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10835723 No.10835723 [Reply] [Original]

How does ZFC set theory bypass Russell's paradox?

>> No.10836100

>>10835723
you just call the thing of all sets a “category” instead of a set

>> No.10836103

>>10836100
err wait no a “class” my bad

>> No.10836187 [DELETED] 
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10836187

By restricting the comprehension scheme (the naive on by "state a property P and I collect all sets with that property into a set Y for you") with one that is restricted to subsets ("state a property P, give me a set X and I collect all sets FROM THAT SET X with that property into a set Y for you").

The above is a bit too restrictive and so another axiom is affed, replacement, that also allows for searching for parameters fulfilling a property and also guarantees that those are sets (you can think of it as "the image of a function that you can define on X also gives set Y")

>> No.10836194

>>10836103
Amazing

>> No.10836195
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10836195

By restricting the comprehension axiom. The naive one being "State a property P and I collect all sets with that property into a set Y for you".
The restriction is to one that is restricted to subsets. "State a property P, give me a set X and I collect all sets FROM THAT SET X with that property into a set Y for you".

The above is a bit too restrictive and so another axiom is added, the scheme of replacement, that also allows for searching for parameters fulfilling a property and also guarantees that those are sets. You can think of it as "The image of a function that you can define on X also gives set Y"

>> No.10836210
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10836210

>>10836195
>>10835723
Come to think of it, it actually relates to the joke in your image. Of course those books at best claim to contain knowledge about the topic business (a subset X among all topics). It ranges not over all knowledge, but apriori over some subset of it (making that restriction explicit would also ruin most of the joke)

>> No.10836452

PS I'm currently cover a bunch of set theory stuff - not at big sets yet, but getting there..

https://youtu.be/SAstOixrpdA

>> No.10836530

>>10835723
the nonexistence of { x : x is not a member of x }
is a theorem of ZFC

>> No.10836809

>>10835723
It's a two-step program. First, as anon >>10836195 has pointed out, we weaken comprehension. However, if there is a set of all sets, then the restricted and original forms of comprehension are obviously equivalent. So we add regularity on top.