Gödels incompleteness theorem proven invalid

[ladyjane]

The Australian philosopher Colin Leslie Dean points out that t Godels theorems are invalid because: his axioms are invalid; he uses impredicative statements; he miss uses the theory of types; and he ends in at least 3 paradoxes

gamahucherpress.yellowgum.com/bo … GODEL5.pdf

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
By
COLIN LESLIE DEAN
B.SC, B.A, B.LITT (HONS), M.A, B,LITT (HONS), M.A,
M.A (PSYCHOANALYTIC STUDIES), MASTER OF PSYCHOANALYTIC STUDIES, GRAD CERT
(LITERARY STUDIES)
GAMAHUCHER PRESS WEST GEELONG, VICTORIA AUSTRALIA
2007

1)he uses the axiom of reducibility in his incompleteness theorem that it invalid- even
Ramsey says this axiom has no place in mathematics

(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)”
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

( 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” - mind.oxfordjournals.org/cgi/repr … 28/823.pdf
2)“does this mean the paradoxes are reinstated. The answer seems to be
yes and no” - 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.
   (helsinki.fi/filosofia/gts/ramsay.pdf)

4. 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
roved 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

2 he uses impredicative statements

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.)

What Godel understood by “propositions which make statements about
themselves”

is the sense Russell defined them to be

'Whatever involves all of a collection must not be one of the
collection.‘Put otherwise, if to define a collection of objects one must
use the total
collection itself, then the definition is meaningless. This explanation
given by Russell in 1905 was accepted by Poincare’ in 1906, who coined the
term impredicative definition, (Kline’s “Mathematics: The Loss of
Certainty”

Note Ponicare called these self referencing statements impredicative
definitions

texts books on logic tell us self referencing ,statements (petitio
principii) are invalid

Godels has argued that impredicative definitions destroy mathematics and
make it false

friesian.com/goedel/chap-1.htm

Gödel has offered a rather complex analysis of the vicious circle
principle and its devastating effects on classical mathematics culminating
in the conclusion that because it “destroys the derivation of mathematics
from logic, effected by Dedekind and Frege, and a good deal of modern
mathematics itself” he would "consider this rather as a proof that the
vicious circle principle is false than that classical mathematics is
false”

[Mr_Reasonable]

Incompleteness is invalid in what context? Observationally verifiable experimental science? Or metaphysically?

[Daobobo]

The claim appears to be that Godel’s system is logically inconsistant… which I’d be interested to see be explicated in one place…

Can you make the contradiction more explicit for me? I can’t follow the logic in this paper very well. He seems to reference a number of very large documents… but I’d really like to see them broken down into one place.

[ladyjane]

you ask

The claim appears to be that Godel’s system is logically inconsistant… which I’d be interested to see be explicated in one place.

apart from using axioms and statements that are not valid Godel is self contradictory

http://en.wikipedia.org/wiki/Gödel’s_incompleteness_theorems

For any consistent formal, computably 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.

Godel showed that we cannot prove an axiomatic system is consistent.

GODEL IS SELF-CONTRADICTORY
But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done

there are at least 3 paradoxes

3 paradox
There is another paradox in Godels incompleteness theorem
As we have seen undecidability in a formal system is dependent on the system PM but the system PM has undecidability

Godel tells us that among those very wide range of formal systems that have undecidability are to be included those systems which arise from PM by the addition finitely many axioms
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 which cannot be decided on the basis of the axioms. [NOTE IT IS CLEAR] This situation 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”
In other words PM is included in those systems which have undecidablity
Thus we have the paradox that while PM is used to find if a formal system is undecidable it is undecidable itself
i.e.
hence in every formal system which satisfies assumptions 1 and 2 [ from P which uses system PM] and is w - consistent there exist undecidable propositions
In other words the very system which is used to find undecidability is included in the set of undecidable systems

[Mr_Reasonable]

I thought that the general point of Godel’s theorem was that open systems can’t be emprically verified. And that closed ones could only be used to describe and decide if things are consistent with it’s axioms and not whether they’re true w/ a capital T about the universe outside the empirically verifiable closed systems. They’re either open, and not complete because speculation doesn’t hold the proper epistemic status for science, or closed and incomplete because they don’t account for all possible observation. This might be a shitty wording of what I think he basically gets at. Let me know if I’m wrong about something. I do not profess to be a Godel expert.

[Daobobo]

I don’t claim to know much about Godel, but …

Godel’s theorem applies to axiomatic systems. But Godel proof itself isn’t actually an axiomatic system… (it’s just done inside axiomatic systems PM and ZF logic- I believe.) So I don’t think it’s necessarily self-contradictory. However, the systems of PM and ZF, by implication, are necessarily incomplete (or if they were to be complete, which they’re not, they would be inconsistent.)

And claiming ZF logic is inconsistent is a big big claim… particularly the principle of reduction? That’s bold.

philosophyprofessor.com/phil … -axiom.php

Something like this only appears to be false if there’s a incredibly large emphasis on hermenutics… that the whole is greater than the sum of its parts, which not everyone buys into.

Furthermore, on the same topic, on page 12 of the document you’re citing:

fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf

Godel goes into axiomatic set theory… and explains that it’s redundant when it is true. This does not mean that it’s false, merely that if it is true, then it is implied by the logical structure of the mathmatical language.

The same is implied here:
helsinki.fi/filosofia/gts/ramsay.pdf

“I, on the other hand, shall adopt the entire original objective
method which will lead to a satisfactory theory in which no such
axiom is required. The method is to treat functions of functions
as far as possible in the same way as functions of individuals… I
propose similarly to determine the symbols which can be determined
as arguments in ’f (ϕbx)’ not by the manner of their construction, but.
by their meanings.”- Ramsay

The point is that while Godel assumes the reduciblity theorem, he doesn’t even need to - the reducibility theorem is trivially true when it’s true at all. That is the “vicious circle” they’re talking about

[ladyjane]

The point is that while Godel assumes the reduciblity theorem, he doesn’t even need to - the reducibility theorem is trivially true when it’s true at all. That is the “vicious circle” they’re talking about

point is godel used it and it is invalid thus making his proof invalid
as 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

you say

Godel’s theorem applies to axiomatic systems. But Godel proof itself isn’t actually an axiomatic system… (it’s just done inside axiomatic systems PM and ZF logic- I believe.

not true godel proves his point by useing system P which is made of peano axioms and the axioms of PM

"in the proof of theorem 1V 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

you say

And claiming ZF logic is inconsistent is a big big claim

the skolem paradox shows zermelo is inconsistent even Fraenkel)
agreed

en.wikipedia.org/wiki/Skolem’s_paradox

• 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. (John von Neumann)

• Skolem’s work implies “no categorical axiomatisation of set theory (hence geometry, arithmetic [and any other theory with a set-theoretic model]…) seems to exist at all”. (John von Neumann)

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

• I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique. (Skolem)

[Mr_Reasonable]

Ok guys, I’m out of this discussion for now because you both know lots more about it than me. But you should really go on. ILP needs good stuff to read. Sometimes it’s hard to learn here, and that’s what I like to do.

Oh and to both of you…

Welcome to the site!

[Daobobo]

-waves- Thanks for the welcome!

Back to the topic:
You wrote this paper, didn’t you? The style of writing is the same. If you’re going to re-draft it, cite me as a source :-p. And don’t use wikipedia as a source in a paper… especially at an MA level…

No- using a trivially true piece of information doesn’t invalidate the proof. In the context Godel is using it, it ought to be trivially true. (If there’s a direct falsification it would be done in symbolic logic, not in prose, at least that’s how it would be done by most readers of the analyitic tradition.)

For example, take the tautology “I am what I am”, and assume that “I am a man” is true.

If I say:

1. I am a man
2. I am what I am

Therefore, I am a man who is a man.

The second premise is trivially true, and therefore worthless- but using the second premise doesn’t make the conclusion false.

In the same way, Godel’s reducibility theorem is supposed to be irrelevant in the context in which it was being used. This use of the word “supposed to be” brings me to my next point-

Ramseys says there’s no reason to assume that the axiom is true? He’s right. However, I think in Godel, it’s quite probable that all the components are in fact reducible. What element, specifically, does he reduce that is irreducible? Explicate.

The components Godel uses for his proof only means one thing:

1. His proof is limited by the limits of the type of logic he was using. His own proof helps define these limits. (apparently, "he uses peano axioms and the axioms of PM "- Sure, I’ll buy that.)

It does not mean he is immediately contradictory. It only means his proof is limited by the limits of PM logic, and therefore, is self-limiting. Note: Self limiting is not the same as self-contradictory. Ex: I would say a snake that eats its own tail is self-limiting. But it’s not self-contradictory in the same way that a married bachelor is.

Lastly… Here’s wikipedia on the same subject you just sent me on… Skolem’s paradox.

“The “paradox” is viewed by most logicians as something intriguing, but not a paradox in the sense of being a logical contradiction (i.e., a paradox in the same sense as the Banach–Tarski paradox rather than the sense in Russell’s paradox). Timothy Bays has argued in detail that there is nothing in the Löwenheim-Skolem theorem, or even “in the vicinity” of the theorem, that is self-contradictory.”

It’s not a contradiction… therefore it’s just interesting, much like Shrodinger’s Cat is called a paradox. It’s not a contradiction - it’s just an interesting point.

[ladyjane]

you say

“The “paradox” is viewed by most logicians as something intriguing, but not a paradox in the sense of being a logical contradiction (i.e., a paradox in the same sense as the Banach–Tarski paradox rather than the sense in Russell’s paradox). Timothy Bays has argued in detail that there is nothing in the Löwenheim-Skolem theorem, or even “in the vicinity” of the theorem, that is self-contradictory.”

the problem is so much hangs on the skolem paradox not being a paradox but the evidence is it is a paradox

but you missed out the counter claims

However, some philosophers, notably Hilary Putnam and the Oxford philosopher A.W. Moore, have argued that it is in some sense a paradox.

The difficulty lies in the notion of “relativism” that underlies the theorem. Skolem says:

In the axiomatization, “set” does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set M of the domain B is nondenumerable in the sense of the axiomatization; for this means merely that within B there occurs no one-to-one mapping of M onto Z0 (Zermelo’s number sequence). Nevertheless there exists the possibility of numbering all objects in B, and therefore also the elements of M, by means of the positive integers; of course, such an enumeration too is a collection of certain pairs, but this collection is not a “set” (that is, it does not occur in the domain B).

Moore (1985) has argued that if such relativism is to be intelligible at all, it has to be understood within a framework that casts it as a straightforward error. This, he argues, is Skolem’s Paradox

and von Neumann and Fraenkel

• 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. (John von Neumann)

• Skolem’s work implies “no categorical axiomatisation of set theory (hence geometry, arithmetic [and any other theory with a set-theoretic model]…) seems to exist at all”. (John von Neumann)

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

• I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique. (Skolem)

AND BUNCH

Bunch notes

“no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur”

then

The good news is that this strongly paradoxical reading is optional. The bad news is that the obvious alternatives are very ugly. The most common way to avoid the strongly paradoxical reading is to insist that the real numbers have some elusive, essential property not captured by system S. This view is usually associated with a Platonism that permits its proponents to say that the real numbers have certain properties independently of what we are able to say or prove about them.

The problem with this view is that LST proves that if some new and improved S’ had a model, then it too would have a countable model. Hence, no matter what improvements we introduce, either S’ has no model or it does not escape the air of paradox created by LST. (S’ would at least have its own typographical expression as a model, which is countable.

any way you believe what fits best your bias and saves mathematics from meaninglessness- thats what it comes down to probably no more than semantics

you say

Ramseys says there’s no reason to assume that the axiom is true? He’s right

then say
the axiom is trivialy true

No- using a trivially true piece of information doesn’t invalidate the proof

as ramsey says

Such an axiom has no place in mathematics.… 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
,

you talk about tautology but ramsey says AR is no a tautology

( 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” - mind.oxfordjournals.org/cgi/repr … 28/823.pdf
2)“does this mean the paradoxes are reinstated. The answer seems to be yes and no” - fds.oup.com/www.oup.co.uk/pdf/0-19-825075-4.pdf )

you ask

What element, specifically, does he reduce that is irreducible? Explicate.

AR is invalid [as ramsey says even russell dumped it following criticism from wittgenstien] and thus so is godels proof

you say

1. His proof is limited by the limits of the type of logic he was using. His own proof helps define these limits. (apparently, "he uses peano axioms and the axioms of PM "- Sure, I’ll buy that.)

It does not mean he is immediately contradictory. It only means his proof is limited by the limits of PM logic, Self limiting is not the same as self-contradictory.

but godel is in paradox with using PM he tells us PM is undecidable yet uses PM to find undecidable systems

godel states

hence in every formal system which satisfies assumptions 1 and 2 [ from P which uses system PM thus AR] and is w - consistent there exist undecidable propositions

as dean shows

In other words the very system which is used to find undecidability is included in the set of undecidable systems

also godel is in contradiction because as dean argues

But here is a contradiction Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he uses to make the proof must be consistent, but his proof proves that this cannot be done

ALSO
the thread pointed out godel uses impredicative statements which every elementary text book on logic say are invalid- poinicare and russell and many other even godel says they make mathematics false

a lot of mathematics is built on impredicative statements mathematicians ignore that as they give them their results but to a philosopher they are invalid- even if they get results

[ladyjane]

YOU HAVE TO ASK YOURSELF WHY

godel used the 2nd ed of principia mathematica

“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)”

note he says he is going to use the axiom of reducibility
but russell did not use this axiom in the 2nd edition because it was invalid
godel would have known this
SO WHY DID HE STILL USE IT IN HIS PROOF

[Daobobo]

Point 1: I didn’t say the axiom of reduciblity was a tautology. It’s not a tautology, I agree. I just used a tautology as an example of something that is trivially true.

I said the axiom of reduciblity probably happens to be trivially true in the context of Godel. Meaning that it just happens to be the case that the elements Godel uses, in fact, just so happen to be reducible.

And unless there’s a specific counterexample, its usage is irrelevant. I’m asking you to give me the counterexample: What statement, specificially, in Godel’s proof cannot be reduced, but that Godel used the reduciblity axiom on? You have to do this before you can claim that he is wrong.

I said:
“What element, specifically, does he reduce that is irreducible? Explicate.”
You said:
“AR is invalid…”

But that’s not an answer to my question!!! You’re just restating your belief. Where is the example I asked you to provide? The example of an element which is irreducible that Godel reduces?

You’re not making arguments.

So this brings me to my next point:

Point 2: So some smart people say Skolem found a paradox. What’s your point? Some smart people also say it is not a paradox.

A true paradox cannot be read as a nonparadox. A paradox has to be paradoxical all the time. Do you follow? If the text is only paradoxical on some readings, and not others… then I’d just say you’re doing the wrong reading of the text.

You say “any way you believe what fits best your bias and saves mathematics from meaninglessness- thats what it comes down to probably no more than semantics”. NO. It’s not semantics. I’m not just believing my bias.

There are two readings of Godel, one which results in a paradoxical Skolem reading (P) and one which results in a merely interesting Skolem reading (I).

Godel has two readings, P or I. Godel is false if read P and true if read I.
G(P) = F
G(I) = T

By conjunction:

G(PvI)= T

Godel, therefore, is true. This isn’t semantics. It’s Logic 211.

Point 3:Quoting people is not a complete argument.

Additional point:
Your quotes don’t even support your argument completely.

“However, some philosophers, notably Hilary Putnam and the Oxford philosopher A.W. Moore, have argued that it is in some sense a paradox.”

Some sense? So in the other sense, Skolem’s claims are just interesting. See above.

Of course, you’re going to say that I can’t use logic to discuss Godel’s proof, because Godel has made claims about things that are undeterminable in logic.

Which brings me to my 4th point:

Who’s Dean? Why should I care what he says?

In fact, he’s wrong.

There’s a distinction between an undecidable system and an inconsistent system. Dean fails to make this distinction. Godel’s proof didn’t claim that all systems are inconsistent, instead it claimed there was always an element within a system that could not be proven. Totally different.

I’m done. I don’t think you’ve read Godel. I haven’t either, but I mean… seriously, you can’t be arguing this. I don’t think the writer of this initial paper is anywhere near the level of ability in logic they need to be at to understand Godel, much less falsify it. The initial paper cites Wikipedia as a factual source in a purportedly MA level paper. You’ve said about 20 things, quoted hundreds of lines of text, and refer me to sources I don’t have access to. You don’t respond to the arguments as presented.

Ban me if you want, guys, but this has got to be flat out the worst debate I’ve ever been in.

[ladyjane]

And unless there’s a specific counterexample, its usage is irrelevant. I’m asking you to give me the counterexample: What statement, specificially, in Godel’s proof cannot be reduced, but that Godel used the reduciblity axiom on? You have to do this before you can claim that he is wrong.

i am sorry i dont have to if you wont take ramseys word for the invalidity of AR i cant help that

as for skolem paradox you beleve what ever gets you through your day but even Abraham Fraenkel co author of ZF says its a paradox

[ladyjane]

There’s a distinction between an undecidable system and an inconsistent system

as dean points out a paradox

In other words the very system which is used to find undecidability is included in the set of undecidable systems

[ladyjane]

hf.uio.no/ifikk/filosofi/njp … node3.html

The Skolem paradox runs as follows. There is a theorem of ZF–and of all other standard set theories–which says that there is at least one nondenumerable set; more specifically, the set of real numbers, commonly called R, is such a set. At the same time, the Löwenheim-Skolem theorem downwards entails that ZF has a denumerable model. Hence, in a denumerable model it is true that there is a nondenumerable set. Paradox.

This paradox arises if we formalize set theory (and hold that set theory only can or should be formalized) in standard first-order logic: in standard second-order logic (where the second-order variables range over the whole power set of the domain) none of the Löwenheim-Skolem theorems holds. For us to obtain the paradox it is essential that we restrict ourselves to first-order logic or a logic with sufficiently similar features (i.e. a logic with the Löwenheim-Skolem downwards property).

NOTE IT IS A PARADOX UNDER THIS CONDITION

For us to obtain the paradox it is essential that we restrict ourselves to first-order logic or a logic with sufficiently similar features (i.e. a logic with the Löwenheim-Skolem downwards property)

some say the reformulation of ZF offered by skolem dissovles the paradox

Formally solving, or dissolving, the Skolem paradox is quite simple. What the theorem of ZF referred to above literally says is that there is no one-one relation from R into N, the set of natural numbers. Intuitively, we take the existential quantifier of the theorem as ranging over all relations N* R, but within a given model the quantifier ranges only over relations in that model. We must distinguish between the N and the R of a given model and the ``real’’ N and R. This is a useful way of putting it: from outside a particular denumerable model for ZF the set of real numbers of that model is denumerable, but from within that model it is nondenumerable since the conditions of the theorem we have referred to are satisfied: in the model there is no one-one relation from R into N.

but skolem insisted still

This ``formal solution’’ to the paradox is the one presented by Skolem himself; but although Skolem knows the paradox to have this straightforward solution he thinks it has far-reaching philosophical implications. The conclusion Skolem (1922) draws from the paradox is that the notions of set theory are not capturable within any logical system. For example, nondenumerability is a set-theoretic notion, and Skolem thinks that since it is true in denumerable models of axiomatized set theory (for Skolem, set theory axiomatized in a first-order language) that there is a nondenumerable set, the notion of denumerability cannot be captured in axiomatized set theory. In later articles, Skolem draws even more radical conclusions from the Skolem paradox: he claims that the set-theoretic notions are essentially relative; that they have no absolute content for any formal system to capture. Now, since Skolem drew this conclusion he must either have been confused on the relationship between first- and second-order logic or have subscribed to the AFOL thesis. Second-order logic is otherwise a logic in which the set-theoretic notions, for all that has been shown, are capturable

earlham.edu/~peters/courses/ … w-skol.htm

nsofar as this is a paradox it is called Skolem’s paradox. IT is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable? … Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it…[but] On this reading, the Skolem paradox would create a serious contradiction … [but] The good news is that this strongly paradoxical reading is optional. The bad news is that the obvious alternatives are very ugly. The most common way to avoid the strongly paradoxical reading .… [but]… Hence, no matter what improvements we introduce, either S’ has no model or it does not escape the air of paradox created by LST

i judge you see the mathematicians that deny skolem is a paradox still say it is paradoxical with no way to fix zermel without ending in again paradox

everything2.com/index.pl?node_id=1234280

The paradox seems to say more about the failures of formal systems to capture human mathematical thinking than about any weakness in set theory.