I've never seen a definition of continuous, which refers to parts ... — Metaphysician Undercover

Yes, you have - I just quoted one to you, verbatim, from Webster's Collegiate Dictionary, Fifth Edition (1936), which is what I happen to have on the shelf here at home. See below for a philosophical definition that explicitly refers to parts.

It is a well known metaphysical principle, that the continuous is indivisible. — Metaphysician Undercover

Sources, please? On the contrary, here is what the SEP article on "Continuity and Infinitesimals" has to say (italics in original, bold mine).

While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit. This means that the process of dividing it into ever smaller parts will never terminate in an indivisible or an atom - that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are divisible without limit or infinitely divisible. The unity of a continuum thus conceals a potentially infinite plurality.

In other words, being continuous is generally (although not invariably) the exact opposite of being indivisible.

Aletheist was probably directed toward theism at an early age. Theism became part and parcel of his metaphysics from the beginning of his life. — Bitter Crank

True in my case, although there are plenty of examples of atheists becoming theists later in life, and vice-versa.

Out of curiosity, I checked my dictionary to see how it defines "continuous." Here is what it says: "Having continuity of parts; without cessation or interruption; continued." Once again, you got it backwards - having parts is necessary for something to be continuous; otherwise, it would be "indivisible," which is a completely different concept.

"Part" implies of necessity, a separation, and this negates any claim of continuity, which is a lack of such separation. — Metaphysician Undercover

Unless you can demonstrate that the concept of "part" necessarily involves separation, rather than just baldly asserting this over and over again as your own idiosyncratic definition, I have no reason to take it seriously. It blatantly begs the question to insist that anything with parts of any kind must be classified as "discrete," rather than "continuous."

They are united as one large continuum and it is false to refer to them as separate infinitesimals. — Metaphysician Undercover

One more time: By definition, infinitesimals are not separate.

To say that a continuum has parts is contradictory. — Metaphysician Undercover

One more time: By definition, a continuum has parts, all of which have parts of the same kind.

... what you seem to be failing to recognize is that "part" also requires separation. — Metaphysician Undercover

No, it does not. Once again, you are rejecting the commonly accepted definitions of terms, and imposing your own idiosyncratic ones.

The true continuum must be indivisible, that's why it cannot be modeled mathematically. — Metaphysician Undercover

No, it must be infinitely divisible - i.e., there cannot be any indivisible parts - and smooth infinitesimal analysis does model this mathematically, whether you recognize it or not.

You mean as a rationalization? Probably so, in many cases. Arguably one's behavior/outlook is one's metaphyics; or more broadly, one's beliefs in general are manifested in one's habits.

What do you include in the scope of someone's metaphysics? For example, I am a theist, and my belief that God is real significantly affects my behaviour, moral outlook, relationships, and overall worldview.

It doesn't matter how you lay the infinitesimal out, as a point, or as a line, there is still the assumed separation between it and other infinitesimals, and therefore it is necessarily a discreteness. — Metaphysician Undercover

Wrong. There is no separation (assumed or otherwise) between infinitesimals. Neighboring infinitesimals are indistinct; the principle of excluded middle does not apply to them.

A continuum cannot have parts, or else it is by virtue of those parts, not continuous, it is discrete. — Metaphysician Undercover

Wrong. A continuum - by definition - is that which has parts, all of which have parts of the same kind. What a continuum cannot have are indivisible parts, like points.

By saying "smoothly overlapping" you are speaking in terms of discreteness. You have identified separate parts which overlap. — Metaphysician Undercover

Wrong. Discreteness requires separation and distinction; infinitesimals, as defined by synthetic differential geometry (a.k.a. smooth infinitesimal analysis), are neither separate nor distinct.

We measure space, time and motion as discrete, because that's the only way we can apply the numbers. — Metaphysician Undercover

I have no problem with measuring continuous things using discrete models; as I have acknowledged previously, they are very useful for that purpose.

So long as you hold this belief, that space, time and motion are continuous you will have paradoxes. — Metaphysician Undercover

No, I have explained how Zeno's paradox dissolves when continuous motion is properly understood as more basic than discrete locations. Besides, a paradox by definition is only an apparent contradiction, not an actual contradiction; beliefs that are paradoxical are not necessarily false.

The concept of "infinitesimal points" is incompatible with continuous motion, it is only compatible with discrete motion. An infinitesimal point must be separate from another infinitesimal point or else it is not a point, and this negates any possibility of continuity. — Metaphysician Undercover

No one is talking about "infinitesimal points" except you. Infinitesimals are not separate dimensionless points, they are lines of extremely small but non-zero length that smoothly blend together so as to be indistinct. A continuum is that which has parts, all of which have parts of the same kind . A one-dimensional line cannot be divided into zero-dimensional points, only shorter and shorter one-dimensional lines.

A series of "timelets" is a description of something discrete. Your quote from John Bell has provided a description of discrete motion, not continuous motion. — Metaphysician Undercover

That would be news to him. I guess you missed the part about the timelets "smoothly overlapping" such that "time is, so to speak, still passing" within each of them, rather than being frozen in a discrete instant.

It's not that motion is continuous, and we are trying to understand it as units, it's that it is not continuous, but we are trying to model it as being continuous. — Metaphysician Undercover

You still have it exactly backwards. Space, time, and motion are all continuous; we only model them as being discrete.

Prompted by some of the discussion in my spin-off thread on "Continuity and Mathematics," I have been reading up on category theory and one of its outcomes, synthetic differential geometry, also known as smooth infinitesimal analysis. I just came across this very pertinent passage on page 277 of John L. Bell's 2005 book, The Continuous and the Infinitesimal in Mathematics and Philosophy (emphases added).

The Principle of Microstarightness yields an intuitively satisfying account of motion. For it entails that infinitesimal parts of (the curve representing a) motion are not points at which, as Aristotle observed, no motion is detectable - or, indeed, even possible. Rather, infinitesimal parts of the motion are nondegenerate [i.e., non-zero] spatial segments just large enough for motion through each to be discernible. On this reckoning a state of motion is to be accorded an intrinsic status, and not merely identified with its result - the successive occupation of a series of distinct positions. Rather, a state of motion is represented by the smoothly varying straight microsegment, the infinitesimal tangent vector, of its associated curve. This straight microsegment may be thought of as an infinitesimal “rigid rod”, just long enough to have a slope - and so, like a speedometer needle, to indicate the presence of motion - but too short to bend, and so too short to indicate a rate of change of motion.

This analysis may also be applied to the mathematical representation of time. Classically, time is represented as a succession of discrete instants, isolated “nows” at which time has, as it were, stopped. The principle of microstraightness, however, suggests that time be instead regarded as a plurality of smoothly overlapping timelets each of which may be held to represent a “now” or “specious present” and over which time is, so to speak, still passing. This conception of the nature of time is similar to that proposed by Aristotle to refute Zeno’s paradox of the arrow; it is also closely related to Peirce’s ideas on time.

As I said before, continuous motion is the most fundamental concept here. It is logically prior to any series of discrete locations - including an infinite one - through which an object passes while traveling from one place to another. In fact, the object's actual path that includes those identified "points" only exists as the result of the motion.

Another possible clue from Timothy Herron's 1997 paper, "C. S. Peirce's Theory of Infinitesimals."

To start we mention that the main attraction of this theory for followers of Peirce beyond the simplification of mathematical practice is in the circumstance that the law of excluded middle does not hold for the points on this extended real line. Hence, there is a strong sense in which points merge together so that they are no longer distinct individuals. Peirce often said that traditional laws of logic, like the law of excluded middle or the law of contradiction, do not apply to points which are merged together in the continuum. The most important parts of the line in synthetic geometry are infinitesimal "linelets" surrounding each point ...

... the consistency of synthetic geometry's infinitesimals is established by formulating it inside topos theory, a subbranch of the theory of categories whose logic is solidly intuitionistic. Thus, we should not expect the law of excluded middle to hold for the objects we can construct with topos theory.

It appears that Peirce would favor this version of a theory of infinitesimals over that of Robinson because it satisfies more of his desiderata. First, the infinitesimal intervals surrounding every point and whose image under a function are linear are the true parts from which the line is built. The points lying on the extended real line are not true atomic elements from which the line is built considering that the law of the excluded middle cannot be used to distinguish between points with infinite precision. We should consider points as the potential elements of the line which are welded together ... It seems that the only way in which Peirce could be disappointed about this model is that it is a projective theory of geometry, one which needs a straight line as a fundamental part of the geometry. It would be very interesting to explore topos theory to see if there are any restrictions on how many points can be placed on the extended real line of synthetic geometry in order that Peirce's desire to fit any cardinality of points on a line can be satisfied within this model of infinitesimals.

Any thoughts on these passages, @fishfry?

Another possible clue in the SEP article on "Continuity and Infinitesimals."

A major development in the refounding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry, also known as smooth infinitesimal analysis (SIA). Based on the ideas of the American mathematician F. W. Lawvere, and employing the methods of category theory, smooth infinitesimal analysis provides an image of the world in which the continuous is an autonomous notion, not explicable in terms of the discrete ... We observe that the postulates of smooth infinitesimal analysis are incompatible with the law of excluded middle of classical logic.

If you want to explain to me how the synthetic continuum is in fact recovered fully by category theory, I would be very grateful. But can you do that? — apokrisis

I certainly cannot do that - at least, not yet - but I just came across one possible clue in the SEP article on "Category Theory."

A slightly different way to make sense of the situation is to think of mathematical objects as types for which there are tokens given in different contexts. This is strikingly different from the situation one finds in set theory, in which mathematical objects are defined uniquely and their reference is given directly. Although one can make room for types within set theory via equivalence classes or isomorphism types in general, the basic criterion of identity within that framework is given by the axiom of extensionality and thus, ultimately, reference is made to specific sets. Furthermore, it can be argued that the relation between a type and its token is not represented adequately by the membership relation. A token does not belong to a type, it is not an element of a type, but rather it is an instance of it. In a categorical framework, one always refers to a token of a type, and what the theory characterizes directly is the type, not the tokens. In this framework, one does not have to locate a type, but tokens of it are, at least in mathematics, epistemologically required. This is simply the reflection of the interaction between the abstract and the concrete in the epistemological sense (and not the ontological sense of these latter expressions.)

A continuum (such as a line) is a type and its parts are tokens, which are instances of it (smaller lines) rather than members or elements of it (discrete points).

Please see my (second) previous response. You can define and mark as many discrete locations between A and B as you like, but this does not in any way affect the continuous motion of the object from A to B, which is logically prior to that mathematical exercise.

Said another way, the object's motion comes first from a logical standpoint. Drawing a line that traces the object's path, and then defining and marking whatever points on that line serve whatever purpose we may have in doing so, only comes afterward.

So you deny that there's an actual half way point between the start position and the end position? — Michael

It depends on exactly what you mean by "an actual half way point." As I said, there are no actual points at all, if by that we mean mathematical (i.e., dimensionless) points. There obviously is a location on the continuous line that is equidistant from the start and end positions, but there is only a "point" there if we define and mark it as such for some particular purpose, such as measuring. The object would "pass through" any point that you wish to define and mark on the line - but that act of defining and marking a point does not somehow create a separate, discrete, intermediate step that the object must now take in order to get from one place to the other.

Of course the points actually exist. There actually is a half way point between the start and the end of a 100m line. There actually is a quarter way point. And so on. Are you denying this? — Michael

Points and lines do not actually exist; they are mathematical abstractions that we use to model things that do actually exist, like objects moving from one place to another. A line is simply the path through space over time that an object would trace if it were to move with constant velocity. In that sense, the concept of motion is more fundamental than the concept of a line; and as such, the object's path through space over time is more accurately modeled by an unbroken continuum than by an infinite series of separate, discrete locations.

Here is a relevant quote, attributed to Stephen Covey, that I just came across.

Most people do not listen with the intent to understand; they listen with the intent to reply.

It stills has to pass through an infinite series of separate, discrete points. — Michael

Only if all of those points actually exist, which is precisely what I deny. The line does not consist of separate, discrete points; it can only be modeled as having separate, discrete points.

Your claim, as I understand it, is that the line does consist of infinitely many separate, discrete points, and thus can only be modeled (or "considered') as continuous. This seems to be our basic disagreement.

you can try your best to represent life experiences using symbols, but try as you might 1 does not in any way describe the experience of going from a bed to a bathroom. — Rich

We agree that the phenomenal experience cannot be modeled adequately by mathematics, or even by other symbols like narratives; but various other aspects of it can be - again, depending on the purpose of the model.

It doesn't matter if you don't consider the movement to be in separate, discrete steps. — Michael

It has nothing to do with how I consider it. The movement does not actually consist of an infinite series of separate, discrete steps. It is simply a single, continuous motion from the start of the 100-m line to its end. This is what I mean when I say that the line itself does not actually consist of infinitely many separate, discrete points; it is simply a single, continuous line.

We must stop using mathematics for describing life experiences. — Rich

I still think that this is an overreaction. We can still use mathematics for describing certain life experiences, depending on our purpose in doing so.

I don't get this. You do pass the half-way point (after 50m). And you do pass the quarter way point (after 25m). And so on, ad infinitum. — Michael

Yes, you pass each of those arbitrarily identified "points"; but each instance of doing so is not a separate, discrete step in the continuous motion of traversing the entire 100-m line.

There is a half way point between the start of a 100m line and the end, and this is true even if we don't plot it, which is why I don't understand aletheist's and apokrisis' objection at the start. — Michael

We do not have to treat every halfway point as a discrete step in the motion from the start of that 100-m line to its end. We can traverse the one full interval (100 m) without individually traversing infinitely many half intervals (50 m, 25 m, 12.5 m, etc.).

Well my view is that the laws of thought are designed to make the world safe for predicate logic - reasoning about the concretely particular or actually individuated. So the three laws combined - or rather three constraints - secure this desirable form of reasoning in a suitable strait-jacket. — apokrisis

Yes, I agree. What remains unclear to me is what it means to say that the principle of identity does not apply to something. Zalamea helpfully formalizes the principles of vagueness and generality on page 21 of his paper; he describes them as failures of distribution of the principles of contradiction and excluded middle, respectively. Is there an analogous way to formalize the principle of identity and/or its failure, which would show what you have in mind here?

Again, this is somewhat of a departure from conventional Peirceanism. I employ the logic of dichotomies (as it is understood from the vantage of hierarchy theory) where definite actuality or 2ns is emergent from the interaction of constraints and free or vague potential. So 2ns comes last in a sense (though this is no contradiction of Peirceanism, just making something further explicit). — apokrisis

I likewise dissent from the most common interpretation of Peirce's cosmogony, in which 1ns came first (so to speak), then 2ns, and finally 3ns. I think that 3ns, with "its really commanding function," is the most fundamental - "the clean blackboard" as a continuum of two dimensions representing the original one that had "some indefinite multitude of dimensions." A chalk mark then represents the spontaneous introduction (1ns) of a brute discontinuity (2ns), but the mark is not really a line - it is a surface whose own continuity is parasitic on that of the underlying blackboard. Only after developing habits (3ns) lead multiple chalk marks to persist and aggregate into "whiteboards" that represent Platonic worlds of possibility (1ns) does the final step occur, when "this Universe of Actual Existence" (2ns) comes about as "a discontinuous mark" on one of those whiteboards.

Why not just do much less rebuking all round and focus on dealing with the substance of any post. — apokrisis

This is probably good advice, and I will try to heed it going forward.

Generality is defined by its contradiction of LEM. Vagueness is defined by its contradiction of PNC. So it would be neat if actuality or 2ns were contradicted by (thus apophatically derivable from) the remaining law of thought. — apokrisis

Sorry to nitpick, but is "contradiction" the right word here? In accordance with the Peirce quote, we started out using "inapplicability," which seems more appropriate to me. So your hypothesis, as I understand it, is that actuality/2ns is defined by the inapplicability of the principle of identity; and I am still wondering which particular formulation of it you have in mind, since there are several. For example, Peirce did say that "Leibniz's 'principle of indiscernibles' is all nonsense" (CP 4.311). In fact, in his definitions of "individual" for Baldwin's Dictionary of Philosophy and Psychology (1911), he wrote the following.

Used in logic in two closely connected senses. (1) According to the more formal of these an individual is an object (or term) not only actually determinate in respect to having or wanting each general character and not both having and wanting any, but is necessitated by its mode of being to be so determinate ...
(2) Another definition which avoids the above difficulties is that an individual is something which reacts. That is to say, it does react against some things, and is of such a nature that it might react, or have reacted, against my will ...
... whatever exists is individual, since existence (not reality) and individuality are essentially the same thing; and whatever fulfills the present definition equally fulfills the former definition by virtue of the principles of contradiction and excluded middle, regarded as mere definitions of the relation expressed by "not." As for the principle of indiscernibles, if two individual things are exactly alike in all other respects, they must, according to this definition, differ in their spatial relations, since space is nothing but the intuitional presentation of the conditions of reaction, or of some of them. But there will be no logical hindrance to two things being exactly alike in all other respects; and if they are never so, that is a physical law, not a necessity of logic. — CP 3.611-613

My other point was that if vagueness/1ns is defined by the inapplicabiity of the principle of contradiction, then actuality/2ns and generality/3ns must be subject to it; and if generality/3ns is defined by the inapplicability of the principle of excluded middle, then vagueness/1ns and actuality/2ns must be subject to it. Likewise, if actuality/2ns is defined by the inapplicability of the principle of identity, then vagueness/1ns and generality/3ns must be subject to it. Otherwise, each characteristic is not distinctive of its corresponding category after all.

Does that make any more sense?

I expected fishfry to tell me where I was wrong about category theory vs semiotics in his own words ... — apokrisis

He never said that you were wrong. He merely said that Zalamea said the opposite of what you said.

He has now told me to f*** off. And you seem to think he is right to do so. — apokrisis

What gave you that idea? I thought that was also unfortunate and unnecessary. Do I need to rebuke him to demonstrate my impartiality?

I'm not aware that Peirce ever made this point about identity. — apokrisis

How would you formulate the principle of identity such that it would not apply to the actual, because nothing that exists is determinate with respect to every predicate? Does it apply to 1ns and 3ns, such that its inapplicability is a distinguishing feature of 2ns as you seem to be suggesting?

Have I ever attacked you personally, in this thread or elsewhere? — aletheist

Yep. You are doing that right now too. — apokrisis

Really? That was not my intention at all. I was just trying to moderate a dispute between two of my favorite PF participants.

And so you would rather chase me off now. — apokrisis

What? Why would that be your response? My whole objective was to bring everyone back to the table; I have no desire to chase anyone off.

everything actual is indeterminate to some degree — aletheist

Yes. And so does that now suitably define 2ns or actuality as that to which the principle of identity does not apply? (And can you find the quote where Peirce said that?) — apokrisis

Ah, good point. Where Peirce said what I said, what you said, or both?

I find that to be the first insult here. I gave a full answer and I get back no useful reply. — apokrisis

I see nothing insulting about pointing out a discrepancy between what you wrote here and what is claimed in a paper that you recommended.

If you or fishfry want to enlighten me otherwise, be my guest. — apokrisis

I have already acknowledged (more than once) that I do not know enough about category theory (yet) to say anything at this point. I was hoping to learn more about it from the two of you.

But don't keep attacking me personally instead of addressing the actual ideas I have attempted to put out there. — apokrisis

Have I ever attacked you personally, in this thread or elsewhere? At this point, I am just annoyed that you seem to have driven off @fishfry, who I thought was making helpful contributions to the discussion. If it now devolves into "Peircian exegetics," as @SophistiCat thinks it already has done, then it will just be the two of us trading thoughts about our favorite philosopher. I was hoping for much more than that when I started the thread.

And yet where does the principle of identity sit as actual individuation if vagueness and generality are the apophatic definition of the PNC and the LEM? — apokrisis

The actual is that which is neither vague nor general by these definitions - both principles apply to it. The tricky part is that this notion of absolute singularity is strictly ideal - everything actual is indeterminate to some degree; no existing object definitively possesses or lacks every predicate.

So category theory seeks an analytic foundations whereas semiosis seeks a synthetic one. — apokrisis

Sorry to repeat myself, but would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context - i.e., as characterizations of the kind of foundation that a particular theoretical approach seeks? That might help me understand why you apparently disagree with Zalamea on whether category theory helps us recover the Peircean notion of a continuum.

Again, @fishfry simply pointed out something that Zalamea claims in the paper that you recommended, which is contrary to your own comments. I took him to be asking you for an explanation of this particular discrepancy, not nitpicking or in any way asserting that Zalamea is right and you are wrong, since he is still in the process of digesting the paper. Then you are the one who responded with the first insult, alleging that he does not understand category theory. If you had left out that one unnecessary sentence, I suspect that a more fruitful exchange would have ensued - one that I would still very much like to see happen.

If you want to explain to me how the synthetic continuum is in fact recovered fully by category theory, I would be very grateful. But can you do that? — apokrisis

Come now - this is not @fishfry's own claim, but Zalamea's, in the very paper that you recommended. The later discussion linking category theory with Peirce's continuum is on pages 38-40. I guess you disagree with Zalamea about this? If so, why? Again, I do not know much about category theory (yet), so I personally have no opinion one way or the other and am open to persuasion.

So category theory seeks an analytic foundations whereas semiosis seeks a synthetic one. — apokrisis

Would you mind clarifying exactly what you mean by "analytic" and "synthetic" in this context?

Possibility comes in two varieties - 1ns and 2ns. Firstness is unconstrained possibility and secondness is constrained possibility. So 1ns is more like the notion of pure potential, and 2ns more like the ordinary notion of statistical probabilty (or even a propensity). — apokrisis

That is not how I understand it, unless by "constrained possibility" you mean the actually possible as opposed to the logically possible. Peirce sometimes even distinguished continuous potentiality as 3ns ("indeterminate yet capable of determination in any special case") from pure/ideal possibility as 1ns ("incapable of perfect actualization on account of its essential vagueness"). I associate propensity more with 3ns as habit than with 2ns as brute actuality. Ultimately it depends on the particular type of analysis that we are doing, since all three categories are part of every phenomenon to some degree.

Firstness is more or less indeterminate or determinate, not more or less vague or precise; only with Peirce's category of Thirdness can we speak of vagueness versus precision (and then there's also vagueness versus generality).

Peirce usually distinguished vagueness (1ns) from generality (3ns). "Perhaps a more scientific pair of definitions would be that anything is general in so far as the principle of excluded middle does not apply to it and is vague in so far as the principle of contradiction does not apply to it."

Thanks for your comments and clarification.

I may have been responding to you, but I thought it obvious I was not referring to you! — tom

Ah, okay; it was (obviously) not obvious to me, since I generally assume - unless there is a clear indication otherwise - that a reply to one of my comments is directed at me. Thanks for clarifying.

Thanks for yet another helpful clarification.

I think it is a bigger mistake to write off the great works of Georg Cantor and Nicolas Bourbaki (that makes at least 8 geniuses) on the basis of your personal inability to comprehend the first thing about it. — tom

What exactly is it that you think I am not comprehending? Sincere question, I am eager to learn.

That does not follow. It must be proved. That's Cantor's theorem. — fishfry

If it is a theorem that has been proved, then it follows, does it not? What am I missing?

Without moving to exclude those ideals which are impossible, we have no way to increase accuracy and success. — Metaphysician Undercover

It is a mistake to treat accuracy and success in the actual world as the only legitimate objectives of inquiry. For one thing, it is inconsistent with what most people mean when they talk about "ideals."

If you want a metaphysics which allows that anything is possible, you go right ahead and adopt that metaphysics, but it's not for me. — Metaphysician Undercover

It is a mistake to confuse mathematics with metaphysics. Many things are possible within mathematics that are not actually possible. I also happen to believe that there are real possibilities that are not actually possible, but that is not at all the same thing as allowing that anything is possible.

I would prefer to exclude things which at first glance may appear to be possible, but which are later shown not to be possible, as impossible ... the way to succeed in inquiry is to narrow possibilities, by eliminating unjustified possibilities. — Metaphysician Undercover

It is a mistake to block the way of inquiry by ruling out possibilities too hastily. I would prefer not to exclude things which at first glance may appear to be impossible, but which are later shown to be possible. The key is to formulate retroductive conjectures that are amenable to deductive explication and inductive evaluation through experimental testing - often in the actual world, but sometimes in the imagination, as for example within mathematics.

Furthermore, I believe that if we adhere to this separation, mathematics will be rendered useless, because we will not be able to use it to actually count or measure anything. If the principles of mathematics have no relation to what it means to actually count, or measure something, then what good are they? — Metaphysician Undercover

This right here is precisely the reason why we have been at such loggerheads throughout this discussion (and others). As I keep saying over and over, mathematics is the science of drawing necessary conclusions about ideal states of affairs; it does not pertain to anything actual, except to the extent that we use it - with varying degrees of accuracy and success - to model the actual. You have an idiosyncratic metaphysical prejudice that requires something to be actually possible in order to be considered possible in any sense. Again, your worldview is too small; there is much more to mathematics than merely counting and measuring things, and the value of pure mathematics - like that of pure science - is not limited to its practical usefulness. Do not block the way of inquiry!

... it is impossible that an infinite set has a cardinality. — Metaphysician Undercover

Not when "cardinality" is defined as a specific property of infinite sets.

I am ready to accept it, as soon as all inconsistencies and contradictions are removed. — Metaphysician Undercover

There are no inconsistencies or contradictions within the hypothetical realm of mathematical set theory.

Those who are "in denial" will always refuse to face the fact that their "well established truths" are actually falsities. — Metaphysician Undercover

Pot, kettle, black.