Oh, yes the Epimenides paradox. That popular say; "All the Cretans are liars". Yes that. Even though it gives me migraine, I still like it.
Not like you need me to tell you. But let me just define it for content sake.
A Paradox is a statement or situation that may be true but seems impossible or difficult to understand because it contains two opposite facts or characteristics. A statement that seems self-contradictory or absurd but in reality expresses a possible truth. For example; "Less is more", "If I know one thing, it's that I know nothing", "This is the beginning of the end", e.t.c.
“Paradox” is derived from the French word "paradoxe" that means “a statement contrary to common belief or expectation.” The Greek word, "paradoxon" means "contrary to expectation".
The ancient Greeks were well aware that a paradox can take us outside our usual way of thinking. They combined the prefix para- ("beyond" or "outside of") with the verb dokein ("to think"), forming "paradoxos", an adjective meaning "contrary to expectation." Latin speakers used that word as the basis for a noun paradoxum, which English speakers borrowed during the 1500s to create paradox.
The two types of paradox are:
Literal Paradox: This types of paradox is usually found in literature, it is known for being able to disclose a deeper meaning than what is actually said. Like John Donne’s “Holy Sonnet 11” states, “Death, thou shalt die” which is logically impossible to expect death itself to die. and, Julius Caesar by William Shakespeare; “Cowards die many times before their deaths; The valiant never taste of death but once”.
Logical Paradox: This type of paradox are a group of antinomies centered on the notion of self-reference, some of which were known in Classical times, but most of which became particularly prominent in the early decades of last century. The Logical Paradox tends to defy logic and is considered unresolvable. The most popular of this type of paradox is "The Liar" paradox by Eubulides, who lived in the fourth century BC.
The classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. Then again; If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on. (Oh, my HEAD!)
In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.
Another very common paradox is the Epimenides paradox. Epimenides was a Cretan who made the immortal statement: "All Cretans are liars." but Epimenides is himself a Cretan; therefore he is himself a liar. But if he is a liar, what he says is untrue, and consequently, the Cretans are veracious; but Epimenides is a Cretan, and therefore what he says is true; saying the Cretans are liars, Epimenides is himself a liar, and what he says is untrue.
The solution to Epimenides paradox paradox is thus: "Not all Cretans are liars"
If we assume the statement "All Cretans are liars" is false and that Epimenides is lying about all Cretans being liars, then there must exist at least one Cretan who is honest. This does not lead to a contradiction since it is not required that this Cretan be Epimenides. This means that Epimenides can say the false statement that all Cretans are liars while knowing at least one honest Cretan and lying about this particular Cretan. Hence, from the assumption that the statement is false, it does not follow that the statement is true. The mistake made by Thomas Fowler (and many other people) above is to think that the negation of "all Cretans are liars" is "all Cretans are honest" (a paradox) when in fact the negation is "there exists a Cretan who is honest". The Epimenides paradox can be slightly modified as to not allow the kind of solution described above, as it was in the first paradox of Eubulides but instead leading to a non-avoidable self-contradiction.
Next is the Burali-Forti paradox (Yes the best for last). It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.
It goes as thus:
- Let Ω be a set consisting of all ordinal numbers.
- Ω is transitive because for every element x of Ω (which is an ordinal number and can be any ordinal number) and every element y of x (i.e. under the definition of Von Neumann ordinals, for every ordinal number y < x), we have that y is an element of Ω because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.
- Ω is well ordered by the membership relation because all its elements are also well ordered by this relation.
- So, by steps 2 and 3, we have that Ω is an ordinal class and also, by step 1, an ordinal number, because all ordinal classes that are sets are also ordinal numbers.
- This implies that Ω is an element of Ω.
- Under the definition of Von Neumann ordinals, Ω < Ω is the same as Ω being an element of Ω. This latter statement is proven by step 5.
- But no ordinal class is less than itself, including Ω because of step 4 (Ω is an ordinal class), i.e. Ω ≮Ω.
(yup, now my head really hurts)
The insight here is that The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate with each well-ordering an object called its order type in an unspecified way (the order types are the ordinal numbers). The order types (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must have an order type
Ω\Omega . It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that the order type of all ordinal numbers less than a fixed
�\alpha is
�\alpha itself. So the order type of all ordinal numbers less than
Ω\Omega is
Ω\Omega itself. But this means that
Ω\Omega , being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals, but the latter is
Ω\Omega itself by definition. This is a contradiction.
If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidable: the offending proposition that the order type of all ordinal numbers less than a fixed
�\alpha is
�\alpha itself must be true. The collection of von Neumann ordinals, like the collection in the Russell paradox, cannot be a set in any set theory with classical logic. But the collection of order types in New Foundations (defined as equivalence classes of well-orderings under similarity) is actually a set, and the paradox is avoided because the order type of the ordinals less than
Ω\Omega turns out not to be
Ω\Omega .
FYI: Burali-Forti paradox is a 'Set theory'.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Resolutions of the Burali-Forti paradox:
Modern axioms for formal set theory such as ZF and ZFC circumvent this antinomy by not allowing the construction of sets using terms like "all sets with the property 'P' ", as is possible in naive set theory and as is possible with Gottlob Frege's axioms – specifically Basic Law V – in the "Grundgesetze der Arithmetik." Quine's system New Foundations (NF) uses a different solution. Rosser (1942) showed that in the original version of Quine's system "Mathematical Logic" (ML), an extension of New Foundations, it is possible to derive the Burali-Forti paradox, showing that this system was contradictory. Quine's revision of ML following Rosser's discovery does not suffer from this defect, and indeed was subsequently proved equiconsistent with NF by Hao Wang.
Hmm...I cannot deny the fact that we hear paradox everyday; P2P conversations, TV, Radio, e.t.c. But one unique feature of paradox is that you just can't wrap your head around it.
References:
https://www.dictionary.com/browse/paradox
https://literaryterms.net/paradox/
https://www.learngrammar.net/a/paradox-definition-types-and-examples
https://en.wikipedia.org/wiki/Liar_paradox
https://en.wikipedia.org/wiki/Epimenides_paradox
https://en.wikipedia.org/wiki/Burali-Forti_paradox
https://www.jstor.org/stable/185640
https://en.wikipedia.org/wiki/Set_theory#Formalized_set_theory
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