As you can see, we can quantify a quality and this is what provides us with the capacity to measure. But to qualify quantity, and this is what set theory does, is a mistaken procedure because it inhibits our capacity to measure, by imposing qualitative restrictions on quantity.

This is false. There are sets of numbers, but number themselves are not at all sets — Ikolos

Of course they are.

I really don't know what you're talking about. Infinity (as in the cardinality and ordinality of sets which can be put into a function with a proper subset of themselves) is a quantity. Finite Numbers can be defined in terms of the cardinality of sets in just the same way.

E.g. 1=0∪{0}={0}={{ }}

This is the 'singoletto'(I don't know the English word, kind of 'single set') not of a number.

the cardinality and ordinality of sets which can be put into a function with a proper subset of themselves — MindForged

The cardinality and ordinality are not at all variables which can be 'put into a function'. You are very confused.

Finite Numbers can be defined in terms of the cardinality of sets in just the same way. — MindForged

This is not even wrong. It has no sense. Numbers are not defined otherwise than by postulating a number and stating which operations preserves that being a number. See basic, high school Math(Peano Axiomatization of Arithmetics).

Of course they are. — SophistiCat

There is no reply to a deep confusion like yours. BELIEVING numbers be sets is at least a funny thing.

Quality is the degree of excellence of a thing. — Metaphysician Undercover

Frankly, a definition so senseless as all the medieval definitions were. You are deeply confused. You can quantify on something, by selecting somehow a unit of quantity, only if there is yet a quantity independently from that selection.

Furthermore it does not give us the 'capacity of measuring', because, in fact, a measure(as you may intend, for the use of terms by you is ambiguous: physical measure? or which one?) presupposes a mode of measuring, and so presupposes a quantity on which that method is to be applied. In physics, this is matter, or whatever you may call it.

But since you are so darkened on thoughts about what a quality is I will give you a fair dispute, stating better your naive argument and than responding to this peter version(be honored: the argument is by Kant):

«Reality in appearances(i.e. matter as we perceive it) always has a magnitude(a scale of degrees, i.e. quality), which is not, however, encountered in apprehension(you do not apprehend the space in which it is by itself) , as this takes place by means of the mere sensation in an instant and not through successive synthesis of many sensations, and thus does not proceed from the parts to the whole; it therefore has a magnitude, but not an extensive one.» p.290 of Critique of pure reason (guyer wood)

Then he continue:

«Now I call that magnitude which can only be apprehended as a unity, and in which multiplicity can only be represented through approxima­tion to negation = 0, intensive magnitude. Thus every reality in the appearance has intensive magnitude, i.e., a degree. » ibid.

Now, if something is to be selected as a discrete unity, as you say, it presupposes a quantity on which this selection is operated. Or do you think we actually create quantity by itself, against the basic postulate of physics?

Kant is saying that we do not apprehend it, in the sense that the homogeneity of the synthesis is a presupposition, and also it renders possible at all to select a unity: for a unity is a unity of a manifold, and if there were no homogeneity it would not be the unity of that manifold.

It is only deniable by pathologically affected(in the brain) people that homogeneity is a spatial property. And quantity in general if we have to be cautious, thus not saying quantity in general is space(which is a big assumption), we say that PRESUPPOSES ONLY SPACE, as its parts are external to one another: this sufficing to provide an account of quantity, on the postulate(common to every physical theory that has any sense) that space is occupied by something(I,e. matter), and THUS a quantity is possible at all, insofar as we can distinguish it from the space itself and thus it is possible to be measured with a certain referential unity.

Hence as the scale of degrees(quality) relies on a spatial property, than the degrees do rely on that to. But that property alone can not give us any quantity, for, by a quantity, we do not intend a mere externality between parts, but an OCCUPATION of (parts of) space. Hence Space(spatial properties we are able to detect) and matter are presupposed by any concept of quality that has any sense.

But neither Space nor matter presupposes any detectable quality by themselves. Quality, furthermore, presupposes a RELATION between space and something, which renders possible to detect some spatial properties or, as you seem to prefer, to select from that properties units, in respect to which establish a scale of measuring. This something is matter. Matter does not imply quality(degrees) but the distinguishability of degrees implies matter. But matter presupposes space. Then quality presupposes space. Either you identify space with the properties we can distinguish and classify under the kind 'spatial' and name IT quantity, or you do not identify space with those properties and call those properties 'QUANTITY' it is the same for our question: quantity it is presupposed by quality.

This is the 'singoletto'(I don't know the English word, kind of 'single set') not of a number. — Ikolos

The word you're thinking of is "singleton" I believe. And no, that's not just the singleton of a set. It was the definition of the number one in set theory. If Zero is defined as the empty set, One can be defined as it's successor; the union of Zero with singleton-Zero.

I'm still confused about what you meant when you said I had a medieval theologian type of thinking...

If Zero is defined as the empty set — MindForged

This is the condition to CONSIDER the 'singleton'(I trust you on the term) as the number one, but not to DEFINE it. Numbers are defined by a postulate and succession: see Peano Arithmetics( 0 is number(POSTULATE). If n is a number, then n+1 is number.(AXIOM) Then 1 is a number, as 0+1=1.)

I'm still confused about what you meant when you said I had a medieval theologian type of thinking... — MindForged

Not referring to you in particular. By 'medieval thinking of infinity'(and conception more or less related to it) I mean a very precise thing that I say to you.

Thinking infinity has THE QUANTITY OF WHICH NO QUANTITY IS GREATER is the MEDIEVAL conception of infinity.

Since Kant gave (1781!!) an explanation on why this is wrong, I cite from him(Guyer Wood translation, p.472)

«I could also have given a plausibleb proof of the thesis by presuppos­ ing a defective concept of the infinity of a given magnitude, according to the custom of the dogmatists. A magnitude is infinite if none greater than it (i.e., greater than the multiplec of a given unit contained in it) is possible.58 Now no multiplicity is the greatest, because one or more units can always be added to it. Therefore an infinite given magnitude, and hence also an infinite world (regarding either the past series or ex­ tension), is impossible; thus the world is bounded in both respects. I could have carried out my proof in this way: only this concept does not agree with what is usually understood by an infinite whole. It does not represent how great it is, hence this concept is not the concept of a maximum; rather, it thinks only of the relation to an arbitrarily as­ sumed unit, in respect of which it is greater than any number. According as the unit is assumed to be greater or smaller, this infinity would be greater or smaller; yet infinity, since it consists merely in the relation to this given unit, would always remain the same, even though in this way the absolute magnitude of the whole would obviously not be cognized at all, which is not here at issue.»

I will, a time or another, explain in details how mathematical infinity relates to what Kant meant by Transcendental CONCEPT of infinity, i.e. the relation between math-logical and epistemological infinity.

Our cranky and inarticulate friend has a point in that there is a difference between a conceptual definition of a number, which describes the properties that anything fitting the definition of a 'number' ought to have, and its particular theoretical construction, such as von Neumann's (which was designed to meet the requirements of the conceptual definition).

But my comment about numbers being sets (everything is a set in the set theory construction of mathematics - obviously) was made in the context of the preceding discussion, which Ikolos does not or will not follow.

This is the condition to CONSIDER the 'singleton'(I trust you on the term) as the number one, but not to DEFINE it. Numbers are defined by a postulate and succession: see Peano Arithmetics( 0 is number(POSTULATE). If n is a number, then n+1 is number.(AXIOM) Then 1 is a number, as 0+1=1.) — Ikolos

How is this different from what I said?

Thinking infinity has THE QUANTITY OF WHICH NO QUANTITY IS GREATER is the MEDIEVAL conception of infinity. — Ikolos

Well I that's definitely not how I described infinity, and it doesn't even seem correct because it's too vague. As there are many sizes of infinity, I don't even think I could pretend to accept "the quantity of which no quantity is greater" as a description of infinity.

↪MindForged Our cranky and inarticulate friend has a point in that there is a difference between a conceptual definition of a number, which describes the properties that anything fitting the definition of a 'number' ought to have, and its particular theoretical construction, such as von Neumann's (which was designed to meet the requirements of the conceptual definition). — SophistiCat

Hm, okay but I don't really get what their point about infinity was supposed to be. It didn't really sound like something which would accurately characterize how the concept is used and understood in modern math. Aleph-null is definitely not a quantity above all others, for instance.

Frankly, a definition so senseless as all the medieval definitions were. You are deeply confused. — Ikolos

It's the #1 definition in my OED under "quality". I really think that it's you who is confused. would you prefer:"a distinctive attribute or faculty: a characteristic trait"?

Now, if something is to be selected as a discrete unity, as you say, it presupposes a quantity on which this selection is operated. Or do you think we actually create quantity by itself, against the basic postulate of physics? — Ikolos

You clearly have this backward. Unity does not presuppose quantity. Quantity follows from unity, that's how we measure, by units. Quantity requires the individuation of units, therefore it requires unity. Unity does not presuppose quantity. So it s quite clear that we do create quantity, as it is minds that individuate units such that something may be counted. Which basic postulate of physics states otherwise?

Hence as the scale of degrees(quality) relies on a spatial property, than the degrees do rely on that to. — Ikolos

Yes, I agree that "degrees" rely on "quantity". And a difference of quality is measured as a difference of degree, that's the point I made. This is what allows us to measure different qualities. When we notice difference, and we produce "degrees" to measure that difference, we produce the means to quantify that quality. The units of measure, "degrees", by which the qualitative difference is divided, may be a completely arbitrary production of the mind. And, this arbitrary division of the qualitative difference, into fundamental units, "degrees", may give us the capacity to measure the quality.

But neither Space nor matter presupposes any detectable quality by themselves. Quality, furthermore, presupposes a RELATION between space and something, which renders possible to detect some spatial properties or, as you seem to prefer, to select from that properties units, in respect to which establish a scale of measuring. This something is matter. Matter does not imply quality(degrees) but the distinguishability of degrees implies matter. But matter presupposes space. Then quality presupposes space. Either you identify space with the properties we can distinguish and classify under the kind 'spatial' and name IT quantity, or you do not identify space with those properties and call those properties 'QUANTITY' it is the same for our question: quantity it is presupposed by quality. — Ikolos

You are misrepresenting the difference which exists within the thing, matter, and replacing it with our measurement of that difference, which is with degrees. Degrees of difference are not necessarily distinguishable because "the degree" may be arbitrary. The premise of "the distinguishability of degrees" misleads you. So we commonly divide space by degree with no implication of matter.

Yeah, his communication skills leave much to be desired, in more ways than one. I am not going to bother with him any further.

everything is a set in the set theory construction of mathematics - obviously) — SophistiCat

This is naive set theory darling. Thank you for your opinions directly expressed on my communication skills. I am sorry you don't understand set theory, but that's not my fault.

how we measure, by units — Metaphysician Undercover

Yes, and what do you measure my friend? Quantity. Hence there is quantity yet, and you measure it. That fact that the methods and units of measurement may be relative(as you correctly say) does not make this less true, i.e. quantity is presupposed REALITER as what is to be measured.

Which basic postulate of physics states otherwise? — Metaphysician Undercover

Independently existing matter, independently from particular modes of perceiving it, and that actually causes any perception to happen.

"the degree" may be arbitrary. The premise of "the distinguishability of degrees" misleads you. So we commonly divide space by degree — Metaphysician Undercover

The degree as a reference of measure is arbitrary, but clearly the degree of complexity of bacteria is inferior to that of an dog as an organism. No one divide space by degree man, that is nonsensical. Topology does not consider differences of space by degrees at all.

The differences in the things, as you say, is a difference which is not merely spatial, but involve matter, hence degrees. Because if you can not recognize the difference(with a possible measure) things are not at all distinguishable, except by logical or topological characteristics. Unfortunately, we are sensible beings.

Aleph-null is definitely not a quantity above all others, for instance. — MindForged

Thank you for being rational. As you can now see, thinking infinitity as a quantity whatsoever is against any basic calculus teaching.

How is this different from what I said? — MindForged

Is different in a very significant way. Because CONSIDER something as something else(with reasons to do that) is a thing, DEFINE(as you said) something is another. Citing you:

that's not just the singleton of a set. It was the DEFINITION of the number one in set theory. — MindForged

Numbers are not defined in set theory. They are considered elements of sets. Numbers are defined by induction in Peano Arithmetics, postulating 0 as a number and posing the axiom of succession.

Yes, and what do you measure my friend? Quantity. Hence there is quantity yet, and you measure it. That fact that the methods and units of measurement may be relative(as you correctly say) does not make this less true, i.e. quantity is presupposed REALITER as what is to be measured. — Ikolos

No, this is clearly not the case. The units of measurement, which are counted as a quantity, are very often created by the human mind, for the purpose of measurement, they are artificial. So "quantity", as the thing measured, is not necessarily presupposed. In the cases where we cannot find individual units to count, we simply create them, giving us the capacity to measure without there being any real quantity which is being measured.

We're talking about measuring quality, which is said to differ by degree, and the problem is that the units of degree are often artificial and may be arbitrary. For instance there are degrees of temperature. The thing measured is a difference, the unit of measurement, one degree, is artificial and arbitrary. So the quantity is artificial, and there is no actual quantity which is being measured, the "quantity", as the measurement, is assigned to it, arbitrarily, it is not something within the thing measured.

You can also look at the 360 degrees of a circle in the same way. That number of degrees around a circle is complete arbitrary. The circle could be divided into an infinite number of degrees, and there could be an infinite number of degrees between each of the four right angles, and any other angle around the circle, expressing a continuity around the circle, rather than the discrete units of degree which are commonly used. This is actually expressed with minutes and seconds. There is no real quantity of measurable units which are being measured around the circle.

Independently existing matter, independently from particular modes of perceiving it, and that actually causes any perception to happen. — Ikolos

That's a postulate of physics?

No one divide space by degree man, that is nonsensical. — Ikolos

Have you not heard of a circle?

1. Infinity is a never ending quantity.

2. Infinity is not a number.

Infinity is a symbol given to an unlimited amount of things.

Btw, countable and uncountable infinity is counterintuitive, insane and nonsensical. Cantor was a lunatic.

In the diagonal proof you have an infinite amount of rows and columns. Supposedly you can make a new infinite number different from the one on the list. How? By taking the first digit and changing it to whatever, then the second digit, and so on. This is nonsense.

First of all the new number has conditioned itself to be infinitely different from the one on the list. In no possible way can the number be the one on the list, however the same can be said by the number on the list ever actually being a quantity. It has and will never end. Thus they're both infinite numbers, and thus they both have the same length. The only difference is the rate at which they grow.

Second of all the rate of growth in infinite numbers is not a valid argument to define N (aleph-null) as an ordinal 'uncountable infinite'. This is because the rate of growth can still be achieved without ever actually having the list to being with. This new number N is in no way different or special from the numbers on the list.

Third of all the rate of growth to infinity does not change at all the fact that they're both infinite. Saying that some N1 is infinite will then have an N2 being infinite proceed from that one makes no sense and is mathematically possible but absolutely irrational, and a waste of time for your brain to acknowledge. But people don't understand these three simple concepts.

Infinity is a symbol given to an unlimited amount of things. — Emmanuele

Maybe it signals an inability to quantify.

Maybe it signals an inability to quantify. — frank

Inability to quantify or a never ending quantity I guess could be considered the same.

Seems to have something to do with quantity, that's true.

The units of measurement, which are counted as a quantity, are very often created by the human mind, for the purpose of measurement, they are artificial. So "quantity", as the thing measured, is not necessarily presupposed. In the cases where we cannot find individual units to count, we simply create them, giving us the capacity to measure without there being any real quantity which is being measured. — Metaphysician Undercover

Planck Length, Planck Scale, speed of light (which is basically a scale constant) are not in any sense arbitrary. Unless Im mistaken, the SI system is based on non-arbitrary physical constants not this nonsensical notion of "qualities" or whatever. And even if it isn't, such a thing is possible but the gain is little for all the work required. See Natural units.

That number of degrees around a circle is complete arbitrary. The circle could be divided into an infinite number of degrees, and there could be an infinite number of degrees between each of the four right angles, — Metaphysician Undercover

That something could be done a different way does not make something arbitrary. There are perfectly sensible reasons to put the number of degrees at 360. It's a highly composite number allowing us to avoid fractions (which are hard for humans to do, hence the preference for decimal expressions), it's not a large whole number so it's fairly easy to do basic math with (particularly division), etc. I'm thinking you're using a weird definition of "arbitrary" or not explaining why it is (supposedly) so.

1. Infinity is a never ending quantity.

2. Infinity is not a number. — Emmanuele

#1 is false in some sense. Unless you ignore modern math, there are a hierarchy of infinities. #2 is extremely wrong, there are many infinite numbers. The transfinite cardinals and ordinals will fit any sensible, non-arbitrary definition of a number.

Btw, countable and uncountable infinity is counterintuitive, insane and nonsensical. Cantor was a lunatic. — Emmanuele

Except the Diagonal argument is a proof by contradiction, the most standard argument type of all.

First of all the new number has conditioned itself to be infinitely different from the one on the list. In no possible way can the number be the one on the list, however the same can be said by the number on the list ever actually being a quantity. It has and will never end. Thus they're both infinite numbers, and thus they both have the same length. The only difference is the rate at which they grow. — Emmanuele

Who cares if it's "conditioned"? That is not a criticism, you're whining with this non-objection. The mere fact (proven by the opposite entailing a contradiction) that such a number necessarily exists is why mathematicians accept Cantor's diagonal argument. "Never actually being a quantity" is, naturally, something you do not demonstrate to be the case. That the number "never ends" does not mean they have the same length, that's a non sequitur. Show exactly where the "conditioned" number will appear in the original set. That you cannot (again, because it would be a contradiction) entails the different sizes of the respective sets.

Second of all the rate of growth in infinite numbers is not a valid argument to define N (aleph-null) as an ordinal 'uncountable infinite'. This is because the rate of growth can still be achieved without ever actually having the list to being with. This new number N is in no way different or special from the numbers on the list. — Emmanuele

Aleph-null is not an ordinal, it's a transfinite cardinal. Literally the smallest, first transfinite cardinal is aleph-null. And thus it is not uncountably infinite, it's countably infinite, because the definition of that is "being capable of being put into a function with the set of natural numbers".

Third of all the rate of growth to infinity does not change at all the fact that they're both infinite. Saying that some N1 is infinite will then have an N2 being infinite proceed from that one makes no sense and is mathematically possible but absolutely irrational, and a waste of time for your brain to acknowledge. But people don't understand these three simple concepts. — Emmanuele

"Both infinite" says nothing more than that they're both the same type of number. That "0.33" and "0.44" are both real numbers doesn't mean they're both the same number. You are under a terrible misapprehension. If the sets were the same size they could be mapped onto each other. That's how size is defined in math, they have to have the same cardinality. But of course, if you cannot even accept Cantor's proof by contradiction that the naturals and the reals cannot be the same size then I'm not surprised you fail to grasp this.

Cut the nonsense. Show exactly where the sets map together to make them the same size. That you cannot means they aren't the same size. Ergo, one is larger and the other smaller. It's funny you say it's mathematically possible yet "absolutely irrational". I'm sure you can defend that characterization as coherent. One would assume mathematics cannot be incoherent, given incoherency in mathematics means triviality; everything becomes a theorem if the math is incoherent, but there's absolutely no evidence that current mathematics is trivial. Show the formal contradiction. You'll win a Fields Medal.

Planck Length, Planck Scale, speed of light (which is basically a scale constant) are not in any sense arbitrary. Unless Im mistaken, the SI system is based on non-arbitrary physical constants not this nonsensical notion of "qualities" or whatever. And even if it isn't, such a thing is possible but the gain is little for all the work required. See Natural units. — MindForged

As I said, and provided examples for, some divisions of degree are arbitrary, I didn't say that all are arbitrary. But the fact that some are, is all that's required to disprove Ikolos' claim that quantity is what is measured. Actually, quantity is the measurement.

That something could be done a different way does not make something arbitrary. There are perfectly sensible reasons to put the number of degrees at 360. It's a highly composite number allowing us to avoid fractions (which are hard for humans to do, hence the preference for decimal expressions), it's not a large whole number so it's fairly easy to do basic math with (particularly division), etc. I'm thinking you're using a weird definition of "arbitrary" or not explaining why it is (supposedly) so. — MindForged

That's not even an argument, the number of degrees in a circle is not arbitrary, it was chosen because it's "easy to do basic math with". The principle divisions of the circle are the four right angles. So the number of degrees in a circle need only be divisible by 4 in order that "it's fairly easy to do basic math with". That the circle consists of 360 degrees, and not 4, 8, 12, 16, 20, 400, 800, or any other number of degrees divisible by 4 is arbitrary. That the number of degrees in the right angle is 90, and not 1, 2, 3, 4, 5, 100, 200, or any other number, is completely arbitrary.

As I said, and provided examples for, some divisions of degree are arbitrary, I didn't say that all are arbitrary. But the fact that some are, is all that's required to disprove Ikolos' claim that quantity is what is measured. Actually, quantity is the measurement. — Metaphysician Undercover

Incorrect, their argument was that some were not "qualities" as you deemed them because they are part of reality. Pointing out that some aren't (as that user already admitted) is very much besides the point when they already admitted so.

That's not even an argument, the number of degrees in a circle is not arbitrary, it was chosen because it's "easy to do basic math with". — Metaphysician Undercover

How is that not an argument? Ease of use is a perfectly legitimate reason to do prefer something. You user particular kinds of screw heads in because they're easier to use for particular kind of screws (this generalizes to tools in general). Hell, the entire reason a radian is defined as 2 x Pi is because it makes calculus and trigonometry easier if angles are measured in radians instead. Again, you're working under a strange definition of "arbitrary" if practicality doesn't count because it's very helpful that a circle has 360 degrees. Also, try to do set a circle equal to 4 degrees and see how the math works out for you. Other numbers have many divisors, but they aren't as useful mathematically (example, 2570 can be cleanly divided by every number 1 - 10 but it's not very useful in geometrical calculations to set it that way).

That's a very good damn response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.

All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.

j = {0, 1, 2, 3, 4 ... M}

N = {1, 2, 3, 4, 5 ... X}

j = numbers on the list
N = Aleph-null
X= Statement of difference
M = hyperreal of j

however

N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M1 ... X ... },
N = {1, 2, 3, 4, 5 ... M1 ... M2 ... X ... },

To say that aleph N is different from j is to say that j ends at some M for it to become X. It doesn't and never will become X (different).

Even mathematically, to state that aleph-null is a thing is to reduce the nature of the problem to cardinality. Which is meaningless. The infinity to be reasoned in any physical sense is to acknowledge that impossibility of convergence to some difference.

Also, the reals are bigger than natural numbers because they're defined in terms of themselves. They are not conditioned to exist to something which will never end. In this case aleph-null is conditioned to exist for when M ever becomes X - It will never do such a thing.

That's a very good response.
Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different.

All the notation that comes after aleph-null is to assume the difference. Here is a mathematical intepretation of my reasoning.

j = {0, 1, 2, 3, 4 ... M }

N = {1, 2, 3, 4, 5 ... X }

j = numbers on the list
M = hyperreal of j
N = Aleph-null
X= Statement of difference

however

N = {1, 2, 3, 4, 5 ... X },
N = {1, 2, 3, 4, 5 ... M ... X2... },
N = {1, 2, 3, 4, 5 ... M ... M2 ... X3 ...},

To say that aleph is different from j is to say that j ends at some M for it to become X. It will never do such a thing.

Now, the real numbers are greater than the naturals because they're both defined by themselves to be so. In this case aleph is defined to exist as long as it satisfies the difference it has with j. It doesn't have one because it cannot happen due to the nature of what is infinity.

Mathematically you can reduce all this into cardinality but it throws out the window the sense of infinity in any physical and reasonable sense, and so it defines things in a finite way similar to a hyperreal. Stating that it exist if one day M stops growing to become X.

Even then the argument behind a hyperreal is more logical then that of aleph-null. Because the hyperreal is taken into account to never be n. And so, to be able to stay at some n.
The difference X involves taking some n of j to be x. Which is to state that M is an n. No logos my buddies, what the heck is going on?

With this I say that these mathematics are physically inconsistent. So you'll never see anything like this in the physical world. To think this is amazing is to be amazed at 2 + 2 = 3 but in a more complex and sophisticated way

Logically I would state that the reason why aleph-null makes no sense is because like both aleph-null and the numbers on the list are infinity, then to say that aleph-null is in any way different from the numbers on the list is to state that the numbers on the list have an ending. This is because for them to actually be different is for both numbers to end in a difference. Such statement may sound illogical at first glance because no matter what number appears it will always be different, however the point is that if we think about it aleph-null is still impossible to fully become different. — Emmanuele

That does not follow. For one to be larger than the other all that need be true is that one set has a greater cardinality. What this will mean is that when you try to place them in a one-to-one correspondence with each other, it fails to be possible to do so. After all, sets that can be mapped together in this way are the same size. What Cantor showed was that it's impossible to map the naturals with the reals on pain of contradiction, it turned out the reals were larger not that the naturals had an end (in the sense of a final member). That's what makes them different, despite being infinite. They're different levels of infinity.

That does not follow. For one to be larger than the other all that need be true is that one set has a greater cardinality. What this will mean is that when you try to place them in a one-to-one correspondence with each other, it fails to be possible to do so. After all, sets that can be mapped together in this way are the same size. What Cantor showed was that it's impossible to map the naturals with the reals on pain of contradiction, it turned out the reals were larger not that the naturals had an end (in the sense of a final member). That's what makes them different, despite being infinite. They're different levels of infinity. — MindForged

I get the point. The ending member is different in a one to one correspondance from T (numbers on the list) to N (aleph). I would just like to believe that from R to N (naturals) the last member is different because that's how they are defined by themselves. If we make a number out of the difference of a specific number is to state that the former was a number at all, even in regards to the cardinality of a set.

I would like to believe that this up here is sufficient to say that aleph as a set was to just be math dribble. And the last member of the set T never existed in order for it to be different. So the definition of the set aleph is not satisfied.

Incorrect, their argument was that some were not "qualities" as you deemed them because they are part of reality. Pointing out that some aren't (as that user already admitted) is very much besides the point when they already admitted so. — MindForged

I can see that one, or both of us, misunderstands what Ikolos was saying. I can restate what I was saying though. I said that qualities can be quantified, but it is a mistake to attempt to qualify quantities. Both qualities and quantities are "part of reality", so this reference is just a diversion. The issue is what the mind is doing when it attempts to quantify a quality, or qualify a quantity.

Ikolos replaced "quality" with "relation", and I had to insist that relation is a quality rather than a quantity, because Ikolos wanted to argue that a relation is a quantity, without the required mental act which quantifies that quality.

How is that not an argument? Ease of use is a perfectly legitimate reason to do prefer something. — MindForged

As I said, there are many other numbers which offer equal, or greater ease of use, so there is no basis for your argument.

Also, try to do set a circle equal to 4 degrees and see how the math works out for you. — MindForged

If a circle were 4 degrees, I see that an acute angle would be less than one degree, and an obtuse angle would be greater than one degree. Forty five would be half a degree, and one eighty would be two degrees. Looks very easy to me. I'm no mathematician so I might be missing something. Where would the mathematical problem be, which would make 360 degrees mathematically easier than 4 degrees?