It's not a label for two different concepts, it's two ways of defining instances of the same concept. Intensionally defined sets are not incomplete, and they are certainly not undefined. I think I really am pulling out of this now.

it's two ways of defining instances of the same concept. — MindForged

But the two ways produce two different concepts which maths tries to then treat in the same way via fudges like transfinite numbers.

They don't produce two different concepts. The extensionally defined set {3,5} and the intensionally defined set "odd numbers which are greater than 1 but less than 7" define exactly the same set. It's the same concept, a well-defined collection. Bye~

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. — Metaphysician Undercover

That's an ideal sphere. Nowhere did I mention an ideal sphere. Now why don't you address the point. You want a collected "infinity"? Take any sphere-like object. The number of possible paths on the sphere is not less than aleph-c.

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. — Metaphysician Undercover

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?

They don't produce two different concepts. — MindForged

But they do: the ‘set of bananas’ and {banana 1, banana 2, ...}. The first is not fully defined.

Or the ‘set of rationals’ - not defined and is undefinable

I think you're overlooking that as the numbers get intellectually bigger, they get physically smaller so it gets easier and easier to fit them inside the brackets until it takes no effort at all.

Or the ‘set of rationals’ - not defined and is undefinable — Devans99

Sez you On what authority? The set of rationals is easily defined. Look, these posts of yours are merely displays of ignorance. Is that what you're about? If you want to learn and find out about these things. Youtube is an excellent entry source for information about them. You have an opportunity to learn. Take it.

"Well, inconceivable is a subjective assessment, it's a far cry from being provably impossible. If you just want to say that you don't believe the past can be infinite because an infinity of elapsed time seems inconceivable to you, you are welcome to it. Does an absolute beginning of time, such that right at the beginning there is no before, seem more conceivable to you?"

It's straightforward to conceive of a beginning of time: an initial state. It maps onto a real number line, with a completed past that contiues to be appended, , a continuously changing present moment, and a potentially infinite future. There are cosmological models consistent with this conception.

Yes, conceivability is subjective, but conceptions can be intersubjectively shared, analyzed, and discussed. Belief is similarly subjective. When there are two mutually exclusive possibilities, one of which is conceivable and the other is not, which should be considered more likely to be true?

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

I do not rule out the possibility of an infinite past, but for the reasons I just discussed, it seems more reasonable to believe it is finite.

The set of rationals is easily defined — tim wood

How do you completely define something that is larger than any given finite number:

- You lack infinite paper to write out a definition
- You lack infinite mental power to visualise an infinite set
- R or ‘the set of rationals’ is merely the selection criteria for the set not a full description of the set itself
- Actually Infinite sets are not fully describable so are NOT DEFINED

Apparently, the concept of human face is undefined, merely because noone ever listed every instance of the class represented by the word "human face".

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.

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.

Yes, conceivability is subjective, but conceptions can be intersubjectively shared, analyzed, and discussed. — Relativist

Conceivability, the way you are using the word, is nothing more than an attitude, an intuition, a gut feeling. While different individuals can hold such attitudes in common, it is not the sort of concept that can be described and transmitted by a rational argument. I, for instance, do not find the beginning of time to be any easier to conceive than an infinite past, and I doubt that you could do much to change my attitude.

But then I do not make much of such attitudes. If one holds time to be an objective feature of the physical world, rather than a subjective attitude, then what does it matter if an infinite or a finite past does not sit well with one's intuitions? We are animals with a lifespan of a few tens of years; we can hardly get to grips with timespans of thousands, let alone billions of years. If we were to trust our intuitions on this, most of us would have had to be Young-Earth creationists, right? But then what are we to do with the powerful intuition that at any moment there must always be before? Or, for those having trouble conceiving of an infinite space, what are they to do with Lucretius and his spear? Intuitions just aren't a good guide to the truth in this case.

I, for instance, do not find the beginning of time to be any easier to conceive than an infinite past, — SophistiCat

An in infinite past leads to logical contradictions so time must have a start:

- the measure problem. Everything that can happen will happen, an infinite number of times
- this breaks probability; everything becomes equally likely
- Reductio ad absurdum, time is finate and has a start

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. — Metaphysician Undercover

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.

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".

Actually, I define terms like "set" "collection", "object", and "infinite", in the ways normally accepted in philosophy. — Metaphysician Undercover

I forget, are mathematicians doing math or philosophy?

Worse, most philosophers who actually study maths too will employ the mathematical definitions of these. It's part and parcel of just using standard mathematics and classical logic.

Again, you keep talking about "their very definition" and pretending you don't simply means "the definitions I happen to use". Words are defined by their users, they don't have free floating meanings so your entire approach is fundamentally ridiculous.

1) A "set" is a well defined collection.
2) A collection which has an unknown cardinality is not "well-defined", in any mathematical sense.
3) If a collection were infinite its cardinality would necessarily be unknown.
4) Therefore an infinite collection cannot be well defined in any mathematical sense, and cannot be a "set". — Metaphysician Undercover

#2 is just an outright misrepresentation. Infinite sets do not have an "unknown cardinality". The cardinality of the set of natural numbers is the transfinite number aleph-null. 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. The natural numbers have this property. Take the even numbers (which are half the naturals) and you can pair them up with the naturals and never fail to establish a pair, e.g.

0 - 0
1 - 2
2 - 4
3 - 6
etc.

No finite sets can have this property, as eventually you'll find they run out of numbers to put in a function. And as Cantor showed using his diagonal argument, on pain of contradiction we know the set of natural numbers is a smaller infinity than the set of real numbers as the reals are uncountably infinite.

Unless you can argue that the notion of a one-to-one correspondence is logically incoherent, you have no recourse against these well established mathematical tools. The idea you tried to pass off earlier that arguments from authority are off the table was ridiculous. Arguments from authority are an informal fallacy, meaning they are only invalid in specific cases. Namely, when the source is not actually an authority on the subject. In this case, my authority is quite literally nearly the entirety of the mathematicians.

#3 is incorrect for the previously stated reason. We know the exact cardinality of the set of natural numbers, real numbers (etc.) And those are infinite sets by the mathematical definitions of these terms. #4 just falls out as false because the premises used to establish it were false.

Unless you can argue that the notion of a one-to-one correspondence is logically incoherent — MindForged

Well one-one correspondence is logically flawed: There are the same number of natural numbers as square numbers? Surely a paradox - a sign we are dealing with a logically flawed concept.

We are comparing two undefined things and we get nonsense.

We are comparing two undefined things and we get nonsense. — Devans99

Is the concept of human face undefined? If not, how do you define it?

- The concept of a human face can be defined
- The ‘set of all human faces’ is a finite list so in principle is also definable (as a set)
- the description ‘set of all human faces’ is not a complete definition of the set (so is undefined)

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,

- The concept of a human face can be defined
- The ‘set of all human faces’ is a finite list so in principle is also definable
- the description ,set of all human faces’ is not a complete definition of the set — Devans99

The set of all human faces is a finite one? Can you show it to me?

If we were to get the name of everyone in the world as write it down we would have the set of human faces.

Forget about human faces. Let's take a simpler concept -- the concept of natural number. Is the concept of natural number undefined? If not, does that mean the set of all natural numbers is a finite one and that you can show it to me? If so, will you please show it to me? Or can you just answer this simple question: what is the largest natural number?

There is no largest natural number X because X+1>X.

The natural numbers are defined, but not as a set, just the description of how to populate a set.

The set of natural numbers is undefined.

"Conceivability, the way you are using the word, is nothing more than an attitude, an intuition, a gut feeling. "
No, I outlined a mapping of a possible finite past, and pointed out there are cosmological models based on a finite past (Hawking, Carroll, and Vilenkin to name 3). I am aware of no such conceptual mapping for an infinite past.

Admittedly, I am basing my view on A-theory of time: only the present actually exists, while the past represents a sequence of all prior existing times. This sequence is completed, and I see no way to conceive of a completed, infinite sequence of ordered events, one following the other.

I invite you to find flaws in my conception of a finite past, or to provide a conception of an infinite past. But please avoid a handwaving dismissal.

Well one-one correspondence is logically flawed: There are the same number of natural numbers as square numbers? Surely a paradox - a sign we are dealing with a logically flawed concept. — Devans99

How is it "surely a paradox"? That they can be put into a one-to-one correspondence shows they are the same size.

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. — Metaphysician Undercover

OP has been arguing against the coherence of infinity, including infinite sets. Qlso, I have repeatedly mentioned the transfinite numbers. I am talking about infinity, transfinite numbers are infinite. A set whose members can be put into a one-to-one correspondence with a proper subset of themselves (like the naturals) are infinite. "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.

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. — Metaphysician Undercover

I've just explained this. Transfinite numbers are infinite. They meet Dedekind's definition of infinity, don't be confused by the name "transfinite". Finite sets can't be put into a one-to-one correspondence with a proper subset of themselves, as you'll end up with members in one of the sets running out because finite sets cannot have part of the set be the same cardinality as the entire set.

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, — Metaphysician Undercover

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.

"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.

That's an ideal sphere. Nowhere did I mention an ideal sphere. Now why don't you address the point. You want a collected "infinity"? Take any sphere-like object. The number of possible paths on the sphere is not less than aleph-c. — tim wood

A "sphere" (or "ideal sphere") is an abstraction, not an actually existing thing. You bring up another abstraction: the number of possible paths being infinite. This is hypothetical; in the real world, you cannot actually trace an infinite number of paths. So in the real world you cannot actually COLLECT an infinity. All you can do is to conceptualize.