The discussion began with Cantors Paradox, that the collection of all sets is not itself a set.
If we use wolfram's encyclopedic mathematics reference as you did, we find that Cantor's paradox is defined as follows:
"The set of all sets is its own power set. Therefore, the cardinal number of the set of all sets must be bigger than itself." (
source)
However, if we go to Wikipedia, as is typical when it comes to technical nuances in academic fields we can do better:
"While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians award this distinction to Bertrand Russell, who defined a similar theorem in 1899 or 1901." (
source)
However, to actually understand what's going on need to realize that the notion of a universal set which was paradoxical wasn't from Cantor but Russell:
"[Cantor's] argument, when applied to Russells universal set as well, generates what is called Cantors paradox"
Hinkis, A. (2013).
Proofs of the Cantor-Bernstein Theorem: A Mathematical Excursion (
Science Networks. Historical Studies, Vol. 45). Birkhäuser.
For Russell's work on the subject, see
here.
So linked is Russell's work with Cantor's paradox that some, such as Doets, "go so far as to claim that Russell's paradox
is Cantor's paradox" (Anellis, I. H. (1991). "The First Russell Paradox" in T. Drucker (Ed.)
Perspectives on the History of Mathematical Logic (pp. 33-46). Birkhäuser. The important point is that Cantor's paradox, whatever one's views on whether it actually is his, doesn't state "that the collection of all sets is not itself a set". Rather, it is essential to understand that the paradox stems from Russell's "applying Cantor's proof to the universal set
U, and that Cantor's "proof" concerned the existence of a largest cardinal. (from Myrvold's "Pierce on Cantor's Paradox and the Continuum").
These are some other ways that demonstrate the bounds of logic
Yes. "If you're hungry, there's food on the table" is a far simpler one, but the utter failure of ~60 years of generative grammar to develop a computational model capable of actually generating "grammar" (or grammatically correct speech/language) is a far more devastating extension.
Other difficulties are posed by quantum mechanics, as the logic of QM (and most formulations from von Neumann onward of quantum logic) violate one or both (and arguably necessarily both) of perhaps the most central components of classical (Aristotelian) logic: the excluded middle and the law of non-contradiction. Unlike Fuzzy logic or other many-valued logics, which can potentially be written off as epistemic rather than ontological in nature, "quantum logic" is the basis for perhaps the most successful scientific theory of all time.
And I was unaware that Gödels undecidability and incompleteness theorems and that the validity of the Axiom of Choice and the Continuum Hypothesis cannot be determined in any Set Theory we presently have were blatantly obvious.
This is what I said was blatantly obvious:
Gödel and others have shown that there are limitations to the use of logic
Even if we use the term colloquially, it is clearly limited. One cannot derive Shakespeare's works or rely on algorithms to generate De Gas' creations.
Gödels 1937 paper linked the two together, showing that they are compatible with the axioms of ZF. Cohens 1963 paper also linked them, showing that they are independent of the axioms of ZF. It is not difficult to see that they are intertwined.
"Kurt Gödel proved in 1938 that the General Continuum Hypothesis and the Axiom of Choice are consistent with the usual (Zermelo-Fraenkel) axioms of set theory. Twenty-five years later, Paul Cohen established that the negations of the Continuum Hypothesis and the Axiom of Choice are also consistent with these axioms.
Taken together, these results tell us that the Continuum Hypothesis and the Axiom of Choice are independent of the Zermelo-Fraenkel axioms."
(
source; emphasis added; italics in original).
Gödel and Cohen proved opposite things, which is why we know the continuum hypothesis is undecidable. The link between the AC & GCH is "set theory", and it is not difficult to see that anything relating to set theory is intertwined with other things relating to set theory. This is in no way the same as stating:
Both of the latter turn out to be the same problem.
The generalized continuum hypothesis (CH) is that the cardinality of every infinite set is a Cantor Aleph with each successive Aleph being the powerset of the previous Aleph.
"The generalized continuum hypothesis is the proposition that for no infinite cardinal
a is there a cardinal
b such that
."
Potter, M. (2004).
Set Theory and its Philosophy: A Critical Introduction. OUP.
You'll recognize the final term above as equivalent to the expression of the power set of any set
a. As a corollary of Sierpinski's 1924 proof, we get:
This corollary is the logical equivalent of the generalized continuum hypothesis, but the proof whence the corollary comes shows that the GCH
entails the AC, not that the two are the same. Also, as Cohen showed that on the assumption that both the continuum hypothesis and the axiom of choice are false, ZF is consistent, the distinctions between both the AC & (G)CH and these together and the ZF axioms are extremely important. Hence ZF vs. ZFC.
Determining whether CH is the case requires AC.
It is impossible to determine whether "CH" is the case. Hence "undecidable".
Cantors statement that a set is the form of a possible thought still holds.
1) Ignoring the translation and modal construal in German, the above expresses a modal proposition which can't be expressed in ZF, ZFC, or any "classical" logic. It requires modal logic.
2) Nothing about the above "still holds". Sets aren't defined this way and Cantor's set theory was abandoned for a reason.
But it is not possible to coherently think the concept because it contains a contradiction.
1) There is no empirical, mathematical, or any other basis for asserting that we can't coherently "think [a] concept" if it is paradoxical. It is far more likely for formal/mathematical conclusions to be paradoxical to "thought" than it is formal paradoxes be inconceivable.
2) Paradoxes abound in formal systems, including logics (see the bit about quantum logic above). It is because of our ability to coherently conceive of that which is paradoxical that we were able to develop modern cosmology, quantum mechanics, particle physics, etc.
3) Were we incapable of thinking coherently about contradictory propositions than we could kiss mathematics and logic goodbye. Proof by contradiction requires us to coherently think of contradictory "concepts" or statements in order to realize that we have demonstrated some theorem or similar result to be true by showing that it contradicts another. Only by first "thinking" the proof in such a way that we can simultaneously conceive of it as coherent and as entailing contradictions can it be a proof.
Cantors Absolute is not a thinkable thought.
If you wish to examine what cognition involves and what we can or can't "think", mathematics isn't the place to go. Check out research in the cognitive sciences.
The Tao that can be spoken is not the eternal Tao.
Prove it.
By applying a genuinely coherent, definite attribute we do indeed define a set. If something leads to a contradiction, such as Cantors paradox or Russells antinomy, it is not coherent in any genuine sense.
The above asserts that a set is a set. The 'genuinely coherent, definite attribute" part is extraneous and misleading. The attribute "not a set" is necessarily as definite, coherent, etc., is "is a set". Hence, we can define a set as "not a set" were we to use your definition. The issue is vastly more nuanced.
It is a semantic construct
The reason for words like "formal", "mathematical", etc., is to indicate that we are working within a system which is entirely syntactical. Logic, set theory, etc., are
designed to lack semantics. Hence the practice derivations in any intro or advanced mathematical/symbolic logic textbook that don't bother trying to give meaning to the symbols. One of the most important goals of mathematics since the early modern period has been the reduction of mathematical "objects" to meaningless symbols that we manipulate within some system. This is why calculus, linear algebra, statistics, etc., can be used in quantum field theory and sociology: the notations, systems, and other formal aspects of mathematical subjects are
given semantic content through application. The philosophy of mathematics is another matter, but it is also irrelevant here as we aren't even dealing with whether mathematical objects are solely epistemological or are ontological.