Well, I'm not sure that's quite the case under intuitionism, where infinity is only a potential, and the only natural numbers that exist are the ones which have been stated, written down or computed. — Marchesk
It's not indefinite, the members of the "set of natural numbers" never increases or decreases, it is exactly what it is and has always been. — MindForged
Apart from Meta, is there anyone willing to defend the notion of words having essential meanings? — Banno
Apart from Meta, is there anyone willing to defend the notion of words having essential meanings? — Banno
Until you demonstrate that "set of natural numbers" is not self-contradictory, such claims are nonsense. And to say that something infinite is not indefinite clearly is contradictory. So carry on with the nonsense. — Metaphysician Undercover
Nothing about "red" inherently makes the mind conjure up a particular range of colors, just ask a pacific islander who doesn't speak a lick of English. — MindForged
whether the words themselves have inherent meaning — MindForged
Infinite sets are not indefinite, why do you keep saying this as if it's an obvious fact that I've conceded? Every object that's a natural number will fall into that set once I've stipulated an intensional definition of that set. — MindForged
You haven't once shown it to be contradictory, you just fall back on saying that anytime you're challenged to defend your position. The set of numbers equal to or greater than zero is a perfect consistent, definitely set. If you don't understand what the members of that set are, then that's because you don't understand the definition. — MindForged
Case in point. When you say "schmets can have infinite members" do you mean "schmets can have infinitely many members" or do you mean "schmets can have infinity as a member"?Okay, so sets are by definition, conceptually finite collections so any attempt to define or talk about infinite collections is incoherent on pain of contradiction. But let's create a new concept and a word to refer to it: "Schmets". Schmets are just like sets, except some schmets can have infinite members provided they are defined appropriately. So now the question is, are there sets or are there schmets? Well, since sets are, by hypothesis, necessarily finite, they aren't very useful in mathematics since nearly every standard and non-standard maths uses infinity in some form or fashion (ultrafinitism doesn't look very promising). So it seems mathematicians are using Schmets and so we can just dispense with using sets in maths.
Stipulating that certain numbers fall into a certain set does not make that so. — Metaphysician Undercover
But the original point I made, on the other thread, is that you cannot stipulate the existence of a set, because "set" is defined by "collection", and collection requires collecting. Do you see the circularity of your begging the question? — Metaphysician Undercover
So I see a number of objects, and I stipulate, those objects are a collection. What makes them a real collection rather than just an imaginary collection? — Metaphysician Undercover
If it is not the act of collecting them into a group, and demonstrating that they have been collected, then what is it? Would you argue that sharing essential properties is what makes them a collection? Who would determine which properties are essential, and which are not? — Metaphysician Undercover
gave a rule that populates members of a set, I do not literally gather abstract objects and place them somewhere. — MindForged
Name some condition which applies to all of them or just create an extensional list of the objects. It's seriously simple. — MindForged
It was whether the words themselves have inherent meaning... — MindForged
OK, but the question was, how does your giving a rule populate a set? Do you apprehend the issue. Suppose I decree, as you suggest, that all red things are members of a set, the set of red things. How does this declaration make certain things members of that set, while excluding other things? — Metaphysician Undercover
You think that it is simple to name some condition which applies to all of a number of objects? This could only work if the condition which is named had an essential meaning, an official definition, allowing that all the things could be judged according to that definition.
A set is defined by said rule applying to the objects in question. It has nothing to do with the process of collecting things. The properties in question are possessed by those objects whether or not I accept they do or if I call it something else. Being in the set of African Americans doesn't depend on anything to do what I call them. It doesn't necessitate an essential meaning, just a conventional one which people roughly agree picks out a certain class of objects. — MindForged
Being in the set of African Americans doesn't depend on anything to do what I call them. It doesn't necessitate an essential meaning, just a conventional one which people roughly agree picks out a certain class of objects. — MindForged
I've just answered this. — MindForged
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