A point must has length > 0 else it does not exist. With this revised definition of a point we can see that the number of points on any line segment is always a finite number rather than Actual Infinity. — Devans99

This is the difference between an intelligible object and a sensible object. The intelligible object is apprehended directly by the intellect, while the sensible object is perceived by the senses. They both exist. So a non-dimensional point, like other mathematical objects, does not need spatial dimension to exist.

I believe the question you were asking in this thread is whether "actually infinite" is a valid intellectual object. If "actually infinite" were proven to be impossible by way of contradiction, or some other logical proof, we'd be obliged to dismiss it as unintelligible, and therefore not a valid object.

Basically the idea is that if you really want to understand the nature of language, two seemingly marginal areas need to be investigated: math and gesture. My intuition is that all three terms - gesture, language, and math - all stand on a continuum of increasing abstraction, and that to understand each, we need to understand the other(s). Or to put it differently, gesture and math stand at opposite ends of a line on which language occupies the centre: they are the limit-points though which language must be understood. — StreetlightX

I don't think that this is the proper way to represent language, as such a continuum, because we need to account for the significant difference of intention, purpose and use, between oral language and written language. Significant differences in intentionality, meaning differences in what language is used for, can be seen to correspond with the two distinct types of language oral and written. Prior to modern civilization, the two can be understood to have been developing individually. We can see that oral language is used mostly for communication, whereas writing is often used as a memory aid.

This is where we ought to be careful not to follow too closely Wittgenstein's denial of private language. Written language is often private, and written language is what gives rise to mathematics, whereas oral language is seldom private, and gives rise to communicative skills. If you read Richard Feynman's "Surely you're joking Mr. Feynman", you'll find where he describes that as a youngster he developed his own system of mathematical symbols, and when he studied physics in university he had to reconcile his symbols with the conventional ones, because the professors could not understand his. Mathematical symbols, which are by nature written, are based in the desire for a private memory aid. Oral words are based in the desire for communication.

So, rather than place gesture and math as opposite ends of a continuum, I would position them as categorically distinct, one for the purpose of communication, the other as a memory aid for private use. I would argue that in prehistoric times, the two were completely distinct. When they came together, to intermingle, such that one person's memory aid could be effectively and efficiently communicated to another, we had the explosive development of modern civilization.

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?

This is what I'm talking about. "Infinity" in the context of limits might mean something else (emphasis on "might"), but calculus still uses multiple levels of infinity as understood in set theory, because we understand calculus through set theory. Hell, even in limits I could just assume the infinit there refers to Aleph-null and the calculation is still going to work. All it needs to mean is that it's larger than whatever I'm working with. And Aleph-null is necessarily larger than any finite number. — MindForged

So the contradiction remains unresolved.

The word 'infinite' is usually only applied to a set, to refer to its cardinality (although it can also be applied to ordinals, but let's not complicate things by worrying about them). — andrewk

That a set could have an infinite cardinality is what I dispute, as contradictory. "Infinite cardinality" contradicts the definition of "set" as a "well-defined" collection. To be "well-defined" in this mathematical context, of a "set", is to have a definite cardinality, and "infinite" means indefinite.

It is only by removing "well-defined" from the mathematical context, and defining the set by a quality (things with the same property for example) rather than by a quantity, that one can say an infinite set is "well-defined". But that is a category error, as mathematical objects are not defined by qualities. For example, the mathematical difference between a circle and a square, is found in the definitions of lengths, angles etc.. That a circle has a curved line rather than the straight lines, of a square, follows as a consequence, a conclusion from the definition. Even the simple "line" is not defined as "straight", or some such quality, it is defined by points and dimension, which are not qualities. A mathematical definition cannot be based in a quality without being unsound.

1. A set is finite if there exists a bijection between it and a natural number. A set is infinite if it is not finite.

2. A set is infinite if there exists a bijection between it and a proper subset of itself. — andrewk

In each of these cases, the thing referred to as an infinite set, ought to be dismissed as not a real set, by failing the criteria of being "well-defined". In the first, cardinality is determined by a bijection with the natural numbers. Without bijection cardinality is indeterminate, the so called "infinite set" is not well-defined, so there is no set. In the second case, the bijection is never complete. The assertion that it is any sort of complete "bijection" is only supported by hidden, undisclosed principles, and is therefore not "well-defined".

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.

This is why you don't quote Wikipedia, especially when it's not a topic you're familiar with. The infinity referred to there is not a number. Limits do not diverge to a number per se (or if it does, it's to some transfinite number), they just increase without bound which meets a colloquial meaning of "infinity". — MindForged

Exactly, this is what the quotation is saying, "infinite" in calculus and algebra is different from "infinite" in set theory. Set theory has transfinite numbers, alephs, but the definition of "infinite" in calculus and algebra is defined in relation to limits.

The point being that there is no clear definition of "infinite" in mathematics as you claim, the definition varies. In geometry for example, a line is endless, infinite. Contrary to your claim, "boundless" is a valid definition of "infinite". Varying definitions inevitably lead to contradiction, like my example of the difference between the way that classical mathematics treats the zero and the negative integers, and the way that "imaginary numbers" treats zero and the negative integers. So your defence, which was nothing more than an appeal to authority is lame and vacuous. Because the various mathematical authorities have various ways of defining the term, we cannot trust that any of them really knows what "infinite" means.

"Infinity," like "existential," is a word with multiple meanings and applications. — tim wood

I first engaged MindForged on this thread because I objected to the claim that "infinite" has "a clear" definition in mathematics. It now seems like we're all in agreement, that it does not. "Infinite" is like "zero", there are various different conventions in mathematics which give these terms different meaning. The result, I argue, is contradiction within mathematics.

Near as I can tell, and charitably at that, you've taken the word out of its usual context, tried to fit it where it doesn't fit, and reported it back as a problem with the underlying concept. What possible use is that? Why would any intelligent person do that? — tim wood

Seems you haven't read my posts. The context we're referring to has been stipulated, "mathematics". If the word has varying definitions within the same context, mathematics, then there's a problem with that discipline.

And mathematicians that I've read are uniform in saying that infinity itself is not itself a number. — tim wood

Try telling this to MIndForged, who steadfastly insisted that infinite is a quantity. I agree that many mathematicians would say that infinite is not a number, but MindForged argued set theory in which infinite sets are allowed to have a cardinality. As such, an infinity has a number, a "transfinite" number.

Before we close this discussion MindForged, remember the reason why I first engaged you on this thread. It was this statement:

The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. — MindForged

I didn't agree with your claim that the definition of infinity in mathematics is clear and unambiguous. So consider this quote from Wikipedia:

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.

Do you still believe that there is one clear definition of "infinity" in mathematics?

If I take the cardinality number aleph-null, the the size of the natural numbers, and remove the element that's the number Zero, the cardinality doesn't change, e.g. — MindForged

That is what is nonsense. There is no such thing as "the size of the natural numbers", unless the natural numbers are not infinite.. If the natural numbers are infinite, they are boundless and therefore cannot have a "size". To say that the natural numbers are infinite, and also that they have a size of aleph-null is just contradiction, because an infinite thing is boundless and cannot have a size. If you assign a size to something you do not consider it to be infinite, (hence the term "transfinite", instead of "infinite"), because to say that a boundless (infinite) thing has a size is contradiction.

I showed the informal proof of it being an infinite set (the one-to-one correspondence argument) and you couldn't even address it. — MindForged

All you have demonstrated is that a so-called infinite set cannot have a definite cardinality. Instead of proving the reality of an infinite set, what this demonstrates is that "infinite set" is self-contradictory.

Seems that way doesn't it?

Did I include anything that shouldn't be there? No! — Banno

Neither did you include anything which should be there. I didn't see any odd numbers. Where's this collection you're referring to? It's easy to speak nonsense. Behold my collection of one hundred pounds of gold nuggets! Want to buy it?

You should have specified what you meant by difference. I assumed you were asking how such sets were any different than a purportedly infinite set, so I gave the difference. If you were talking about the difference as in subtraction, then the answer is infinity. If I subtract any finite number from an infinite number, it's not going to change the cardinality. It's only finite numbers whose cardinality decreases when removing finite numbers of elements. If I take the natural numbers and remove the element Zero, it can still be put into a one-to-one correspondence with the even numbers, so this just provably doesn't change the size of the set. — MindForged

If you knew the precise cardinality of an infinite set, you'd be able to tell me the relationship between the cardinality of a finite set and that of an infinite set. Obviously you know of no such relationship, as subtracting a finite number from an infinite set does not change its cardinality. There is no such relationship. Therefore my suspicions are confirmed, you really do not know the cardinality of an infinite set. Your claim was a hoax. And so your assertion that "infinite set" is not contradictory is just a big hoax.

And as I said, I don't care if it's a set according to your definition. — MindForged

I know you feel this way, that's why I've proceeded to, and succeeded in demonstrating that "infinite set" is contradictory according to your definition, and the one used by mathematicians. Clearly an "infinite set" is not a well-defined collection in any mathematical sense, because the cardinality of such a set is not at all well-defined. Therefore it cannot be a well-defined collection, mathematically, and cannot be a mathematical "set".

You don't seem to understand the issue. You have stated that the cardinality of the set of naturals between 1 and 100 is 100, and that the cardinality of the naturals between 1 and 200 is 200. So I can conclude that the difference between these two cardinalities is 100.

You have also stated that you call the cardinality of the complete set of natural numbers, "aleph-null". If it is true that you know precisely and clearly the cardinality of the complete set of natural numbers, then you ought to be able to tell me the difference between the cardinality of 100 and aleph-null, as well as the difference between the cardinality of 200 and aleph-null, and also show how the difference of 100 which exists between the cardinalities of 100 and 200, is reflected in the difference between the difference between the cardinality of 100 and aleph-null, and the difference between the cardinality of 200 and aleph-null.

Otherwise I will conclude that you were not telling the truth when you asserted that you know precisely the cardinality of the so-called "set" of natural numbers, and also I will conclude that this so-called "set" is not well defined in any mathematical sense, and so is not a "set" at all, under our definition of the term.

"Transfinite" is more of an artefact in mathematical language from times where there was some dispute about the numbers, no mathematician nowadays thinks such numbers are anything but infinite. — MindForged

OK, then I suggest you quit using "transfinite", because you are only introducing ambiguity. Why then did you say: "The cardinality of the set of natural numbers is the transfinite number aleph-null." If "transfinite" is just an artefact, and transfinites are really infinite, then infinite sets really have no distinct cardinality, they are simply "infinite".

I'm somewhat confused about the relevance to infinite sets. The set of natural numbers between 1 and 100 (call it "A") has a cardinality of 100. The set of natural numbers between 1 and 200 (call it "B") has a cardinality of 200. Set A cannot be put into a one-to-one correspondence with B since the cardinality of B is greater than that of A.

Neither A nor B can be put into a function with a proper subset of themselves (again, any subset will run out of numbers to pair with the parent set) and are therefore finite; try to match up 100 things with 200 things and you'll be able to see that's it's impossible to pair up one thing in one set with exactly one thing in the other set for all the members. This is exactly the difference between finite and infinite sets. Infinite sets can have parts of the set have the same cardinality as the entire set because you never can "run out" of members to pair up. That was the point of my earlier example with the Natural numbers and the Even numbers. — MindForged

Your claim was that an infinite set has a precise and known cardinality. If this is the case then you can show me the relationship between the cardinality of an infinite set, and those other two finite sets, and how the difference between the cardinality of the two finite sets is expressed in the two relationships between each finite set, and the infinite set.

So go ahead, give it a try, demonstrate to me that you know precisely, the cardinality of an infinite set. Show me the difference in cardinality between the set of natural numbers between 1 and 100, and the set of all natural numbers, and the difference in cardinality between the set of natural numbers between 1 and 200, and the set of all natural numbers. Then show me how the difference in cardinality between the set of natural numbers between 1 and 100, and the set of natural numbers between 1 and 200, is expressed in the difference between these two relationships.

Are you going to make your divine declaration "I have collected all of the odd numbers", and therefore you have collected them?

I've noticed that no one has reached the end of pi yet, why do you think that you can reach the end of the odd numbers?

Care to prove that?

The cardinality of the set of natural numbers is the transfinite number aleph-null. — MindForged

OK, now we're getting somewhere. You were not talking about "infinite", or "infinity", you were talking about transfinite numbers. Why didn't you say so in the first place? This thread appears to be concerned with the "actually infinite". Transfinite numbers are something completely different, and I guess that's what caused the confusion, you did not properly differentiate between these two, nor did you let me know that you were talking about transfinite numbers rather than infinity.

This is demonstrated by simply looking at the mathematical means of determining the cardinality of a set, namely when we known sets have the same size as other sets. Any set which can be put into a one-to-one correspondence with a proper subset (meaning sharing some of its members but not having all of them of itself) is what defines an infinite set. — MindForged

Wait, now you're claiming that this demonstration which you produced earlier shows that a transfinite number is infinite. Care to explain, because I really do not see any demonstration of that.

We know the exact cardinality of the set of natural numbers, real numbers (etc.) — MindForged

Come on, give me a break. If you're not joking about this, then how gullible do you think I am? If you actually believe that you know the exact cardinality of the set of natural numbers, then show me the precise relationship between the cardinality of the following sets. The set of natural numbers between 1 and 100, the set of all natural numbers, and the set of natural numbers between 1 and 200,

Let's just collect all the odd numbers and ignore him. — Banno

That would be a never ending task, so you'd never have that collection. Perhaps you like to think that the impossible is possible Banno, but that's contradiction.

An arbitrary quantity of elements referred to as a whole and which gain membership in said whole by means of sharing a common property we pick out or by being subject to the same stipulated rule. — MindForged

That's fine, but if there's an infinite amount of such an element, I don't see how this qualifies as " "quantity". Don't you know that "quantity" is defined as a measurable property of something, or the number of something" Infinite is neither a measurable property nor is it a number, so you really haven't given me a definition which allows for infinity.

Really, I wish you would give more thought to what you say MindForged. How could "infinite" signify a quantity? Any such so-called "quantity" would clearly be indefinite and therefore not a quantity at all.

On one hand, I know that the professional mathematicians do not define sets in a way which assumes they must be finite collections. On the other hand, I was running into a wall where the insistence was that the very meaning of "collection" entails finitude. — MindForged

Try looking at it this way Mindforged. Let's assume that a collection may be infinite and then describe what it means to be a collection, keeping in mind that a collection may be infinite. I can think of many examples of what a collection cannot be under this stipulation, ( it cannot for instance, be a bunch of things collected together in a group, because this implies finitude), but I cannot imagine what being "a collection" could actually mean with this particular criterion. Can you help me, by describing what a collection would actually be, if we allow that collections may be infinite.

Um, no. Literally you're entire argument is that "collection" and "set" are necessarily finite because of the definition your use. Your argument is without any force because it's indisputable that mathematicians don't use your definitions of these terms. It's entirely besides the point to try and claim they're incorrect for doing so by the means you're doing it. It's like saying "marriage" is definitionally between men and women and so the idea of gay marriage is a contradiction. — MindForged

Actually, I define terms like "set" "collection", "object", and "infinite", in the ways normally accepted in philosophy. It's your argument which doesn't hold any force because it's nothing but an infinite regress of defining terms to support your conclusion (begging the question). You argue for the coherency of "infinite set', and you do this by claiming that any of the descriptive words used to define "set", allow for infinity. So things like "objects", and "collections", which are known by philosophers to be necessarily finite, because by their very definitions these things are necessarily bounded, you assert may by infinite, in order to support your claim of an infinite set.

But are you prepared to provide real support for your claim? Tell me which of the following you disagree with, and back up your disagreement with solid principles. A "set" is a well defined collection. A collection which has an unknown cardinality is not "well-defined", in any mathematical sense. If a collection were infinite its cardinality would necessarily be unknown. Therefore an infinite collection cannot be well defined in any mathematical sense, and cannot be a "set".

Is it ever reasonable to believe in something that is inconceivable? What would one actually be believing in? — Relativist

That's the problem with those mathematicians who believe in contradictory things like "infinite sets". They believe in these "inconceivable" concepts because they find them useful. However, it is always unreasonable to believe in a conception which is inconceivable, so whatever use these people find those concepts to be, it is really self-deception.

Really, there's no evidence any of standard mathematics entails a contradiction, provided you actually use the definitions mathematicians actually use. — MindForged

I've explained to you how "infinite set" is clearly contradictory. Also it's quite obvious that the waythe concept of "imaginary numbers" treats the negative integers contradicts conventional mathematics. You can rationalize these contradictions all you want, trying to explain them away, but that is just a symptom of denial, it doesn't actually make the contradictions not contradictory.

That's an ideal sphere. Nowhere did I mention an ideal sphere. — tim wood

An ideal sphere is what is necessary to produce the infinity you referred to. Without the ideal sphere there is no such infinity, and your example is useless, so you really were referring to an ideal sphere. There is not an infinity of possible paths on a sphere-like object because each path is different and the full extent of possible paths may be exhausted.

Ok, integers greater than two; that's a distinct cardinal And also the irrationals are a distinct cardinal. Now it's time for you to start making sense. Can you do that? Make sense or make your case? — tim wood

Actually it's you who is not making sense. Infinite cardinality is nonsense. Cardinality is a measurement and the infinite cannot be measured. You, like MindForged, suffer the symptoms of denial, rationalizing to cover up the true fact that "infinite set" is contradictory.

It can't be all that obvious, since so many mathematicians and scientists have failed to observe the contradiction, and some of them have been reputed to be quite bright. — andrewk

Honestly, I don't think mathematicians care about contradiction within they're work. What is important is that the prescribed methods work. Mathematicians, and scientists, follow like sheep, the methods taught to them, without questioning the underlying principles, that is there discipline. Without that discipline there would be no such thing as mathematics or science. It's not an issue of how bright they are. Philosophers are wont to question these things, but it takes a major shift in strategy for a philosopher to tell a mathematician what to do.

Do you believe that "infinite" refers to an indefiniteness, and that "set" refers to a definiteness. If so, you should see the contradiction. Do you not think that it is contradictory to say that the same energy moves in the form of a wave, and in the form of a particle, at the same time (wave/particle duality)? Do you not think that the way that classical mathematics treats zero and the negative integers is contradicted by the way that "imaginary numbers" treats these? These, amongst others, are contradictions which are employed by very bright people in their daily practise.

We must all be grateful that this thread has finally come to light, so that the said mathematicians and scientists can be freed from the delusion under which they have been labouring. — andrewk

The problem is that there have developed philosophies such as dialectical materialism, called dialetheism, which support the acceptance of contradiction. So in principle, the use of contradiction is justified. From the angle of philosophy, many philosophers who recognize the existence of such contradictions, instead of trying to root them out, and replace them with acceptable principles, instead produce epistemologies which justify, and allow for the acceptance of contradiction.

Really MU? There's no such thing as a sphere? — tim wood

That's right, "sphere" is conceptual only. Take any object which appears to you to be a sphere, and examine it under a high power microscope and you will see that it really is nota sphere.

Infinite sets very obviously contradictory? How about the set of numbers greater than two? The set of irrational numbers between zero and one? — tim wood

Correct again, such named "sets" cannot really be sets by way of contradiction. Do you agree that a set is a "well defined" collection of objects, and accordingly is an object itself? An object has definite boundaries and cannot be infinite. Objects such as "numbers greater than two", and "irrational numbers numbers between zero and one" are not well defined because the cardinality is unknown. You cannot have a "well-defined" set in which the cardinality is an unknown factor.

I'm still not sure I understand your meaning of the term 'judgement'. Could you perhaps give an example of judgement, and then in contrast, an example of proposition? — Samuel Lacrampe

Let's say "the earth is round" is a proposition. If I claim that what this proposition means, is that the planet we live on, is the shape of a circle, then this is my judgement of what "the earth is round" means. Once I have judged what "the earth is round" means, I can make a further judgement as to whether that proposition is true or false.

E.g. the Earth was round before earthling subjects existed. Thus the judgement "the Earth is round" is objective. — Samuel Lacrampe

As far as I know, only subjects make judgements. So a judgement cannot be objective (of the object), because a judgement is always the property of a subject. One makes a judgement concerning an object, but that judgement is not properly "of the object" because it is property of the subject, it is "of the subject". And this is evident from the fact that such judgements vary in accuracy.

The surface of a sphere is a finite quantity. It is also unbounded. It's reasonable to plot a path on the surface of a sphere. We do it all the time. What would you say the sum of the distances of the possible paths on the surface of a sphere is? And the surface of the sphere is just exactly a collection of those paths. I guess it's aleph-c and maybe greater, but not less. — tim wood

The problem is that spheres are only conceptual, just like infinities. So the question is, does a concept, like "infinity", have actual existence.

Please cite some. I always did like a good contradiction, and if you're right then very likely there are not just a lot of them, but an infinite number of them. — tim wood

I'm starting with "infinite set" which is very obviously contradictory. A "set" is limited, restricted, by the defining terms of the set. "Infinite" means unrestricted, unbounded, or unlimited. Therefore "infinite set" is very clearly contradictory. Once you grasp this obvious contradiction, then I might be able to show you some other, more complex contradictions within mathematics, but if you cannot see the contradiction here, in this very simple example, I don't see any point in giving any other examples.

My initial points were that infinity isn't inherently off the table when talking about reality, as the OP and another user were arguing that infinity is a contradictory concept (which is just flatly untrue); so if anything in reality is infinite or not is an empirical matter, there's no strictly logical argument against it being instantiated. Anyway, sorry if I was unclear! — MindForged

If you're referring to me, saying that someone is arguing that infinity is a contradictory concept, then this is wrong, it's not what I've been arguing. What I have been arguing is that "infinite set" is a contradictory concept.

"Collection" does not refer the process of collecting things. If I talk about the collection of stars in the sky and I call that a set, no one thinks I've literally gathered the stars in the sky. They readily understand I'm mean that there's a condition each of those objects share (that is, "being in the sky") and that I'm grouping them into a collection. — MindForged

You haven't actually grouped those stars into a collection though. That collection is completely imaginary, in the mind only . That's the point of the thread, such a collection is not an "actual" collection it's an imaginary collection.

Now the problem with an infinite collection is that it is impossible to actually collect an infinite number of things. So not only is that collection imaginary, but it is impossible due to contradiction. It's very easy to name impossible collections. The difficult thing is to determine whether such a collection is actually possible or not.

They refer to well defined groups of objects related by some common property, condition or rule and are referred to as a whole as a "Schmet" because OBVIOUSLY that's not a "set", supposedly. — MindForged

The problem with your analogy here is that you are concentrating on the defined "common property", and claiming that this constitutes a "group", "a whole", but neglecting what in reality are the criteria for "a group", or "a whole". You seem to think that you can define an object (a whole) into existence. Your "Schmet" has existence as a group, a whole, an object, because you say that it does.

That's fine, I have no problems with that, as that's the way that concepts exist as objects, they exist as definitions. So we can give intelligible objects existence in that way. The problem is that with "infinite set" you are attempting to create a contradictory concept, and this must be disallowed as unacceptable. To have "a group" or "a whole" which is infinite is contradictory, so the existence of that concept, as an intelligible object, must be disallowed as actually unintelligible.

And unlike you, my definitions are actually used by virtually all modern mathematicians. — MindForged

That's an appeal to authority. Do you think that just because mathematicians accept and use this concept, therefore it is not contradictory. You're only fooling yourself, as modern mathematics is full of contradiction. In mathematics there is no real principle by which an axiom is judged as acceptable or not. They are generally accepted on pragmatic principles. So when they are well disguised, as is the case with "infinite set", contradictions are accepted by mathematicians quite readily. It seems that mathematicians do not subject their axioms to the same scrutiny that philosophers do, and that's how such mistakes occur.

Objects quantified over are not assumed to exist. — MindForged

If it's an object then it exists. To be an object is to exist. There is no non-existent object, that's contradiction. You're just trying to find a semantic loophole, but you are really digging yourself deeper into a hole of contradiction.

A set is a well-defined collection, often characterized by sharing some property in common or holding to some specified rule. — MindForged

You agree with me that a set is a collection, so we have no disagreement over the definition of a set. Our disagreement is over what constitutes a collection. I think that things must be collected to be a collection. You seem to think that things which are by some principle "collectible" are a collection. Clearly you are wrong, a collection must be collected, and collectible things do not constitute a collection. "Collection" often refers to the act of collecting. So it is quite clear that a collection does not exist until the things are collected in the act of collection. That is why an infinite set is utter nonsense, because it is absolutely impossible to collect an infinity of things.

"My" definition (in actuality, the mathematical definition) of sets are clear and they allow for infinity. — MindForged

Absolutely not. Until you demonstrate how an infinity of things may be collected into a collection, your definition of sets does not allow for an infinite set. You are in denial, refusing to understand the words of your own definition.

You are confusing determining if an object belongs to a set with whether or not the object does in fact belong to a set. — MindForged

Obviously, if the object has not yet been collected it does not belong to the collection. What could constitute the act of collecting other than determining that the object belongs to the set? It's nonsense for you to think that an object could belong to a collection without having been collected. Therefore there is no such thing as an object belonging to a set without having been determined as belonging to that set. It's your nonsensical way of thinking which is making you believe that belonging to a set is something other than having been determined as belonging to that set.

You are making up definitions of sets, I'm literally using the standard mathematical definition which in fact captures many of our intuitions about collections and does so without any contradictions. — MindForged

No, clearly we agree on the definition of a set, it is a collection. But nothing can belong to a collection without the act of collecting (collection), by which the thing is collected. And it's also very clear that an infinity of things cannot be collected. Therefore, according to the definition of 'set" which we both agree on, an infinite set is absolutely impossible.

Putting non-existent things in a set in no way commits one to their existence (goodbye existential import). The set of Harry Potter characters is only populated by non-existent things. — MindForged

That's false. To put something into a set is to assign it some sort of existence. If Harry Potter characters are non-existent then the set of Harry Potter characters is an empty set. If you assert that the set of Harry Potter characters is not empty then you assert the existence of Harry Potter characters.

It's question begging because no one is using your definition of infinity which is defined in a way so as to preclude being actual, nor does the definition of a set preclude it from being infinite. — MindForged

Yes, as I explained, the definition of "set" precludes the possibility of an infinite set. A set is a collection. It is impossible to collect an infinite number of things Therefore an infinite set is impossible. Some people, like you, just like to deny the obvious. That means that you are in denial.

There's no understanding "the" definition because there is no one definition. — MindForged

Well, so much for your "clear" definition of infinity in mathematics then. You seemed to be so certain of that point. I'm glad you now see that you were wrong about it.

Incorrect. If two things hare a property they share it whether or not I judge them to. Two red objects share the property of being red even if no one exists to recognize such. So to speak of sets having members based on a shared property in no way requires a judgement to make it so. — MindForged

As I said, that something has the property of being red, is a judgement. Whether an object is red or not requires a definition of "red", and a judgement as to whether the thing fulfills the criteria of being red. That definition, and that judgement, are necessary because "having the property of being red" is a relation between the universal "red", and the particular object which is said to be red. Otherwise "red" might be defined in any way, and any object might be red. Or do you think that "red" has determinate meaning without a definition?

You're doing it again. It's not a mechanistic process that occurs over time nor is it necessarily done by an agent. Sets don't exist in the mind. The "set of numbers greater than 500 trillion but smaller than 1 quadrillion" is simply too large to be conceptualized in the mind, but it's obviously a perfectly legitimate set. — MindForged

It seems like you're redefining "set' to suit your purpose. No longer does "set" refer to a collection, it refers to things which are collectible, potentially collected. That's the issue of the thread, things which can be potentially collected together do not make an actual collection. And in the case of infinity, an infinite number cannot even be potentially collected together, because the definition of infinity makes collecting an infinite number impossible. So all you are doing with your "infinite set" is asserting that the impossible is possible. That's nonsense.

The "set of moments after the present moment" is unbounded but no one gets up in arms about defining such a collection of moments as a set. — MindForged

There is no such "set". The moments after the present moment have not yet come into existence so you cannot collect them into a set, nor can they be members of "a set" in any way or fashion, as they are non-existent. You are claiming to have a set of things which do not exist, but that's impossible so it's pure fiction, nonsense.

Again, what is the non-question begging argument for this? — MindForged

It's clearly not a matter of begging the question. It's a matter of understanding the definition of "set", and understanding the definition of "infinite", and realizing that it is impossible to have an infinite set. These two are incompatible, by definition, so talking about infinite sets is contradictory nonsense. Of course we all know that because of the many paradoxes which are known to arise from the assumption of infinite sets, but some like you, choose to ignore this obvious fact.

They aren't "collected" in a mechanistic process, i.e. going out and declaring "You go in this set" and such. Just sharing a property is enough, and it happens to be perfectly compatible with there being infinite collections. — MindForged

I think you are mistaken here. That something has a particular property is a judgement. The thing is a particular the property is a universal. Therefore if "sharing a property" is what is required to be a member of a set, then a judgement is required in order that things be of the same set. So the declaration "you go in this set" is exactly what is required in order that a thing be a member of a particular set.

You seem to either believe that sets just naturally exist without ever being created by human minds, or else that things automatically jump up and join any set which they are supposed to be a member of, without being counted into that set. So either the green grass is naturally a member of the set of green things without that set ever being created by a human mind, or else the green grass jumps into the set, of its own power, as soon as "the set of green things" is named by a human being. Both of these, I tell you are nonsense.

Nonsense based on what argument? — MindForged

Take any set of a series of natural numbers, 1 - 10, 1 - 20, 2 -40, whatever. If that set has two or more members, then the subset of the even numbers has less members than the original set. This is always the case, and by inductive reason we can state such a law, that this is always the case. The infinite set is specifically designed for no reason other than to break this law, therefore it is unreasonable, nonsense.

As it is an unbounded (open) set, it is not truly a "set", as a collection of objects, it is a boundless collection which is not a collection at all. A collection, or "set" means that the members are collected together in a group. If the collecting is not complete, then the described collection (set) does not exist. To call it a collection, or set, is contradictory nonsense.

How this is rambling, I don't know. It's literally just lining things up. — MindForged

To me you have just demonstrated the logical deficiency which the concept of "infinite" introduces into set theory. You have demonstrated that the set of natural numbers is equivalent (in the sense of having the same number of members) as the set of even numbers. That's nonsense, and that's what the concept of infinity introduces into mathematics, nonsense.

It's nonsense because it's a totally useless piece of trivia. Infinite sets have the same number of members as other infinite sets ... a nonsense number ... an infinite number.

A set is infinite if it's members can put into a one-to-one correspondence with a proper subset of itself. So we know the natural numbers are infinite because, for example, there's a function from a set to a proper subset (read: non-identical) of itself like the even numbers. For every natural number, you're always able to pair it up with an even number and there's no point at which one of the subset cannot be supplied to pair off with the members of the set of naturals.

That's pretty clear, it's exactly the same reason I can, without knowing the exact number of people in an audience, know that if every seat is occupied, then there's no empty seats (each seat can be paired off with a person). — MindForged

I see no clear definition of infinity here, just a rambling description of a particular type of set, which you call an infinite set. That description doesn't tell me what it means to be infinite, it tells me what it means to be an infinite set.

The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. — MindForged

Care to provide that "clear" mathematical definition of infinity?

2. The universe has a boundary. In that case, as Aristotle asked, what happens if we go to the boundary and poke a spear through it? — andrewk

What's wrong with the idea that the universe has a boundary? That seems to be a natural and intuitive idea, the universe being a thing, and things have boundaries. You wouldn't be able to poke a spear through it though because that would be to violate the boundary, put a part of the universe beyond its own boundary. That's impossible.

The problem though is the nature of the boundary. Boundaries are not what they seem to be. We sense boundaries, see them for example, as the edge of objects. But in reality objects overlap through things described by fields, like gravity. So the boundary which is seen as the boundary of spatial extension, is not the true boundary because things really extend beyond this apparent boundary.

We find a true boundary in the nature of time, as the boundary between past and future. This is the boundary of physical existence. What you do at the present has everlasting existence in the past, as what you have done. So you may stir the pot of the past, your actions having influence on what has occurred, but you cannot poke your spear into the future as that is impossible.

Language, thought, and communication, are just like morality, success requires effort. To think that being moral comes naturally to a human being is to think a false thought. And the same is true of communication, to think that communication comes naturally to a human being is to think a false thought. This is because communication, like morality, requires acting correctly and that requires effort.

The question is where do we put an end to this line of thought? Acting correctly, using language correctly, thinking correctly, don't come naturally they require effort. But if thought can go both ways, toward the correct, or toward the incorrect, then thought itself must come naturally, and it is only correct thought which is what requires effort. That is, unless thinking itself is a good, correct act, which requires effort to avoid non-thinking, then we must look even deeper to put an end to this line of thought.

My opinion, for what it's worth, is that the shoutbox is generally full of posts with low quality philosophical content, and if left on the front page will attract new members with a taste for such.

Now while Kant is full of ambiguity on this point, one essential discovery was that there were experiences where this guarantee could be broken: experiences where thought did not conform of its objects, becoming untethered to them and generating 'transcendental illusions'; these illusions were generated internally by thought itself, precisely to the degree that were not anchored in an object which would lend these thoughts the force of necessity that would relate them to something concrete in the world. The notion of transcendental stupidity is simply an extension and renovation of this Kantian idea, one oriented not toward truth, as in Kant, but toward significance: a question of relating thought less to an 'object' than to a problem. So yeah, the question of metaphysics here is almost entirely irrelevant. — StreetlightX

It all looks like metaphysics to me. You have described a separation, a distinction between the object of thought and the good ("significance", what is valued in relation to intention)). If this separation is upheld then the object of thought (intelligible object) becomes an illusion, as transcendental stupidity sets in.

In your terminology, is a judgement that same as a proposition, that is, a sentence that can be either true or false? — Samuel Lacrampe

No. a statement, sentence, or a proposition (as a type of statement), is a collection of words which needs to be interpreted. And then, what is taken as the meaning is judged as true or false. That judgement is subjective, attributable to the subject..

I admit that in this example, it is hard to judge if it is closer to a square or a circle. But the challenge here is due to the challenging example and not due to judgements always being subjective. Here is another easier example. In this drawing, is E closer to D or G? The objectively right answer is "E is closer to G than D". This statement is clearly objective. — Samuel Lacrampe

If agreement between us. concerning our judgements, makes our judgements "objective", then you are using a different meaning of "objective" than I, which I defined as "of the object". Agreement on judgements about the object doesn't make the agreement "of the object".

You cannot mean that, can you? Since only subjects can judge, all judgements are carried out by subjects, including the judgement that "2+2=3 is wrong". Are you saying that this judgement is therefore subjective? — Samuel Lacrampe

Yes, that is what I mean. I define subjective as of the subject, and objective as of the objective. Judgement is something that subjects do therefore all judgements are subjective.

Again, objectivity implies the possibility for either right or wrong, where as subjectivity cannot be neither right nor wrong. And for a given shape, it is either right or wrong that it is closer to a circle or a square. — Samuel Lacrampe

I don't see this as an acceptable definition of "objective". That a given shape is closer to a square or closer to a circle is a judgement, and therefore subjective. And that this judgement is a right or wrong judgement is a further judgement, and is therefore subjective.

Consider an octagon. It is not a square and it is not a circle. To say that it is either right or wrong that it is closer to being a square than to being a circle is nonsense, because to judge it as closer to one or the other, is also nonsensical. Judgements concern what is and is not, so we can judge it as being close to a square and also as being close to a circle. But each is a different scale of judgement so we would have to refer to a third scale to say whether it's closer to being a circle than it is to being a square. All such judgements are dependent on the scale which is chosen.

I think that the idea with the microdosing of acid is that you are not really supposed to notice effects. If depression is the problem, then there is an issue with "noticeable" effects, and one issue is the likelihood of mania. Whenever there are "noticeable" effects of medication, there is a question of the criteria whereby the effects are judged as negative or positive.