If a set contains the sets [1] and [1,2], does it contain the number 1 once or at least twice?

Once, but it doesn’t really matter. If I understand your question.

My question has to do with Russell’s paradox, which rests on the assumption that a set is defined by its members.

In fact, the more I think about it, the more Russell’s paradox appears to be much ado about nothing. The smallest set that contains the set [1] is either the set [1] (if sets are defined by their members) or the set [[1]] (if they are not). So either sets by definition contain themselves, or they by definition do not. If sets are defined by their members, the notion that the set of all squares does not contain itself because it’s not itself a square is bullshit: the only reason it’s not a square is that it’s multiple squares—supposing that there’s more than one square. If the set of all squares cannot contain multiple squares, because multiple squares are not a square (i.e., one square), then the set of all squares can only exist if there’s no more than one square!

I wouldn’t call it much ado about nothing, as it led to Russell’s theory of types, which is very useful to the philosopher. I didn’t realize that was what you were getting at.

However, it should be noted that Russell’s paradox rests upon what i would consider tenuous grounds. The set of all objects that are not part of the set of squares is not a very useful set to begin with. But Russell was on a quest for the basis of mathematics, and so it mattered to him. I think this is an important question for mathematicians, but not for philosophers.

A number is a name derived by an ordered naming convention. When you read the Platonic dialogs, you sometimes find the character Socrates making as distinction between a class and a member of a class, for example “cat” and “a cat”. Now, the terms “once” or “twice” are still numeric naming conventions, only with adjectives added, which means you are in a self-referential rut.

Now, if you really want to read someone who was a complete idiot with class mechanics, try Russell and Whiteheads theory of types. By burying one in a mountain of words, they did not get out of their ability to not contradict themselves, they only made one forget it.

Predication is the inverse function of abstraction. When one is careless with names, one gets easily confused.

If your dealing with a combat designed trench from WW1, and it’s a straight line with a curve in it… let’s use ~ to represent it, then the left side is set 1 and the right side is set 2, then the method you use to exploiting the trench will determine if 1 or 2 can simultaneously exist under two systems of evaluation and usage at the same time. It’s because the two sets can overlap for some functions and yet dramatically differ in other respects. Side 1 and 2 can use the same forward observer and radioman and medics for example. However, the placement of machine guns and lines of fire can differ qualitatively on either side, and the way you use reinforcements or sending men over the top WHILE maintaining a static defense, or repelling a enemy oblique attack can differ significantly in set one or two. Set 1 can be independent with independent values from set 2 while still being inseparable from other aspects and considerations from set two, which can have a unifed personality of a whole of 1 and 2 that some usages of 1 alone doesn’t recognize or concern itself with.

Rommel’s infantry book ‘Attack’ is a good place to consider this from. It’s where I am getting my visualizations. Companies in sections of a battalion’s line, each a unit, while being part of their neighbors and a whole, while each having unique characteristics and mutually overlapping responsibilities. The hierarchical scheme to categorize everything would be daunting but achievable, but hardly enjoyable for a philosopher to do. There is a reason why society left it to Lt. to figure out, and not the regular joe who none the less after a while learns it as well as anyone else can learn it from experience.

for what its worth

from a programming perspective, you can pick either

sometimes it makes sense to treat ‘1’ as instantiable, where there is some property of one instance distinguishable from the property of another instance… (location in computer memory, for example) sometimes it might make sense to think of the 1 in [1] as different from the 1 in [1,2], as evident by… lets say the presence of neighboring numbers

or it can make sense to think of 1 as universal… great from a memory management perspective, it doesn’t make any sense in this regard to have multiple copies of 1, because anything that you ‘should’ be doing with ‘1’ is consistent across instances…as many times as you print ‘1’, or add ‘1’, or subtract ‘1’ etc… the properties of the instance of ‘1’ do not need to be changed

the choice that ‘should’ be made depends on the context in which ‘1’ is defined…

…but that doesn’t mean it’s the choice that was made, as redundant instances of ‘1’ in some system can be operationally equivalent and effectively universal

Don’t you think , Set in Question is not perfect example for Russel’s paradox.

OR is that what Russel’s Paradox is all about .

[b]Imperfect example for Bad example is always a Bad example for Bad example.

OR

Perfect example for Bad example is always Good example for Bad example !![/b]

very confusing thought. .

Now, can there exist a person who can be self respectful without holding respect to others who don’t deserve it.Yes, can be the answer.

In today’s philosophical world , Self respect means holding respect for his own dignity , and Equality means holding respect for all despite of all odds.This kind of definition is what has created paradox in our today’s democracy.

How can your class stay educated, when you give consideration for Uneducated class identity.The educated class will loose its identity called : Educated class.

Moderation is really necessary in definition.

Number theory as an example on set theory,bit confusing always.most of them avoid that habit.(just a suggestion)

The set 0 contains [0] The set 1 contains [1 and 0] The set 2 contains [1,2 and 0] hence the entire number system can be derived from its relation to nothing.

Each number has its own unique set even 0. And that’s how you prove that the number line is correctly axiomatic. Alternatively you could get out more, meet girls and stand an outside chance of getting laid.

Some would say that each number is its own unique set.

Well yes that’s why set theory underpins the whole of number theory and that underpins the whole of Euclidean and non-Euclidean maths.

If each number weren’t its own set then it would not exist as a viable axiom. Which is why 0 is so useful, because it gives us a reference with which to define the whole number line, even though in reality of course nothing does not exist.

In the same way abs |-4| = 4 is a way of describing magnitude. Ie how far am I from 0? Although ironically this defines it using the opposite, the idea that whichever side of the number line they are they have the same magnitude, if you see what I mean.