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

Sure but I'm assuming that none of the other users here are doing constructive mathematics. In classical, standard mathematics, what I said is completely true. I'm all for discussing alternative maths and logics (I'm a logical pluralist), but I ignored it for simplicity.

However, as a matter of fact most constructivists nowadays do accept that the set of natural numbers (and any other countably infinite set) is actually infinite.

Apart from Meta, is there anyone willing to defend the notion of words having essential meanings?

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

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.

Apart from Meta, is there anyone willing to defend the notion of words having essential meanings? — Banno

I wouldn't defend such a thing. We've been through this before ...numerous times. In fact, it's you who always argues for "essential meaning" under different names. Our last disagreement, you insisted that a proposition is an essential meaning, while I argued that the meaning must be interpreted by an individual human being; interpretation is subjective.

Apart from Meta, is there anyone willing to defend the notion of words having essential meanings? — Banno

I don't think the meanings of words is exclusively in their usage to the extent that we can ignore what the brain is doing cognitively. There's an cognitive component to meaning which makes the usage possible. I don't know about the essential part, since language is fluid and evolves. But if you tell me a word, it does cause mental activity in me.

Do words have inessential meanings then? Curious how we manage to communicate ideas and concepts with such alacrity.

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

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. 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.

Ehh, I think the issue there is what the brain does cognitively when we think about words is that we've already internalized "this means that because that's how people in my life have used it to mean". 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.

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

Right, using 'red' to denote a color is arbitrary, but the debate is over whether words have an internal meaning in the head of speakers, or the meaning is from the language use in a community, and thus external. Therefore, the beetle in the box we can't talk about, and the impossibility of private language.

That's not the debate I was asking for in the OP. It was whether the words themselves have inherent meaning that straying from is necessarily changing the subject in a ridiculous way.

whether the words themselves have inherent meaning — MindForged

To quote MCU Thor: "All words are made up."

I agree.

Nope~

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're as bad as Banno with your divine proclamation. "I have collected all the natural numbers" and therefore you have collected them. Stipulating that certain numbers fall into a certain set does not make that so. This I thought was the subject being discussed here. Does a "collection" require collecting? You seem to think that a collection can be produced by stipulation. I am asking you to explain how this could be the case, but all you are doing is going around in a circle.

This appears to be your argument. Since I can stipulated the existence of a set, therefore a collection can be produced by stipulation rather than by collecting. 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?

I am asking for something more than this. Can you explain what a collection which does not require collecting actually is? Let's assume, as you suggest. that it is a collection by stipulation. 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?

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

This is the issue. Your insistence that X is a perfect, consistent, definite set, does not make it so. "Set" was defined by "collection". Now we need to determine what makes something a perfect, consistent, definite collection. 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?

So I guess my question to discuss is this: Is there some crucial, essential meaning to words or concepts (whatever) which, if you ever define differently than, you've inherently changed the subject or something?

Effective discourse depends on intersubjectivity: both subjects need to associate the same concept to each word. In practice, we often don't - but we can arrive at a common definitions through discussion. Of course, there are some commonly accepted definitions for many things - even then, there can be different senses of a word.

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.

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"?

Here's a list of mathematical objects that are potential topics of discussion, each corresponding to a more general Cantorian definition of set but which have important distinctions:

Finite collections of finite objects (e.g. {1,3,5}
Infinite collections of finite objects (e.g. the set of real numbers)
fine collections of infinite objects (e.g.: {aleph-0, aleph-1})
infinite collections of infinite objects (e.g. the set of cardinal numbers)

Stipulating that certain numbers fall into a certain set does not make that so. — Metaphysician Undercover

I gave a rule that populates members of a set, I do not literally gather abstract objects and place them somewhere.

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

You're using a colloquial usage of collection, and not even the only colloquial use of that words. People have spoken of the collection of stars in the sky, only a child would think they literally meant they gathered the stars into the sky as opposed to a condition that applies to some class of objects

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

Name some condition which applies to all of them or just create an extensional list of the objects. It's seriously simple.

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

It's not essential properties, it's just whatever properties you declare the set to be constructed based on. The "set of all red things" is, quite obviously, populated by all the objects that have the property of being red.

gave a rule that populates members of a set, I do not literally gather abstract objects and place them somewhere. — 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? What if there are some things which I would say are red but someone else would say are orange? Is it the case that since I am the one who declared the set, these things are members of the set, because I think that they are red? Don't we need an official definition of "red", an essential meaning of the word? Otherwise my declaration that all red things are members of the set of red things is meaningless, and cannot populate a set because there is nothing here to indicate what it means to be red, and no one to judge which objects fulfill that condition called being red.

Name some condition which applies to all of them or just create an extensional list of the objects. It's seriously simple. — MindForged

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. Even then, the judgement could be mistaken. So this requires two things, essential meaning and flawless judgement. A capable human being might be able to produce a flawless judgement, but where do we get the essential meaning from? Who determines exactly what it means to be red, such that one might be able to understand this essential meaning, and judge for that condition without making a mistake.

It was whether the words themselves have inherent meaning... — MindForged

They have inherent meaning, which is assigned by us. And, for our own purposes and convenience, we change those meanings as it suits us. Language belongs to the people, and all that.... :wink: :up:

Yes, we need a degree of consistency, otherwise communication between us would be impossible. But it is also the case that word meanings are not set in stone. They are regularly and continually updated or changed, by us, the owners of language and words. Surely all this is as it should be? :chin:

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

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.

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.

I've just answered this. It has nothing to do with judgements. This is akin to saying that because I haven't looked at all the even numbers I can't declare them to be part of the set of even numbers. So long as they fall under the same rule or property specification then my directly "collecting" them is entirely optional, and really irrelevant.

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

You're just repeating your affirmation without answering to my objection. Isn't the act of applying a rule to objects, and determining whether or not the rule applies to them, an act of collecting the ones which the rule applies to? How could a rule be applied to an object without someone applying it to that object? As much as you might insist that "a set is defined by said rule applying to the objects in question", a rule does not apply itself to an object, it must be applied. Do you not agree that someone must apply the rule, and this involves the process of interpreting it, and judging the objects, as I described in my last post? You can't just declare that all red things are members of the set of red things, and expect therefore there is a set of red things, because someone must interpret and define "red", and judge which things fulfil the conditions in order to produce that set.

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

This is nonsense. What about all those people of mixed ancestry whom some would say are African American, and others would say are not. There is no "set of African Americans" unless "African Americans" has an essential meaning by which every individual might be judged. Otherwise there'd be a number of individuals who may or may not be judged to be members of the set, and the so-called set would not be a set at all due to indefiniteness.

I've just answered this. — MindForged

No, you surely did not answer this. You've gone from saying that there could be a collection without an act of collecting, to saying that a set exists as a set because you stipulate that it does, to now saying that a rule applies to an object without actually being applied to that object. Each time, you claim that something comes into existence, (a collection, a set, the applying of rules to objects), without the act which is implied by the descriptive terms.