Mathematicians are in deep shit for 2 reasons

[ladyjane]

he australian philosopher colin leslie dean points out Mathematicians are
in deep shit for 2 reasons

1. skolem discovered a paradox which makes set theory inconsistent

of which freankel and most mathematicians at the time saw

math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Neither have the books yet been closed on the antinomy, nor has
agreementon its significance and possible solution yet been reached." –
(Abraham
Fraenkel)

and that

"most mathematicians followed fraenkels skepticiam

[in regard to skolem relativism attempt at resolution - which is at
present is not accepted]

(John von Neumanns states

“At present we can do no more than note that we have one more reason here
to entertain reservations about set theory and that for the time being no
way of rehabilitating this theory is known.”

of which a few mathematician also agreed

now rather than solving the paradox before moving on with set theory
mathematicians just ignored it and used set theory for all sorts of
proofs
Now mathematicians are in deep shit for there is now so much invested in
set theory that the skolem paradox threatens the very foundations of
mathematics

so some mathematician now try to argue away the paradox by saying it is
not a contradiction
but
skolems paradox want go away it is at present unable to be disproved
and modern maths is buried so much in shit for useing set theory they cant
get out

mathematician have so much invested in godels incompleteness theorem
much maths is reliant on it
but at the time godel wrote his theorem he had no idea of what truth was
as peter smith admitts

Gödel didn’t rely on the notion
of truth

but truth is central to his theorem
as peter smith kindly tellls us

assets.cambridge.org/97805218/57 … xcerpt.pdf

Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim ‘The
sentenceGT itself is unprovable in theory T’. So G T is true if and only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,

you see godel referes to true statement
but Gödel didn’t rely on the notion
of truth

now because Gödel didn’t rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless

this puts mathematicians in deep shit because all the modern idea derived
from godels theorem have no epistemological or mathematical worth for we
dont know what true statement are

en.wikipedia.org/wiki/G%C3%B6del … rems#Fir….

Gödel’s first incompleteness theorem, perhaps the single most celebrated
result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves

basic arithmetical truths, an arithmetical statement that is true, but not
provable in the theory, can be constructed.1 That is, any effectively
generated theory capable of expressing elementary arithmetic cannot be
both consistent and complete.

without a notion of truth we dont know what makes those statements true
thus the theorem is meaningless

and modern mathematics is in deep shit for useing a meaningless theorem

[Mr_Reasonable]

What kind of deep shit?

[IrvingWashington]

Wait, if maths isn’t incomplete thats kinda good isn’t it? Maths isn’t meaningless, it’s complete damnit!

[Mr_Reasonable]

When someone gives me a check for half the money I need for vacation, then I add it to the half that I’ve already got, then that means I can take my vacation. Right?

[IrvingWashington]

Yeah, actually thats precisely the point. All of this stuff, ladyjane, is like saying science is in deep shit because of some issues in the philosophy of science. Whereas, as somebody once said, maybe Richard Feynman, “philosophy of science is about as useful to scientists as ornithology is to birds.” Do working mathematicians really care if Godel’s theorem doesn’t work? Just because you’ve got some issues in the philosophy of mathematics, or at the foundations of mathematics, doesn’t mean mathematicians can’t keep on working away making the world work. Just like because philosophy of science can’t decide what makes a theory scientific or not doesn’t mean scientists can’t continue creating new theories. It’s all, practically, irrelevant.

[Mr_Reasonable]

Yeah there’s no deep shit lady. You’re trying to trick us.

[MagsJ (Surya Loka)]

@Smears

You’re trying to trick us.

…and bore us, at the same time!

[Pandora]

Colin, you are creepy.

[gib]

@ladyjane

without a notion of truth we dont know what makes those statements true
thus the theorem is meaningless

You don’t get Godel’s theorem, Colin.

Godel did use a system to prove GT true, it was just outside T.

[ladyjane]

You don’t get Godel’s theorem, Colin.

Godel did use a system to prove GT true, it was just outside T

peter smith cambridge academic who wrote a book on godel says

quote

Gödel didn’t rely on the notion
of truth

but truth is central to his theorem
as peter smith kindly tellls us

assets.cambridge.org/97805218/57 … xcerpt.pdf

Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim ‘The
sentenceGT itself is unprovable in theory T’. So G T is true if and
only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,

now because Gödel didn’t rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless

[Mr_Reasonable]

I wrote a post that says they’re wrong.

[ladyjane]

I wrote a post that says they’re wrong.

then i sugest you contact peter smith at cambridge and take it up with him

[gib]

@ladyjane

peter smith cambridge academic who wrote a book on godel says

quote

Gödel didn’t rely on the notion
of truth

Peter Smith is full of crap.

[ladyjane]

Peter Smith is full of crap.

then contact him at cambridge and take it up with him

in the meantime
godels theorem is mathematically meaningless
you might like to look at colin leslie deans book outline why mathematically godels theorem is invalid
gamahucherpress.yellowgum.com/bo … GODEL5.pdf

[Mr_Reasonable]

Sorry Dean, but what you’re saying isn’t true. Prove it to me.

[ladyjane]

Sorry Dean, but what you’re saying isn’t true. Prove it to me.

hey dont you take the word of an expert on godel
peter smith says

Gödel didn’t rely on the notion
of truth

if u dont agree go take it up with the expert himself

[Ed3]

Hi All,

Skolem’s “Paradox” is unproven, and, for reasons I have previously pointed out, highly unlikely to be valid.

However, I too disapprove of ZF axiomatic set theory.

The reason for this is that there are at least two axioms which are not axioms at all. They are theorems. Additionally, one of those axioms has the misnomer of “Axiom of Infinity”. This theorem reads that there exists an inductive set. What were they thinking???

I have other personal problems with ZF but I am in the minority in those regards.

John von Neumann is certainly one the best mathematicians of the 20th century as the founder of modern game theory, a genuine mathematical savant, and a holder of one of the positions at the Institute of Advanced Study at Princeton. Though, I don’t think that a lot of people know that he was probably the most immoral person of the 20th century. He strongly advocated a first strike nuclear attack on the Soviet Union and he was in a position of influence in the military.

As far as the Godel theorems are concerned, ironically Peter Smith offered the Turing machine proof to the Godel theorems to which I have on numerous occasions referred.
His proof was previously posted on the web, but it has been removed. I emailed him recently to find out where it was and he returned my email with the name of his book which contains his proof. He seems like a nice guy.

[ladyjane]

As far as the Godel theorems are concerned, ironically Peter Smith offered the Turing machine proof to the Godel theorems to which I have on numerous occasions referred.
His proof was previously posted on the web, but it has been removed. I emailed him recently to find out where to find it and he returned my email with the name of his book which contains his proof. He seems like a nice guy.

note this same peter smith says

Gödel didn’t rely on the notion
of truth

and points out godel referes to statements that are true
now because Gödel didn’t rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless
ALL VERY SIMPLE AND CLEAR

YOU SAY

olem’s “Paradox” is unproven, and, for reasons I have previously pointed out, highly unlikely to be valid

read suber
earlham.edu/~peters/courses/ … htm#coping

A widely held interpretation is that of Thoralf Skolem himself. He believed that LST showed a relativity in some of the fundamental concepts of set theory. Uncountable cardinalities in particular have no meaning apart from specific sets of axioms. A set may be uncountable within a certain formal system and countable when viewed from the standpoint of an unintended model of that system. This means that there simply are no sets whose cardinality is absolutely uncountable. For many, this view guts set theory, arithmetic, and analysis. It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them.

If we want to insist on the non-relativity of our set theoretic notions, and if we hold that our formal systems to date fail fully to capture the secret of the real numbers, then we must choose between the unattractive options (1) that the theory of real numbers is inconsistent, hence has no model, and (2) that the secret of the real numbers cannot be captured by any first-order formal system, i.e. that every attempt will fail either by having no model or by “incurring” a merely countable model. LST puts us to the choice between inconsistency and non-categoricity.

But the ambiguity is stronger still. The plural permissible models are not always isomorphic with one another. The isomorphism of models is a technical concept that we don’t have to explain fully here. Essentially two models are isomorphic if their domains map one another; their elements have the same relations under the functions and predicates defined in the interpretations containing those domains.

If all the models of a system are isomorphic with one another, we call the system categorical. LST proves that systems with uncountable models also have countable models; this means that the domains of the two models have different cardinalities, which is enough to prevent isomorphism. Hence, consistent first-order systems, including systems of arithmetic, are non-categorical.

We might have thought that, even if a vast system of uninterpreted marks on paper were susceptible of two or more coherent interpretations, or even two or more models, at least they would all be “equivalent” or “isomorphic” to each other, in effect using different terms for the same things. But non-categoricity upsets this expectation. Consistent systems will always have non-isomorphic or qualitatively different models.

We don’t even approach univocal reference “at the limit” or asymptotically, by increasing the number of axioms or theorems describing the real numbers until they are infinite in number. We might have thought that, even if a certain vast system of bits sustained non-isomorphic models, we could approach unambiguity (even if we could not reach it) by increasing the size of the system. After all, “10” could symbolize everything from day and night to male and female, and from two to ten; but a string of 1’s and 0’s a light-year in length must at least narrow down the range of possible referents. But this is not so, for LST applies even to infinitely large systems. LST proves in a very particular way that no first-order formal system of any size can specify the reals uniquely. It proves that no description of the real numbers (in a first-order theory) is categorical.

Very Very Serious Incurable Ambiguity: Upward and Downward LST

If the intended model of a first-order theory has a cardinality of 1, then we have to put up with its “shadow” model with a cardinality of 0. But it could be worse. These are only two cardinalities. The range of the ambiguity from this point of view is narrow. Let us say that degree of non-categoricity is 2, since there are only 2 different cardinalities involved.

But it is worse. A variation of LST called the “downward” LST proves that if a first-order theory has a model of any transfinite cardinality, x, then it also has a model of every transfinite cardinal y, when y > x. Since there are infinitely many infinite cardinalities, this means there are first-order theories with arbitrarily many LST shadow models. The degree of non-categoricity can be any countable number.

There is one more blow. A variation of LST called the “upward” LST proves that if a first-order theory has a model of any infinite cardinality, then it has models of any arbitrary infinite cardinality, hence every infinite cardinality. The degree of non-categoricity can be any infinite number.

A variation of upward LST has been proved for first-order theories with identity: if such a theory has a “normal” model of any infinite cardinality, then it has normal models of any, hence every, infinite cardinality.

and suber note that mathematician say it is not a contradiction but cant prove it is not

Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it.

[denali]

and points out godel referes to statements that are true
now because Gödel didn’t rely on the notion
of truth he cant tell us what true statements are
thus his theorem is meaningless
ALL VERY SIMPLE AND CLEAR

I would spend more time trying to take you seriously if you would stop reiterating a fallacious notion of mathematics based on a bastardized confusion of mathematical proof and its interpretation. Godel doesn’t need to tell us a damn thing about what’s true, he just needs to show that formal systems possess with the “power” to represent integer arithmetic possess a certain property, i.e. incompleteness with respect to the provability of statements OR inconsistency. “Truth” hasn’t a thing to do with a proof; hence, no “meaninglessness” creeps in (not that meaningless is a necessary consequence of that anyway, but I’m moving into philosophy here…). I’m not going to bother saying this again before I start blocking you out. 50% of the stuff you come up with is extremely naive; and being a finite being, this doesn’t recommend me looking into the stuff that actually might be significant.

It’s only “simple and clear” if you can’t figure out what a theorem is.

philosophy of science is about as useful to scientists as ornithology is to birds.

This quote is gold, and so applicable here.

[ladyjane]

Godel doesn’t need to tell us a damn thing about what’s true,

sorry if he cant tell us what true is then we dont know what any of this means
and it then is meaningless babble

if i said there are gibbly statements which cant be proven with out telling what makes a gibbly statement
you would say i am talking meaningless nonsence

thus with out telling us what makes a statement true all this is meaningless nonsence
en.wikipedia.org/wiki/G%C3%B6del … s_theorems

Gödel’s first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that:

For any consistent formal, recursively enumerable theory that proves basic [b]arithmetical truths,[/b] an arithmetical [b]statement that is true,[/b] but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. 

Godel did is find a general method that enabled him to take any theory T
strong enough to capture a modest amount of basic arithmetic and
construct
a corresponding arithmetical sentence GT which encodes the claim ‘The
sentenceGT itself is unprovable in theory T’. So G T is true if and only
if T can’t prove it

If we can locate GT

, a Godel sentence for our favourite nicely ax-
iomatized theory of arithmetic T, and can argue that G T is
true-but-unprovable,