us first sketch the main ideas of the

proof … the formulas of a formal system (we limit ourselves here to the

system PM) …” ((K Godel , On formally undecidable propositions of principia

mathematica and related systems in The undecidable , M, Davis, Raven Press,

1965,pp.-6)

Godel uses the axiom of reducibility and axiom of choice from the PM

Quote

http://www.mrob.com/pub/math/goedel.htm

“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),

and the 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)

AXIOM OF REDUCIBILITY

(1) Godel uses the axiom of reducibility axiom 1V of his system is the

axiom of reducibility “As Godel says “this axiom represents the axiom of

reducibility (comprehension axiom of set theory)” (K Godel , On formally

undecidable propositions of principia mathematica and related systems in

The undecidable , M, Davis, Raven Press, 1965,p.12-13. Godel uses axiom 1V

the axiom of reducibility in his formula 40 where he states “x is a

formula arising from the axiom schema 1V.1 ((K Godel , On formally

undecidable propositions of principia mathematica and related systems in

The undecidable , M, Davis, Raven Press, 1965,p.21

( 2) “As a corollary, the axiom of reducibility was banished as irrelevant

to mathematics ... The axiom has been regarded as re-instating the semantic

paradoxes” - http://mind.oxfordjournals.org/cgi/reprint/107/428/823.pdf

2)“does this mean the paradoxes are reinstated. The answer seems to be

yes and no” - http://fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf )

3) It has been repeatedly pointed out this Axiom obliterates the

distinction according to levels and compromises the vicious-circle

principle in the very specific form stated by Russell. But The philosopher

and logician FrankRamsey (1903-1930) was the first to notice that the

axiom of reducibility in effect collapses the hierarchy of levels, so that

the hierarchy is entirely superfluous in presence of the axiom.

AXIOM OF CHOICE

Godel states he uses the axiom of choice “this allows us to deduce that

even with the aid of the axiom of choice (for all types) … not all

sentences are decidable…” (K Godel , On formally undecidable propositions

of principia mathematica and related systems in The undecidable , M,

Davis, Raven Press, 1965. p.28.)

(“The Axiom of Choice (AC) was formulated about a century ago, and it was

controversial for a few of decades after that; it may be considered the

last great controversy of mathematics…. A few pure mathematicians and many

applied mathematicians (including, e.g., some mathematical physicists) are

uncomfortable with the Axiom of Choice. Although AC simplifies some parts

of mathematics, it also yields some results that are unrelated to, or

perhaps even contrary to, everyday "ordinary" experience; it implies the

existence of some rather bizarre, counterintuitive objects. Perhaps the

most bizarre is the Banach-Tarski Paradox “–

http://www.math.vanderbilt.edu/~schectex/ccc/choice.html)

Godel used impredicative definitions - but these are regarded as an evil

in mathematics

Quote from Godel

“ The solution suggested by Whitehead and Russell, that a proposition

cannot say something about itself , is to drastic... We saw that we can

construct propositions which make statements about themselves,… ((K Godel

, On undecidable propositions of formal mathematical systems in The

undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes,

“it covers ground quite similar to that covered in Godels orgiinal 1931

paper on undecidability,” p.39.)

yet Godels has argued that impredicative definitions destroy mathematics

and

make it false

http://www.friesian.com/goedel/chap-1.htm

Godel uses the theory of types

In Godels second incompleteness theorem he uses the theory of types- but

with out the very axiom of reducibility that was required to make the

theory of types work.

As he states “ we shall depend on the theory of types as our means for

avoiding paradox” (K Godel , On undecidable propositions of formal

mathematical systems in The undecidable , M, Davis, Raven Press, 1965,

p.63 of this work Davis notes, “it covers ground quite similar to that

covered in Godels orgiinal 1931 paper on undecidability,” p. 46.) Russell

propsed the system of types to eliminate the paradoxes from mathematics.

But the theory of types has many problems and complications .One of the

devices Russell used to avoid the paradoxes in his theory of types was to

produce a hierarchy of levels. A big problems with this device , is

that the natural numbers have to be defined for each level and that

creates insuperable difficulties for proofs by inductions on the natural

numbers where it would more convenient to be able to refer to all natural

numbers and not only to all natural numbers of a certain level. This

device makes virtually all mathematics break down.

(http://planetmath.org/encyclopedia/AxiomOfReducibility.html) For example,

when speaking of real numbers system and its completeness, one wishes to

quantify over all predicates of real numbers…, not only of those of a

given level. In order to overcome this, Russell and Whitehead introduced

in PM the so-called axiom of reducibility – but as we have seen this Axiom

obliterates the distinction according to levels and compromises the

vicious-circle principle in the very specific form stated by Russell. But

The philosopher and logician Frank Ramsey (1903-1930) was the first to

notice that the axiom of reducibility in effect collapses the hierarchy of

levels, so that the hierarchy is entirely superfluous in presence of the

axiom. But in the second incompleteness theorem Godel does not use the

very axiom of reducibility Russell had to introduce to avoid the serious

problems with the theory of types.

(http://www.helsinki.fi/filosofia/gts/ramsay.pdf)

(http://planetmath.org/encyclopedia/AxiomOfReducibility.html)

thus we see that as promised godel uses the system of PM but this systems

axioms are invalid thus making godels incompleteness theorems a complete

falure and invalid

Oct 2, 2007, 3:04:16 AM10/2/07

to

elsiemelsi wrote:

> The Australian philosopher colin leslie dean argues what Gödel proved he

> proved this with flawed and invalid axioms

And "philosopher" (ho! ho! ho!) Dean is plain wrong, for the reasons

that have been repeatedly explained here. Repeating elementary

mistakes doesn't make them any less silly mistakes.

All together now .....

Whatever you think about the cut-down-version of PM that Gödel proves

the incompleteness of, Gödel's reasoning applies equally to other

theories that don't involves types, choice or the like. As Gödel says,

loud and clear.

Whatever you think about the cut-down-version of PM that Gödel proves

the incompleteness of, Gödel's reasoning *about* this theory doesn't

draw on type structure or choice or any other fancy feature *in* the

theory. As Gödel says, loud and clear.

Elsiemelsi might like to heed the excellent advice "when in a hole,

stop digging". But why do I think that won't happen!

Oct 2, 2007, 3:34:15 AM10/2/07

to

sorry godel tell us all that he is going to use the axioms of PM -in his

proof

read his lips

proof

read his lips

Oct 2, 2007, 3:51:31 AM10/2/07

to

Peter smith you once said godel did not use the axiom of reducibility i

proved you wrong by pointing out he used it in formula 40

YOU WHERE WRONG HERE

proved you wrong by pointing out he used it in formula 40

YOU WHERE WRONG HERE

Godel uses the axiom of reducibility axiom 1V of his system is the axiom

of reducibility “As Godel says “this axiom represents the axiom of

reducibility (comprehension axiom of set theory)” (K Godel , On formally

undecidable propositions of principia mathematica and related systems in

The undecidable , M, Davis, Raven Press, 1965,p.12-13. Godel uses axiom 1V

the axiom of reducibility in his formula 40 where he states “x is a

formula arising from the axiom schema 1V.1 ((K Godel , On formally

undecidable propositions of principia mathematica and related systems in

The undecidable , M, Davis, Raven Press, 1965,p.21

godel says he uses impredicative statements

“ The solution suggested by Whitehead and Russell, that a proposition

cannot say something about itself , is to drastic... We saw that we can

construct propositions which make statements about themselves,… ((K Godel

, On undecidable propositions of formal mathematical systems in The

undecidable , M, Davis, Raven Press, 1965, p.63 of this work Dvis notes,

“it covers ground quite similar to that covered in Godels orgiinal 1931

paper on undecidability,” p.39.)

when i asked you to show us the propositions that make statments about

themselves you did not

thus showing us you dont know which one they are-if you dont know which

ones they are you cant say he does not use them in his proof

Even though you have written a book on godel on two point your

incompetenace has been pointed out

now you are telling us that even though godel tells us he is going to use

the system of PM he in fact does not

ie so what is this axiom used in formula 40 he tells us is the axiom of

reducibility

Oct 2, 2007, 4:07:30 PM10/2/07

to

Geeze dude, you got an axe to grind over Godel....What, did he steal

your girlfriend in grad school?

your girlfriend in grad school?

On Oct 2, 2:11 am, "elsiemelsi" <cyprin...@nosam.yahoo.com> wrote:

> The Australian philosopher colin leslie dean argues what Gödel proved he

> proved this with flawed and invalid axioms- axioms that either lead to

> paradox or ended in paradox -thus showing that Godel's proof is based upon

> a misguided system of axioms and that it is invalid as its axioms are

> invalid."

>

> GÖDEL'S INCOMPLETENESS THEOREM. ENDS IN ABSURDITY OR MEANINGLESSNESS

> GÖDEL IS A COMPLETE FAILURE AS HE ENDS IN UTTER MEANINGLESSNESS CASE STUDY

> IN THE MEANINGLESSNESS OF ALL VIEWShttp://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf

>

> For those who say godel did not use the axiom of reducibility, axiom of

> choice, theory of types -or impredicative definitions

> Godel states that he is going to use the system of PM -which uses the

> axiom of reducibility, axiom of choice, theory of types

>

> quote

> " before we go into details lets us first sketch the main ideas of the

> proof ... the formulas of a formal system (we limit ourselves here to the

> system PM) ..." ((K Godel , On formally undecidable propositions of principia

> mathematica and related systems in The undecidable , M, Davis, Raven Press,

> 1965,pp.-6)

>

> Godel uses the axiom of reducibility and axiom of choice from the PM

>

> Quotehttp://www.mrob.com/pub/math/goedel.htm

> "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),

> and the 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)

>

> AXIOM OF REDUCIBILITY

> (1) Godel uses the axiom of reducibility axiom 1V of his system is the

> axiom of reducibility "As Godel says "this axiom represents the axiom of

> reducibility (comprehension axiom of set theory)" (K Godel , On formally

> undecidable propositions of principia mathematica and related systems in

> The undecidable , M, Davis, Raven Press, 1965,p.12-13. Godel uses axiom 1V

> the axiom of reducibility in his formula 40 where he states "x is a

> formula arising from the axiom schema 1V.1 ((K Godel , On formally

> undecidable propositions of principia mathematica and related systems in

> The undecidable , M, Davis, Raven Press, 1965,p.21

> ( 2) "As a corollary, the axiom of reducibility was banished as irrelevant

> to mathematics ... The axiom has been regarded as re-instating the semantic

> paradoxes" -http://mind.oxfordjournals.org/cgi/reprint/107/428/823.pdf

> 2)"does this mean the paradoxes are reinstated. The answer seems to be

> yes and no" -http://fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf )

>

> 3) It has been repeatedly pointed out this Axiom obliterates the

> distinction according to levels and compromises the vicious-circle

> principle in the very specific form stated by Russell. But The philosopher

> and logician FrankRamsey (1903-1930) was the first to notice that the

> axiom of reducibility in effect collapses the hierarchy of levels, so that

> the hierarchy is entirely superfluous in presence of the axiom.

>

> AXIOM OF CHOICE

> Godel states he uses the axiom of choice "this allows us to deduce that

> even with the aid of the axiom of choice (for all types) ... not all

> sentences are decidable..." (K Godel , On formally undecidable propositions

> of principia mathematica and related systems in The undecidable , M,

> Davis, Raven Press, 1965. p.28.)

> ("The Axiom of Choice (AC) was formulated about a century ago, and it was

> controversial for a few of decades after that; it may be considered the

> last great controversy of mathematics.... A few pure mathematicians and many

> applied mathematicians (including, e.g., some mathematical physicists) are

> uncomfortable with the Axiom of Choice. Although AC simplifies some parts

> of mathematics, it also yields some results that are unrelated to, or

> perhaps even contrary to, everyday "ordinary" experience; it implies the

> existence of some rather bizarre, counterintuitive objects. Perhaps the

> most bizarre is the Banach-Tarski Paradox "-http://www.math.vanderbilt.edu/~schectex/ccc/choice.html)

>

> Godel used impredicative definitions - but these are regarded as an evil

> in mathematics

>

> Quote from Godel

> " The solution suggested by Whitehead and Russell, that a proposition

> cannot say something about itself , is to drastic... We saw that we can

> construct propositions which make statements about themselves,... ((K Godel

> quantify over all predicates of real numbers..., not only of those of a

> given level. In order to overcome this, Russell and Whitehead introduced

> in PM the so-called axiom of reducibility - but as we have seen this Axiom

Oct 4, 2007, 2:57:39 PM10/4/07

to

"elsiemelsi" <cypr...@nosam.yahoo.com> writes:

> Peter smith you once said godel did not use the axiom of reducibility i

> proved you wrong by pointing out he used it in formula 40

> YOU WHERE WRONG HERE

Is there any reason you provide exactly the same unconvincing

quotations post after post? Do you really believe that you'll change

someone's mind this go-round?

Give us some new material, would you? Perhaps you (or Dean or

whoever) can have a go at Cantor or something.

--

Jesse F. Hughes

"We will run this with the same kind of openness that we've run

Windows." Steve Ballmer, speaking about MS's new ".Net" project.

Oct 4, 2007, 2:55:32 PM10/4/07

to

Peter_Smith <ps...@cam.ac.uk> writes:

> elsiemelsi wrote:

>> The Australian philosopher colin leslie dean argues what Gödel proved he

>> proved this with flawed and invalid axioms

>

> And "philosopher" (ho! ho! ho!) Dean is plain wrong, for the reasons

> that have been repeatedly explained here.

I do wonder why elsiemelsi sells Dean so short. Why refer to him as a

mere philosopher, when he is also a damn fine erotic poet? "Is your

cunt hairy 2?" was a particularly fine example and with very

interesting font choices, too.

Dean is a renaissance man.

--

Jesse F. Hughes

"It is a brilliant proof you, you math haters!!!"

-- James S. Harris

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