I need to go from x = y to saying that for all z, x in x iff z in y. — fishfry
Without sarcasm I say that it gives me a good feeling that reason, intellectual curiosity and communication have won the day finally. — TonesInDeepFreeze
If you mean that it would help for my posts to link to yours, then I'll hope not to forget doing that each time. — TonesInDeepFreeze
My preference regarding you is that you don't gloss my posts and jump to conclusions that I've said something I didn't say but that you think I must have said in you own confusions or lack of familiarity with the concepts or terminology. — TonesInDeepFreeze
I am hopelessly behind composing posts in at least a few threads. Even years behind in threads that I just had to let go because I really should be spending my time on other things more important than posting. — TonesInDeepFreeze
doesn't always explain himself, or is just typing stuff in. — fishfry
Then you tell [the kids on the playground] to line up by height. Now you have an ordered set of kids. Or you tell them to line up in alphabetical order of their last name. Now you have the same set with a different order.
It's an everyday commonplace fact that we can have a set of things in various orders.
Now maybe you are making the point that everything is in SOME order. The kids in the playground could still be ordered by their geographical locations or whatever.
But sets don't have inherent order. — fishfry
But what about that rock? If it's the one that is the crank's head, then it is indeed empty and there is only one ordering of the set of its particles, which is the empty ordering. — TonesInDeepFreeze
The elements of sets have no inherent order. — fishfry
Sets have no meaning whatsoever, other than that they obey the axioms of set theory. — fishfry
This was in response to your denial of the empty set. Tell me exactly -- and be extremely clear and specific, please -- tell me what other rule of set theory is contradicted by the empty set. — fishfry
I have explained to you the ontology of sets many times. They are mathematical abstractions. — fishfry
You know, I am not sure I agree that sets are universals. My understanding is that "fish" is a universal, and the particular tuna that ended up in this particular can of tuna I bought at the store today is a particular instance of the category or class of fish.
Sets are not like that at all.
I did ask you a long time ago to explain what you meant by universals, and you snarked off at me. And now you come back at me claiming that sets are universals. Explain to me what you mean by that.
The concept of a set is a universal. The set of rational numbers is a particular set, of which there is exactly one instance. — fishfry
LOL. Oh man you're crackin' me up. The set of rational numbers most definitely has a cardinality of ℵ0
ℵ
0
, because of Cantor's discovery of a bijection between the rational numbers and the natural numbers. — fishfry
Exactly and well put. I've given the crank that same explanation. He will never understand it, because he wants to not understand it. If he found himself understanding it one day, then he would face the crisis of seeing that he's been confused and in the dark for years and years (decades?). — TonesInDeepFreeze
...the set whose members are all and only the bandmates in the Beatles... — TonesInDeepFreeze
However, we may speak of the set of particles of the rock... — TonesInDeepFreeze
OK, so here we have the issue. Remove the examples of real world objects (schoolkids etc.) as "the elements", and what exactly is an element? — Metaphysician Undercover
It cannot be a particular thing, because it does not obey the law of identity, so it is some sort of universal, an abstraction. — Metaphysician Undercover
But what type of abstraction is it, one which we pretend is a particular? — Metaphysician Undercover
Why is it pretended that these are particulars? Maybe so that the set can be subjected to bijection, and have cardinality. The question then is whether the elements are truly individuals, or just pretend individuals. — Metaphysician Undercover
Isn't that exactly what meaning is, obeyance of some rules? — Metaphysician Undercover
Now, we know what a set is, something which obeys the rules of set theory, the real issue though is what is an element of a set. — Metaphysician Undercover
It seems you are having problems understanding the inherent difficulty of the empty set. — Metaphysician Undercover
I think we'd better have clear agreement on what an element is before we approach that more difficult problem of the empty set. — Metaphysician Undercover
Yes, but you also claim that sets have no meaning. — Metaphysician Undercover
An abstraction with no meaning is contradictory. That's why I can't understand your teachings about set theory. — Metaphysician Undercover
Any abstraction is a universal because its applicable to more than one particular set of circumstances. Whatever it is that any multitude of particulars has in common, is a universal. — Metaphysician Undercover
You appear to be suggesting a third category other than particular and universal, an abstraction which is not a universal. Care to explain? — Metaphysician Undercover
Bijection is a problem, because it requires that the elements are individuals, particulars, — Metaphysician Undercover
which I argue they are not. This is why we need to clear up, and agree upon the ontological status of an "element" before we proceed. — Metaphysician Undercover
I explain in detail. And it's a stupid thing to say that I just type stuff. But in post or even a series of them, I can't fit in an explanation all the way back to the basics of the subject, so if one doesn't have the benefit of a context of adequate knowledge, it's not my fault that I can't supply all that needed context in even several posts. — TonesInDeepFreeze
Now that we got the axiom of extensionality straightened out, it's apropos to get the rest of the dissension worked out.
It starts with these good posts: — TonesInDeepFreeze
I wasn't clear; I didn't mean a URL link; I meant a reply link. Does the link in this post do what you want? — TonesInDeepFreeze
Fairy tale characters are an abstract universal. They are general, and they don't actually exist.
Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.
Fairy tale characters are abstract universals, and Cinderella is an abstract particular.
In your world you don't have any abstraction at all. I think you're taking a point too far. — fishfry
I never said any such thing. I've said the opposite — TonesInDeepFreeze
Actually I am wrong about that TonesInDeepFreeze showed me the error of my ways. All sets satisfy the law of identity. If I have a set X, I may write X = X by way of the law of identity. I do not need the axiom of extensionality for that. Perfectly clear to me now. — fishfry
The law of identity applies to sets. So this line of argument is null and void. — fishfry
Yes, very good. A group is any mathematical structure that obeys the axioms for groups. A set is any mathematical object that obeys the axioms for sets. — fishfry
The relevance of all this to the principles of excluded middle and contradiction is as follows. Peirce wrote that “anything is general in so far as the principle of excluded middle does not apply to it,” e.g., the proposition “Man is mortal,” and that “anything” is indefinite “in so far as the principle of contradiction does not apply to it,” e.g., the proposition “A man whom I could mention seems to be a little conceited” (5.447-8, 1905). If we take Peirce to have meant LEM and LNC, then it appears that he wanted to deny the principle of bivalence (according to which all propositions are true or else false) with regard to universally quantified propositions, and that he meant to claim that existentially quantified propositions are both true and false. But why think that “Man is mortal,” which seems to be straightforwardly true, is neither true nor false? And why think that one and the same proposition, “A man whom I could mention seems to be a little conceited,” is both true and false? Once we see what Peirce meant by “principles of excluded middle and contradiction,” we see that this is not what he was claiming. — Digital companion to C. S. Peirce
Sets are subject to the law of identity. — fishfry
YOU have problems with the empty set. I have no such problems. — fishfry
An element is a set in a set theory without urelements. We say x is an element of y if we can legally write x∈y
∈
. Nothing could be simpler. — fishfry
Cinderella is a particular fairy tale character. She doesn't exist either, but she is an INSTANCE of the category of fairy tale characters.
Fairy tale characters are abstract universals, and Cinderella is an abstract particular. — fishfry
The sophist crank says, "the principle you stated, the elements of a set are not things".
I never said any such thing. I've said the opposite. — TonesInDeepFreeze
Moreover, I did not say that an element of a set cannot be a concrete thing. The set of pencils on my desk has only concrete things as members. — TonesInDeepFreeze
I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same. — TonesInDeepFreeze
I never said that 24 orderings are the same or that they are equal. That would be a ridiculous thing to say. Indeed it is my point that they are not the same. There are 24 different orderings. Of course they are not all the same orderings. The crank is so mentally inept that he can't distinguish between (1) there are 24 different orderings that each have the property of being an ordering of a certain set and (2) all those orderings are the same. — TonesInDeepFreeze
The crank is so mentally deficient that he can't see that it's not a contradiction that "there are 24 orderings of a set" does not imply "all those orderings are the same". — TonesInDeepFreeze
Now like a child with an attention disorder, the crank asks me whether the members of a set are abstractions or concretes, after I explicitly said that they can be either, and I gave explicit examples. Is the crank not able to read? — TonesInDeepFreeze
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