Unsolvable contradictory?

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In summary, a computer science student obtained a second-hand item from one of his professors who mentioned it during a Logic lecture. This item is defined as the set M, which contains all elements that do not belong to themselves. This leads to a contradiction, as M can both belong and not belong to itself. This paradox sparked a discussion among mathematicians, as it goes against the logical axiom that something is either true or false. This is known as Russell's paradox and highlights the limitations of naive set theory. To address this issue, mathematicians developed Zermelo-Frankel set theory, also known as ZF, which includes additional axioms to avoid such paradoxes.
  • #1
kuengb
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I got this little thing second-hand from a computer science student. One of his professors mentioned it in the Logic lecture. Define the set M as follows:
[itex]M:=\{x \mid x \notin x\}[/itex]

This brings up a strange contradictory since
[itex]M\in M \Rightarrow M\notin M [/itex]
and
[itex]M\notin M \Rightarrow M\in M [/itex]

As my information is correct there was a big discussion among mathematicians when these lines were written down the first time since it somehow contradicts the logic axiom that something is either true or false. Is that true (or false:smile: )? Does anyone know something about this?
 
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  • #2
it's just russell's paradox that states naive set theory is not the thing you want to use. some things are too big to be sets, or if you like, just saying a set is a collection of objects with a rule for belonging or not belonging is not sufficient. see zermelo frankel set theory aka ZF
 
  • #3


The concept of M, as defined above, is known as Russell's paradox and has been a topic of discussion among mathematicians and logicians for many years. It is indeed a contradictory statement, as you have pointed out. This paradox arises from the fact that the set M is a set of all sets that do not contain themselves, which leads to the question of whether M contains itself or not.

The resolution to this paradox lies in the fact that M is not a well-defined set. This means that it does not follow the standard rules of set theory and cannot be considered as a set in the traditional sense. In fact, it has been shown that this type of self-referential definition leads to logical inconsistencies.

This paradox highlights the importance of carefully defining concepts and sets in mathematics and logic. It also serves as a reminder that not all statements can be proven or disproven, and that some concepts may be beyond our understanding.

To answer your question, the statement about M being either true or false is not entirely accurate in this case. It is more accurate to say that M is neither true nor false, as it does not follow the rules of logic and cannot be treated as a well-defined set. Therefore, it is not possible to assign a truth value to M in this context.

In conclusion, while the concept of M may seem unsolvable and contradictory, it is a valuable example of the limitations of our logical systems and the importance of precise definitions in mathematics and logic.
 

1. What is an unsolvable contradictory?

An unsolvable contradictory is a statement or problem that cannot be resolved or proven to be true or false due to conflicting evidence or logic.

2. How do you identify an unsolvable contradictory?

An unsolvable contradictory can be identified by looking for conflicting information or evidence that cannot be reconciled.

3. Can an unsolvable contradictory be solved?

No, an unsolvable contradictory cannot be solved as it is inherently contradictory and cannot be proven to be true or false.

4. Why are unsolvable contradictories important in science?

Unsolvable contradictories are important in science as they highlight the limits of our current understanding and can lead to new discoveries and advancements in scientific knowledge.

5. How do scientists deal with unsolvable contradictories?

Scientists may attempt to find alternative explanations or theories to reconcile conflicting evidence, or they may acknowledge the unsolvable contradictory and continue to explore other avenues of research.

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