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Nobody is going to claim that things like 'Ex Falso Quodlibet' or the Zermelo-Frankel constructions represent natural human logic. They are formalizationsformal dodges that avoid confusing aspects of naive logic on purpose.

One important example: The idea that you cannot have a set of all sets is not reasonable to most humans, naively. It has to be motivated by a need to evade paradoxes, and they eventually accept it, but it clearly contradicts a very natural impulse.

TheWe go so far as to have different set-theories (e.g. Zermelo-Frankel and Godel-Bernays-von-Neumann) that do or do not allow for a universal set, because not having one seems too counter-intuitive to some mathematicians. In the latter, you can have sets that include all the sets, but you still can't have a set of all sets, because Russel's paradox still can't be permitted.

So the already artificial notion of 'too big a collection to be contained', the closest intuition we can usually impart for why there should not be such a set, actually fails to capture what is going on. There is a real gap here between the formalized solution and our vocabulary that humans don't actually seem to be able to accommodate fully.

But in the end, the purpose of formalization is to improve the system in some way. If it captured all the confusing parts and all the potential paradoxes, it would not actually achieve anything.

Nobody is going to claim that things like 'Ex Falso Quodlibet' or the Zermelo-Frankel constructions represent natural human logic. They are formalizations that avoid confusing aspects of naive logic on purpose.

The idea that you cannot have a set of all sets is not reasonable to most humans, naively. It has to be motivated by a need to evade paradoxes, and they eventually accept it, but it clearly contradicts a very natural impulse.

The purpose of formalization is to improve the system in some way. If it captured all the confusing parts and all the potential paradoxes, it would not actually achieve anything.

Nobody is going to claim that things like 'Ex Falso Quodlibet' or the Zermelo-Frankel constructions represent natural human logic. They are formal dodges that avoid confusing aspects of naive logic on purpose.

One important example: The idea that you cannot have a set of all sets is not reasonable to most humans, naively. It has to be motivated by a need to evade paradoxes, and they eventually accept it, but it clearly contradicts a very natural impulse.

We go so far as to have different set-theories (e.g. Zermelo-Frankel and Godel-Bernays-von-Neumann) that do or do not allow for a universal set, because not having one seems too counter-intuitive to some mathematicians. In the latter, you can have sets that include all the sets, but you still can't have a set of all sets, because Russel's paradox still can't be permitted.

So the already artificial notion of 'too big a collection to be contained', the closest intuition we can usually impart for why there should not be such a set, actually fails to capture what is going on. There is a real gap here between the formalized solution and our vocabulary that humans don't actually seem to be able to accommodate fully.

But in the end, the purpose of formalization is to improve the system in some way. If it captured all the confusing parts and all the potential paradoxes, it would not actually achieve anything.

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source | link

Nobody is going to claim that things like 'Ex Falso Quodlibet' or the Zermelo-Frankel constructions represent natural human logic. They are formalizations that avoid confusing aspects of naive logic on purpose.

The idea that you cannot have a set of all sets is not reasonable to most humans, naively. It has to be motivated by a need to evade paradoxes, and they eventually accept it, but it clearly contradicts a very natural impulse.

The purpose of formalization is to improve the system in some way. If it captured all the confusing parts and all the potential paradoxes, it would not actually achieve anything.