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Post by Eugene 2.0 on Mar 29, 2021 15:34:49 GMT
I browsed enough literature, and still got no answer to this. The most confusing word is class which can be used beyond math language. While sets is what usually being exploited in many math discussions, classes are something to be little caprice and make a little bit trouble, especially for such rookies in math like me.
Ok, generally: the class is a singular propositional function that performs on this function when and only when this function is true, i.e. for elements which satisfy to this function and make it true. Nonetheless such a definition, as some note, blocks a view that two classes are constructed on its elements, i.e. if all the elements of two classes are similar, then those both classes are the same.
Also, if f(x) and g(x) are both singular propositional functions, and xєR, so as f(x) so g(x) are both true when x belongs to R, then f(x) and g(x) are the same classes with different properties: f and g.
May somebody know what's the difference between classes and sets, and how to define what is what?
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Post by jonbain on Mar 30, 2021 12:50:16 GMT
no difference
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Post by Eugene 2.0 on Mar 30, 2021 18:06:04 GMT
This questions has been interesting me because of Bertrund Russell's paradox in many fields. Particularly he discovered a paradox in the set theory and Frege's program of Logicism aka the foundations of mathematics. Here are some of the formulations of the paradox: a) let's suppose there is a set S such as no elements of S belongs to S, i.e. it's the set S that doesn't contain itself. So whatever X is, if X belongs to S, then X doesn't belong to X. Then if to interpret X as S we will have a paradox: S belongs to S, and S doesn't belong to S. b) let's T is a kind of relation between R and S (that occurs) when and only when R doesn't relate to S. So, whatever R and S, each time when it's fair that "R has a relation T to S" is equivalent to "R doesn't have any relation R to S". If to interpret both R and S as T, we'll have: "T has a relation T to T" and this equivalent to "T doesn't have a relation T to T". Russell, and some others, claimed that to avoid such a paradox we should differ classes from sets by saying that it's impossible to take the set of all sets, while it's possible about the classes, because the classes is something to be more natural. (I guess?) |
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Post by jonbain on Mar 30, 2021 19:18:29 GMT
This questions has been interesting me because of Bertrund Russell's paradox in many fields. Particularly he discovered a paradox in the set theory and Frege's program of Logicism aka the foundations of mathematics. Here are some of the formulations of the paradox: a) let's suppose there is a set S such as no elements of S belongs to S, i.e. it's the set S that doesn't contain itself. So whatever X is, if X belongs to S, then X doesn't belong to X. Then if to interpret X as S we will have a paradox: S belongs to S, and S doesn't belong to S. b) let's T is a kind of relation between R and S (that occurs) when and only when R doesn't relate to S. So, whatever R and S, each time when it's fair that "R has a relation T to S" is equivalent to "R doesn't have any relation R to S". If to interpret both R and S as T, we'll have: "T has a relation T to T" and this equivalent to "T doesn't have a relation T to T". Russell, and some others, claimed that to avoid such a paradox we should differ classes from sets by saying that it's impossible to take the set of all sets, while it's possible about the classes, because the classes is something to be more natural. (I guess?) your starting premise is a contradiction, all that follows is therefore sophistry prove this by using a simple REAL example "there is a set S such as no elements of S belongs to S" there is a nation of people, and none of the people that belong to that nation, belong to that nation this is why programming a computer language is vital because it reveals proper logic to the mind of course many programmers are frauds too |
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Post by Eugene 2.0 on Mar 30, 2021 19:50:03 GMT
This questions has been interesting me because of Bertrund Russell's paradox in many fields. Particularly he discovered a paradox in the set theory and Frege's program of Logicism aka the foundations of mathematics. Here are some of the formulations of the paradox: a) let's suppose there is a set S such as no elements of S belongs to S, i.e. it's the set S that doesn't contain itself. So whatever X is, if X belongs to S, then X doesn't belong to X. Then if to interpret X as S we will have a paradox: S belongs to S, and S doesn't belong to S. b) let's T is a kind of relation between R and S (that occurs) when and only when R doesn't relate to S. So, whatever R and S, each time when it's fair that "R has a relation T to S" is equivalent to "R doesn't have any relation R to S". If to interpret both R and S as T, we'll have: "T has a relation T to T" and this equivalent to "T doesn't have a relation T to T". Russell, and some others, claimed that to avoid such a paradox we should differ classes from sets by saying that it's impossible to take the set of all sets, while it's possible about the classes, because the classes is something to be more natural. (I guess?) your starting premise is a contradiction, all that follows is therefore sophistry prove this by using a simple REAL example "there is a set S such as no elements of S belongs to S" there is a nation of people, and none of the people that belong to that nation, belong to that nation this is why programming a computer language is vital because it reveals proper logic to the mind of course many programmers are frauds too I wouldn't be disagree with you, about how real those examples, but they are just mathematical. It's true that probably there's not much examples we might have. The most famous example is the liar paradox. "There is a set S of liars who say that they say "I am a liar" such as no elements belong to this set S". Quite similar formulations are within as the logicism so in the core of the naїve set theory. |
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