Is the axiom of reducibility invalid

[ladyjane]

Is the axiom of reducibility invalid

Russell Ramsey and Wittgenstein regarded it as illegitimate Russell abandoned this axiom and many believe it is illegitimate and must be not used in mathematics

Ramsey says

Such an axiom has no place in mathematics, and anything which cannot be
proved without using it cannot be regarded as proved at all.

This axiom there is no reason to suppose true; and if it were true, this
would be a happy accident and not a logical necessity, for it is not a
tautology. (THE FOUNDATIONS OF MATHEMATICS* (1925) by F. P. RAMSEY

yet godel uses it in his incompleteness theorem

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

“ [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, 1 by substitution “

following ramsey russell and wittgenstien the Australian philosopher colin leslie dean claims because the axiom of reducibility is invalid then godels incompleteness theorem is invalid as he uses AR

so is the axiom of reducibility invalid

[ladyjane]

well anthem says the axiom is invalid

[Anthem]

From what little I know about it, yes, it’s invalid.

A bigger question is ‘so what?’

[ladyjane]

A bigger question is ‘so what?’

as godels uses it to prove his theorem
his theorem must be invalid as his proof is invalid

[Anthem]

And what about that?

[ladyjane]

And what about that

go read
deans 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

[Anthem]

What does it matter that Goedel might be incorrect? So what? Biggest math fraud in history? I know a good deal of math and I’ve never learned anything explicitly attributed to him. All I knew of him was he was one of Einstein’s friends in IQ, and apparently also in real life.

[ladyjane]

What does it matter that Goedel might be incorrect?

dean uses godel as an example to show his point that all products of human thinking end in meaninglessness
he has proved this for mathematics and science as well- your maths is meaninglessness rubbish
go read his book

http://gamahucherpress.yellowgum.com/books/philosophy/Absurd_math_science4.pdf

The absurdities or meaninglessness of mathematics and science: paradoxes and contradiction in mathematics and science which makes them meaningless, mathematics and science are examples of mythical thought, case study of the meaninglessness of all views

The absurdities or meaninglessness of mathematics and science: paradoxes and contradiction in mathematics and science which makes them meaningless, mathematics and science are examples of mythical thought, case study of the meaninglessness of all views
The absurdities or meaninglessness of mathematics and science: paradoxes and contradiction in mathematics and science which makes them meaningless, mathematics and science are examples of mythical thought, case study of the meaninglessness of all views

[Anthem]

Then why are you here?

If all human thought is meaningless, please, spare us more of yours.

Anyway, what a funny conundrum Mr. Dean gets himself into…

All human thought is meaningless. My paper is a product of human thought. Therefore, my paper is meaningless. But, if my paper is meaningless, then that means its conclusions are too. Therefore, I can say nothing about anything. The end.

[ladyjane]

All human thought is meaningless. My paper is a product of human thought. Therefore, my paper is meaningless. But, if my paper is meaningless, then that means its conclusions are too. Therefore, I can say nothing about anything. The end.

yes dean admitts he ends in meaninglessness
but so does every one else
and he has shown your maths ends in meaninglessness to
thesis and its antithesis
nihilism and its opposite
skepticiam and its oppoiste
all end in meaninglessness
and godel was a fine example to give of the point

yes you are corrects
thend
then end of human arrogance
and all philosophy
and all human babbling

[Anthem]

You miss the point. But anyway, are you trying to bring down all of philosophy and science? You need good grammar to do that, dearie.

By the way, I was looking through old posts you’ve made at other philosophy forums…do you have a life? Seriously you need to get laid. Anyway, other people seem to think the axiom of reducibility is valid as long as it doesn’t create a contradiction. In certain systems maybe it doesn’t create any contradictions. I have no idea, this is not my area of expertise, and obviously it’s not yours either.

[ladyjane]

Anyway, other people seem to think the axiom of reducibility is valid as long as it doesn’t create a contradiction

russell wittgenstien and ramsey said it is invalid and not because it creates a contradiction
Euclid’s fifth postulate ie the axiom of parallel lines is invalid in non-euclidian geometry
ie it is invalid not because it creates contradiction but because it is just invalid

http://en.wikipedia.org/wiki/Parallel_postulate

The main reason that such a proof was so highly sought after was that the fifth postulate isn’t self-evident unlike the other postulates

The independence of the parallel postulate from Euclid’s other axioms was finally demonstrated by Eugenio Beltrami in 1868.

Where Saccheri and Khayyám had attempted to prove Euclid’s fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries which result.

The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and spherical geometry (the obtuse case).

Geometry that is independent of Euclid’s fifth postulate

In geometry, the parallel postulate, also called Euclid’s fifth postulate since it is the fifth postulate in Euclid’s Elements, is a distinctive axiom in what is now called Euclidean geometry. It states that:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Euclidean geometry is the study of geometry that satisfies all of Euclid’s axioms, including the parallel postulate. A geometry where the parallel postulate cannot hold is known as a non-euclidean geometry. Geometry that is independent of Euclid’s fifth postulate