• What is the difference between actual infinity and potential infinity?
I wonder if this is even true. Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem. I suppose that it would be very difficult to distinguish whether the structure was created purely for beauty, or for utility.

I didn't claim an absolute platonic approach to mathematics, definitely not. But it appears to me that once we get to think about some mathematical problem, even if that raised within the context of solving a problem about our world, or certain application, etc.., we can easily figure out many offshoots that can be developed on purely theoretic basis, those offshoots can later on have applications, but I would guess that many were coined even before seeking some application that was raised to instigate them. Lets take a very simple example, lets take the negative integers, those were contemplated by the Chinese as well as the Greek mathematician Diophantus way before the Arabs made full use of it in commerce and related business. Actually Diophantus objected to their existence as being "false", this shows that he knew them but saw no application for them and rejected them along philosophical lines, much as many people rejected imaginary numbers, irrational numbers, and "transfinite" numbers, non-standard naturals etc.. Yes you can start with a shape in application and then try to figure out its rules in the platonic world then problems will raise and their solution can be approached in the mathematical realm and many offshoots of that approach may later turn to have applications. As I said Non-Euclidean geometry was an offshoot of the challenge caused the the "parallel postulate" of Euclid, now this is a pure mental problem, it was not related to utility, many geometricians in their endeavor to prove the fifth postulate actually discovered the roots for non-Euclidean Geometry. The equi-interpretability of non-Euclidean Geometry with the Euclidean was established therefore establishing the independence of the fifth postulate (the parallel postulate), this is a pure platonic problem, however it turned to have a utility later one in Einstein relativity theory. Same to be spoken about Riemannian n-dimensional Geometry, which raised from within solving problems that seem to me to be purely related to theoretic mathematics, rather than being instigated by some particular application in the real world, later on Relativity theory used this multi-dimensional space of Riemann. Regarding shapes ellipses, and others might have been raised from some problem about the real world as contemplating the idea of a shape nearer to that of an egg, or the inclined cut section of a cone, etc.. but that also can be seen as an offshoot of contemplating closed figures beginning from one with ideal symmetry, i.e. the circle, to ones less symmetrical and so on, one can Platonically think of a whole spectrum of these.

I think (though I don't have a proof ) that many mathematics even if initiated in application, would have a pure platonic intermediates bringing many possible structures, then many of those would fade away because they don't have applications, while those that have, will continue also raising problems about them in the platonic world leading to many offshoots, some of which would have applications, and so on..

I don't think the whole of mathematics, i.e. every step of it, was instigated by some utility in a direct manner to solving a problem in the real world, neither do I maintain that the whole of mathematics did or even could have proceeded in an absolutely purely platonic manner. Its a mixture of both that we have.
• What is the difference between actual infinity and potential infinity?
So we're back to this question of art (beauty, aesthetic), or utility. Do mathematicians create all sorts of shapes, forms, and structures simply because they are beautiful, and have them lying around for possible use, or do they create them to serve as solutions to particular problems. You seem to choose the former, that mathematicians create a whole arsenal of beautiful shapes, simply because they are beautiful, then physicists and cosmologists might choose from this collection of designs, those which are suitable to them. I think that mathematicians create their forms with purpose, as potential solutions to particular problems.

Honestly I think its both cases. Some structures were actually contemplated due to their own beauty in a platonic world, while others raised secondary to observations and need for application as you depicted. I in some sense do agree with you that we'll have infinite possibilities if we were to contemplate just purely, but there are definitely some scenarios that are more attractive platonically speaking than others.

Example of "mathematics prior to observation" is that the orbit of planets which suits more of an ellipse. Ellipses where there on board since ancient Greek, and their study didn't arise from contemplating planet orbits as you think. No they actually were studies on our earthly structures which are simply about inclined sections of cones. Then Kepler picked what is already available and matched it with observations about planets movements.

Other examples include Riemannian n-dimensional geometry, this was contemplated before relativity theory and other recent theories of physics which use many dimensions. Also non-Euclidean geometry was long contemplated by Al-Tusi and also by various mathematicians long before relativity theory called for their use, and they did arise from the pure study of geometry in the platonic realm, mainly becuase of the non-proof of parallel postulate. Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures.

On the other hand I agree with you that there are other situations where the mathematics had been created AFTER the observations had been made, i.e. "observation prior to mathematics" direction: like mathematics about the DNA double-helix structure, and also like quantum mechanics, Dirac delta function, etc...

So in real practice both lines are occurring, the pure investigation of those entities in the platonic laboratory and on the other hand the on-demand construction of mathematical entities to match needed application. We can say that mathematics can work to enrich our knowledge about the world by detecting behaviors in the later that we already knew of in the platonic world (in approximate manner), and also the other direction is also true, that observation in our real world as the source and the motive to contemplate certain platonic structures, so our world enriches mathematics also. It is a bilateral movement. And I think this bilaterality is important. And it (the bilaterality) should be observed if we are to have mathematics help enrich our knowledge about our world.

That said. I think we need to unleash both directions!
• What is the difference between actual infinity and potential infinity?
This is a very interesting subject which you bring up here, but my opinion is somewhat opposite to what you say. I think that mathematics allows us to make many very accurate predictions based on statistics and probabilities, without having any accurate description of the mechanisms involved. So for example, Thales apparently predicted a solar eclipse in 585 BC. I think it's common that we observe things, take note of the patterns of specific occurrences, thereby becoming capable of predicting those occurrences, without understanding at all, the motions which lead to those occurrences.

• What is the difference between actual infinity and potential infinity?
It is not an issue of the human capacity to observe, because we already extend that capacity with instruments. Nor is it an issue of the "physical means of measurement", because we create and produce these, the instruments for measuring, as required. Therefore we ought to consider that the problem, which is causing this limit to appear before us, is a manifestation of the principles by which we interpret the information.

Its nice to be informed of that. But my knowledge about those issues is damn sketchy. And so I have no say in such subjects. Thanks for your informative reply.

I was always under the impression that mathematics can supply us with descriptive arsenal that help us discover matters easily. Due to human nature people often don't see (i.e. overlook) what they don't seek. I think that without having descriptive account on "orbits" like those of Ellipses, Parabolas, and hyperbolas that mathematics beforehand supplied us with, it could have been very difficult to observe how the planets moves, and it would be very difficult to predict their movements. Possibly similar things might apply with the uncertainty principle. I don't know really.
• What is the difference between actual infinity and potential infinity?

That's not what I've asked about in my last comment. I wanted to know how the "uncertainty principle" is the error of applying an unsound mathematical system to particle physics? I wanted to know what are your objections to the uncertainty principle? and why you think it is the mathematics involved in it that are the source of the problem? I thought the source of the problem is our "physical" means of measurement not the mathematical side of it. Can you elaborate on this specific issue, I mean exactly that related to the uncertainty principle.
• What is the difference between actual infinity and potential infinity?
It's called the uncertainty principle.

Oh! but that's cornerstone in Quantum mechanics, isn't it? I always hear about a lot of strange conclusions in quantum theory like all possible worlds being actuated, and all intermediate states being there, even between life and death, etc.. I don't know if these are actually of any importance. But do you think that all of those strange results are due to the nature of the platonic realm in which the mathematics of that mechanics is coined? I thought that uncertainty principle had nothing to do with the mathematics involved, it has something to do with inability of have complete form of measurement which is due to the nature of the objects studied and not to the mathematics involved in them. Not sure, really. Can you clarify the picture to me?
• What is the difference between actual infinity and potential infinity?
ou can see these undesirable consequences in the particles of particle physics

I want examples of those, I mean of the undesirable consequences in the particles of particle physics. Which known examples you are referring to?
• What is the difference between actual infinity and potential infinity?
The undesirable consequences only become apparent in application, because the premises concerning the nature of an object are inconsistent with what an object really is. You can see these undesirable consequences in the particles of particle physics.

EXAMPLES?
• What is the difference between actual infinity and potential infinity?

I don't need to name any, they are all unsound. We've discussed the fact that the axioms lack truth, in how they describe objects. The axioms are the premises, and soundness requires true premises. The premises are not true, therefore the conclusions are not sound

I never admitted that they are not sound. They are indeed sound of what they are describing in the platonic sense. And if platonic sense proves to be indispensable for discovering our reality, by then this would prove it to be sound. So the question of soundness of those axioms and its relation to application is still unsettled. But if they were unsound as you claim, then they must bear wrong theorems, i.e. we need to see MANY arithmetical consequences of those theories that violate true arithmetic. Why we are not seeing any? What's available in practice witness to the contrary direction, i.e. the arithmetic sentences proved in them are true, and actually it is provable that any consistent finite fragment of ZFC is arithmetically sound. You gave a metaphysical argument against set theory which I don't totally agree with.
• What is the difference between actual infinity and potential infinity?
You seem to be forgetting about all the wasted time spent using that theory with unsound premises to create conclusions which are inconsistent with the sound theory

Name me ONE conclusion that ZFC proved about arithmetic that is not sound?
• What is the difference between actual infinity and potential infinity?
That a conclusion from a theory with unsound premises happens to be consistent, or "the same" as a conclusion from a theory with sound premises, might be completely coincidental. You seem to be forgetting about all the wasted time spent using that theory with unsound premises to create conclusions which are inconsistent with the sound theory, to focus on one conclusion which coincidentally happens to be consistent, in an attempt to justify use of the unsound theory.

If you are working within a FINITE fragment of ZFC, then the result is always arithmetically SOUND (that if ZFC is consistent). It's a matter of technicality.
• What is the difference between actual infinity and potential infinity?
However, I would say that any proof which utilizes "infinite", or "infinity", is not a sound proof.

No! not always, if the proof is carried in a FINITE fragment of ZFC, and the proved statement is an arithmetical statement, then this is already known to be SOUND, i.e. any finite fragment of ZFC (even though it speaks about infinite sets) if it proves an arithmetical statement, then that arithmetical statement is part of TRUE arithmetic, i.e. it conforms to a proof that only relies on finite objects.

Not only that! It is expected after knowing Wiley's proof of Fermat's Last Theorem (which he actually did it in a theory even stronger than ZFC! Even though mildly so) that it can even be carried in Peano arithmetic, which is of course part of TRUE arithmetic, that's what experts on the proof say, so suppose for the sake of discussion that this happens, then that would be a clear example of a theorem of PA (a theory solely about the finite world which is HIGHLY applicable, actually the most applicable theory ever) had came to be proved first via ZFC, and that knowing that proof in that higher system served as a guide to proving it the lower reliable system. So a theory basically about the infinite did help us understand provability within a theory about the finite, a kind of a detour though it to simplify matters!
• What is the difference between actual infinity and potential infinity?
I just want to give an example of a sentence that is highly related to the finite mathematics, that can find a solution in a system that speaks of infinite objects that mathematicians seems to agree upon. That of Fermat's last theorem! This can be solved in ZFC. It's not yet know if it can be solved in PA. However the theorem is clearly about arithmetic, and its formulated in the language of PA, so it is not essentially about any infinite object. But a theory speaking about infinite objects (i.e. ZFC) can prove it. Now I'm not claiming here that ZFC had contributed to the argument of the proof of that theory, certainly not. But seeing that it is provable in ZFC and yet not known to be provable in PA yet, speaks a lot of that issue.
• What is the difference between actual infinity and potential infinity?
It still requires will power. It may take a long time to convince people that a particular habit is bad, but once it is recognized as bad, without the will power to stop they will continue to do it.

It is not enough to point it as bad, with mathematics you must demonstrate an alternative system with superior utility, something that is better. Once a system with superior utility is at hand, the exchange would be immediate, you won't need any will power. People will readily exchange older cars for new more efficient ones if they can afford to. Its a pragmatic argument.
• What is the difference between actual infinity and potential infinity?
I find myself in full agreement with what you said! However, I do think that imperfections would sooner or later show themselves, no matter how much useful they are. And at that point the habit will break. Sometimes its accidental! Some genius come and discover a system which immediately show superior utility, thus overshadowing the existing system and thereby exposing all of its hidden imperfections by comparison. The switch would be immediate! However, if we want to plan matters, then I'd agree, it won't be so easy, without demonstrating superior utility of the new system, it would be a very hard task to shed the older well accustomed one. On the other hand, there is always the quest of extending theories in mathematics and finding better alternatives, this is an ongoing practice really, even finding ones that suit better metaphysical principles, even those are in development, I know that no one would buy them just for that, but if they conform to reality more than others, they'll be convincing first to philosophers, then gradually over time they would find better utility. So I don't think the matter is so grim.
• What is the difference between actual infinity and potential infinity?
The tool might be the most primitive, awkward tool, but if it brings us success in what we are doing, then we are not inclined to look for a better one, That success misleads us because it hides the fact that we really need a better tool, by making it appear like we have the tool we need

Yes, I would agree with that of course. But it need not be so really. The issue here is that if we have something useful, even if it is not the best, we ought not to reject it, we initially accept it, on pragmatic basis, but at the same time we look for a better system. But until this *better* system is in our hands, we keep working with what we have, despite is metaphysical (or other) weaknesses. I think this is a correct policy.
• What is the difference between actual infinity and potential infinity?

Now how to decide if this is true or not. We simply try it and check. Here the stronger system would not be just a facilitator as it was the case in role 2, no, here it would prove additional theorems about the reliable sector of our world, sentences that the original reliable system cannot prove. Now we go and check if those additional proved sentences have applications, if so, then this would mean that the added theory is increasing our knowledge about our real world, and thus the axioms are true! This is challenging! If it fails and proves misleading, then we REJECT the extended system from being a part of useful mathematics, and only keep it as a piece of beautiful analytic school of art (Mathematics for Mathematics).

Lets see how this fair in practice. Now we know that every *finite* fragment of ZFC is arithmetically sound!!! That is: it doesn't prove false arithmetical statements! Now true arithmetical statements are always considered as "possibly applicable" because they are statements about finite objects, they are the kind of statements you've desired as giving unlimited possibility of measuring!!! Now in applications if we are going to apply a fragment of ZFC then this would be of course FINITE, i.e. well take finitely many theorems of ZFC and work within their deductive closure, now ALL arithmetical sentences proved in such a fragment are guaranteed to be TRUE, and so potentially applicable!!! That's why ZFC practically speaking works as a foundational system for useful mathematics.
• What is the difference between actual infinity and potential infinity?

It is sound! because it is the pure conclusion of the original sound theory. I'm just using theory A as a facilitator and then I'm checking it again in the pure theory B, so there is no harm. For example you have a certain argument for a proof in theory B (the reliable theory), but there are some missing steps, you translate the argument to theory A, sometime theory A manage to fill the gaps, then you go back to theory B itself and check if that filling is correct. There is no problem with that approach as long as we are not depending solely on what theory A is doing. To facilitate matters is an important practical tool. Its no magic, we know why it works, because it increase the expressive power the logical deductive system, it supply you with more tools to solve analytic problems, many times the difficulty in proving a theorem (in the original reliable theory) is not due to you being in the wrong direction, no, its due to poor analytic tools, those are supplied by the more powerful theory that extends A (even if the additional premises of A are unreliable), and that's why it manage to fill the gaps, but we always go back to the original theory and check the complete proof which is solely this time coming from theory B (the reliable theory). Theory B is the final arbiter. We start from B and we end by B; theory A is just an intermediate step quickening provability in B, just a facilitator. That should be right!
• What is the difference between actual infinity and potential infinity?
I don't understand why you call these theories, which are not based in sound premises "stronger theories". They are clearly weaker.

*stronger* is a logical term. Theory A is stronger than theory B if and only if every statement provable in B is provable in A, but not every statement provable in A is provable in B.

You refer to such a theory as a "technical guide", and say that they are aimed at practice. So lets say that they are like hypotheses. We apply them in the attempt to prove whether they are true or false. So we must be willing to reject them when they are proven to be false.

Yes! Of course. If the stronger theory proves to mislead us about provability in the weaker theory most of the time, then we reject it. On the other side if most of the times it proves to be helpful, i.e. assist us in proving theorems of the weaker theory, then we would say that it is playing a conservative extension role over the weaker theory most of the time, then we adopt it as a guide only.
• What is the difference between actual infinity and potential infinity?
If it doesn't make sense in relation to the real world, then it cannot be a true premise. Therefore the proofs which are derived are unsound

Unfortunately you are not following what I'm saying. I'm speaking of two roles that a stronger theory (deductively speaking) can play. The FIRST role is in proving sentences in a language about finite objects that are NOT provable from some known finite theories having SOUND principles. Here you can make your objection above. But your objection above doesn't extend to the SECOND role. The second role is that stronger theories do facilitate provability within SOUND theories themselves!!! I need you to concentrate here, suppose you have a sound theory, by that I mean a theory whose axioms are coined in relation to the reality, i.e. they are realistic rules, and so they are sound, i.e. they meet reality, now supposing that this theory is consistent, then provability of a consistent theory whose axioms are sound, is always sound, i.e. its theorems always conform to reality, since provability (in consistent systems) preserves soundness of axioms. Now I'm speaking about those kinds of theories which we think that they'll have many applications in the real world. Now even for those theories, the stronger theories which are not reality based (like those speaking about the infinite), still do possess much expressive analytic power that can enable them to prove theorems INSIDE the realistic theories in a much shorter manner, here with this situation the proofs are sound of course, because they are only shortening proofs of a sound theory. This technical assisting role is very important. You and others in philosophy might underestimate it, because this second role is in principle dispensable! But there is a great difference between "in principle" and "in practice", I'd agree that they are in principle dispensable, but in practice they are not, because we are humans, so theorems of sound axiom systems that are provable from very long proofs will not be discovered by the human mind, while the assisting stronger systems would enable discovering those theorems because they can prove them in shorter steps, and then afterwards we can go back to the original sound theory and find the long proof of those theorems. So the stronger imaginary (potentially unrealistic) theories can play the role of a technical guide of the weaker more realistic ones. Truly the FINAL acceptance of the theorems would be by establishing their provability in the sound systems by finding those very long proofs, but finding these long proofs can be assisted by provability in the stronger theories, even if the stronger ones are not realistic. The reason is because the stronger ones have more expressive analytic power. Its a pure technical matter.
• What is the difference between actual infinity and potential infinity?
Therefore I believe that "infinite object", or "infinite objects" represents a misunderstanding of the natural boundaries which objects have. Unless "infinite object", or "infinite objects" can be given some real meaning, as referring to something real in the world, supported by real principles, it's simply nonsensical talk..

It doesn't matter whether its makes sense in relation to the real physical world or not! The point is that it can serve as a strong reducer of proof lengths of theories that only speak about objects related to our real world, thereby it can aid our analytic deductive reasoning about our real world, it makes it technically easier to analytically follow up truthful assumptions about our own world, because it "shorten" proofs. That rule of making it easier to follow up the analytic consequences of truthful assumptions about our real world, is a very useful tool, as as far as this property is concerned then it makes full sense to introduce them as Tools to help understand our reality in a much faster and easier way than without them.
• What is the difference between actual infinity and potential infinity?
but this is a self-defeating assumption because it assumes an object which is immeasurable, and the purpose of assuming the infinite is to make all things measurable

I read all of the above account of yours carefully. It is really nice. I should say that it added a lot to my knowledge. So Thank you for that.

There is a lot of depth to what you are saying. I won't here make my final stance with or against it, since I need myself more time to examine it in a deep sense. For now I'll outline the potential difference between what you've presented and the commonly held views in foundations of mathematics.

What you said runs against standard set theory. For example you adopted the strategy of assuming the infinitude of observations, of possible qualities (largely its a stance made because of our incompleteness rather than being a truthful statement about the real world, as you maintain). However you refused to put the infinite among those qualities, and you emphasize that its not, it is just a pure quantity. The problem is that the common modern understanding in set theory of the infinite runs to the opposite of that. It views a lot of grades of infinitude, and the absolute allowance of all observations is something that NO formula in set theory or logic can describe. We may paraphrase it as saying that there no limit to measurability made inside theories, but this here applies to various degrees of the infinite as well as to finities. So you are not differentiating between the 'absolute capacity' of measurement, which is sometimes ironically called by some set theories as the absolute infinite, [which you call the "infinite" by the way], and the various grades of the infinite, the latter ones are using your terms qualities, and they can be measured in an effective manner, while the former one (the absolute infinite) is what you cannot measure nor can formalize it as an axiom, and using your terms I would describe it as not really a quality, its a pure quantity (using your terms), this absolute infinite is something that no set theorist tries to capture by its axioms or theories, its an unlimited tendency of measurements. But the scope of measurement in your case is limited to the finites, while for the traditional set theorist it doesn't stop their, it can encounter various grades of the infinite, he will answer you that you are confusing $\omega$ [the first infinite] with the absolute infinite, and this is wrong. The scope of allowable measurements with the set theorist is vastly larger than the narrow one you are allowing. That's the difference.

However, my argument above (the one you've answered to) is not that deep. It only says that theories that have capture SOME infinite objects, are vastly stronger (deductively speaking) than theories that only capture the finite, that's why technically speaking, those stronger theories can help even prove some theorems of strictly weaker theories that only speak about the finite, not only that it can prove theorems spoken about in their language that those weaker theories cannot prove, and those sentences are of the kind speaking about infinite objects and relations between them and properties of them, so they are (generally speaking) the kind of sentences expected to have application in our finite world. This mean that theories speaking about infinite (as well as finite) objects can aid in measurements of the actual world via proving those sentences of them that are concerned with the finite realm of them. It supply us (technically speaking) with more and more sentences about finite objects, and so enrich our knowledge base and potential to make descriptions in our finite world. Its a pure technical issue. So they are useful and can make contributions to our finite world, although they are theories that have the capability of speaking of infinite objects (pure unities with infinitely many qualities).

Just being a pure unity with infinitely many qualities, it doesn't mean that it is unknowable, or that it is not measurable! That's the basic message that set theory is telling us. The earlier philosophers who thought that, like Aristotle, were mistaken!!! Cantor clearly showed how to make descriptions of various grades of the infinite, and he successfully did so, without being encountered with any inconsistency so far (it has been more than a century since then!).

Since contemplating SOME of the infinite as objects, or as qualities in your terms, did enrich our parcel with sentences about finite sets and numbers, which can possibly have applications, then it enriches our potential for application, so we are justified in doing such contemplation. Its a technical utility point of view.
• What is the difference between actual infinity and potential infinity?
Hi Zuhair, I'm having a little difficulty understanding some parts of your post, but I'll provide my interpretation with some criticism, and you can tell me where I misunderstand.

Hi Metaphysician Undercover! I also cannot understand what you've wrote. I think we are departing a part.

My argument was a very simple argument. I was simply speaking about Peano arithmetic "PA". Peano arithmetic only speaks about natural numbers, which you can in some sense imagine them as indices of the quantity of members in finite sets, so we are speaking here about measurements related to FINITE objects, and so the language of PA doesn't encounter any mention of the infinite, although it allows for the potential of having infinitely many naturals, but it doesn't speak of that infinitude of naturals as an object on its own right. The only objects that PA speaks about are naturals which are in some sense measures of finite objects. So generally speaking PA would be the kind of a theory that is expected to have applications about objects in our finite (or potentially infinite) universe. So all sentences written in the language of PA are statements about finite objects, so they all speak about the state of affairs related to finite objects, as we regard them to be potentially applicable!

The problem is that MOST of sentences written in the language of PA are not provable in PA. So we are missing a lot of sentences that might have useful application in our real world, because PA cannot prove them. However those arithmetical sentences can be proven from theories that encounter speech about existence of infinite objects, like set theory for example, so ZFC can prove arithmetical sentences which cannot be proven in PA. Notice that I'm speaking about arithmetical sentences, i.e. sentences about natural numbers, i.e. statements about measurement of the FINITE, so those are statements that can have applications in our real world, and some of those sentences are provable in ZFC while PA cannot prove them!

Not only that there are sentences that PA happen to prove,i.e. theorems of PA, but yet the proofs of those sentences are too long to be first discovered depending on PA alone, that it is much easier to prove in ZFC, because the proofs are much much shorter. In this way ZFC can help us discover theorems of PA itself.

You see what I'm trying to say here. That's why we revert to ZFC, because it is more powerful and more expressive than PA, that we can even prove theorems of PA itself using ZFC in a much easier way! And also using ZFC we can in addition prove sentences written in the very language of PA itself, that PA itself cannot prove. And those sentences can express facts about measurements of finite objects, and so might have applications.
• What is the difference between actual infinity and potential infinity?
Can you describe this "need" for me? If mathematics prior to the 19th century got along fine without speaking about the infinite, where does this need to apply the infinite, in set theory, come from?

This is indeed a very big subject. And what you've said is reputable. Indeed some imminent mathematicians took that stance, and even a much more aggressive stance of even not allowing for existence of large finites (see finitism and ultrafinitism) let alone infinities. One can indeed write whole volumes on that subject. However, I'll present a possible counter-view:

If the theories about the infinite realm proves to be indispensable for harvesting some useful arithmetical sentences about the naturals, i.e.that no theory limited to the finite realm can effectively prove them, then this would make the case for mathematical investigation of the infinite!

Not only that, sometimes those infinite theories might not be indispensable for the above sake, but being more powerful they make proofs about such sentences much easier to have, and so one can contemplate many versions of equivalent systems about the finite world, much easier than when working with theories restricted to just finite objects. It is this ease that is also important, since it would have heuristic value to discoveries in the finite realm.

Of course there is still the objection that such additional sentences about the finite world that theories encountering infinite objects prove, these would prove to be FALSE arithmetical sentences, and so be misleading rather than useful, and would be deemed as waste of time and efforts. That's of course possible, but until such an argument is provable, the window is still open for mathematical investigation of the infinite, under the hope that it might play the above-mentioned role, and so we need to give it the benefit of doubt!

Of course there are other more radical objections to your line of view, like the mathematics for mathematics viewpoint, and like the other direction objection that is our physical world itself being of ACTUAL INFINITE reality and that our current physical theories and observations being erroneous about that aspect, etc... I didn't want to go to those, because I honestly think that the bulk of evidence supports a finite (or at most potentially infinite) outcast of our universe, and that mathematics ought to be useful in understanding that universe, and therefore I approached it from that perspective as given above.
• What is the difference between actual infinity and potential infinity?
The mathematician might say 'we have to be able to apply the infinite, it's part of mathematics', but really all that the infinite is, is a thing of beauty, a beauty which is negated by any misguided attempt to apply it.

I'm not really sure of that? But as part of history of mathematics, mathematics prior to the 19th century were very cautious when speaking about the infinite. They actually almost avoided it, and only used finite kind of numbers and entities (albeit infinitely many). Not only that! Most of hard core mathematics can be encoded in very weak systems of set theory, however those systems need the infinite most of the time, but they are very weak infinite systems like first and second order arithmetic.
• What is the difference between actual infinity and potential infinity?
Do you agree that this is the basis of that dualist separation between the real world and the Platonic world? The human mind apprehends the world as consisting of numerous possibilities. In order for it to understand each, every, and any possibility, the mind assigns to itself, the capacity to understand infinite possibilities. But that assignment is wrong, because the human mind is restricted by the real world, being a part of the human body, and so its capacity to understand is really restricted. So the human mind has created this dualist premise, and all these dualist principles, in an attempt to give itself the capacity to understand anything, and everything, when in reality it doesn't have that capacity. That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it.

Yes. Actually I find myself in total agreement with what you said in this particular last comment. Of course we are speaking about mathematics judging it according to its possible usefulness. Some mathematicians might reject to look at mathematics from this perspective, seeing it external to what really constitute its cor content.Seeing it like judging chess by how much profit it brings to the chess player? They might insistent that mathematics is the discipline devoted to some imaginary world were there is no impediment to carrying out strict rule following scenarios, so giving itself the maximal freedom in doing so, and enjoy that practice all for itself without caring about whether it would be applicable or not. Something like the position of "Art for Art" and "Art for people". Your views here suite "Mathematics for science", while some mathematicians might insist on "Mathematics for Mathematics". But I agree with you that the importance of mathematics is related to its role in furthering our understanding of the world, and if it didn't have that rule, it wouldn't have had all that fame and seriousness in studying it, it would have been something like chess, a mental game, a kind of sport, or even art.

I agree with the duality policy. The real issue is how to judge when a mathematician is going a stray? I mean as far as possible contribution to knowledge is concerned (i.e. application). I think a real foundation of mathematics must help direct mathematicians towards producing more beneficial mathematical theories. But how to judge this? I think this is a very important question? We need a foundation for applicable mathematics! But I'm almost very sure that a lot of mathematicians, possibly the most, wouldn't care the hell for that, they'll view it as too restrictive, and favor diving deep into the world of logically obedient rule following scenarios, no matter how wildly far their imaginative worlds are from reality. Sometimes I think this is like the dualism of religion and state in secular states. Let the mathematicians dive deep into the imaginary platonic world they like, and let science work with its strict observance to reality moto. The important matter is not to confuse both. We only need to coordinate both at applications!

That Platonism is self-deception. It was far a good cause, but when it runs its course and we see that it is impossible for it to give us what it was designed to give us, we need to get rid of it.

The real problem is even if it is false, still the logically obedient strict rule-following themes it negotiates can prove to be extremely useful, even if in part. The real problem is that we'll never know at which stage it will "run out its course"? Possibly one day foundations for 'applicable' mathematics would issue, having clear cut edge between what is beneficial and what is not?! Perhaps by then this platonic dream would vanish! perhaps?! but I don't really know where such a thing would start? or even if it could start really? Until such alternative is found, we'd better keep the current dualist stance.
• What is the difference between actual infinity and potential infinity?
These objects might be completely fictional, imaginary, and not intended to represent the real world at all.

This is TRUE of many mathematical disciplines. For example a lot of set theory stuff is so imaginary that it might not even find any application at all. However, no one can really tell. Even imaginary numbers turned to have applications, even non-Euclidean Geometry turned to have applications. The problem is that we don't know really what our reality adheres to, or even what discourse about obviously imaginary objects could be useful in applications about the real world. There must be some shared realm for those applications to exist. The problem is that if we take Quines-Putnam indispensability argument, then even those non-spatio-temporal features of mathematical object might need to be accepted as part of reality, even though not a physical concrete kind of reality, but some kind of reality there!? The mathematician usually do not bother with these philosophical ground. All of what he cares for is the analytic consequences of his assumption, which for clarity and simplicity they are usually stipulated outside of the confines of space or time or both, or within the confines of some imaginary world that has its own space and time characteristics, as well as its own part-whole relationship with respect to eternity issues in it. Most mathematicians work primarily in a Platonic world! Philosophy comes later!
• What is the difference between actual infinity and potential infinity?

Yes, I think there is an intermediate position. Mathematics is producing rule following obedient fictional objects and scenarios. However, those happen to have applications in the real world. I suspect that the matter is not accidental. There is seemingly some common grounds between imagination and the real world. Some rules about arithmetic works fine when applied to real objects, and it really succeeded in increasing our understanding of the real world around us. On the other hand obviously there are rules that are not applicable to the real world like having infinitely many numbers, etc... I think logically obedient rule following imaginative scenarios do have some common grounds with reality.
• What is the difference between actual infinity and potential infinity?
Then it would be impossible to create a reasonable hierarchy like you were talking about, if the meaning of tight and loose could vary.

You can! The hierarchy would be more of a mold, a frame, that suites a generality purposes. Of course in the particular application the hierarchies would differ.

By the way do relativity theory speak about rules about the mathematical objects used to write its laws with? Aren't those mathematical objects a part of the theory? I don't think relativity theory assumes that numbers for example have a mass, or that they move with a speed less than light, etc.. Those mathematical objects are fixed, eternal, unchangeable. It's the physical objects that the rules of relativity theory applies to. I don't think that the mathematical objects and rules that it uses has anything to do with relativity theory. Imagine that number 1 for example will rut with time? That's crazy! Isn't it.

I agree that mathematical objects are ideal. And when using them in applications one must be cautious about hidden mathematical assumptions that might causing blurring or even faulty theories.

We use possibly fictional objects to display the mathematical rules with, because this is the most evident way in which it can be presented. Most of these rules, as well as the objects manipulated are non-spatio-temporal. But I think we can have pseudo-spatio-temporal objects representing mathematical worlds, thus in some sense approximating the real world. But I think also that nothing of the rule physical world law about physical objects would be applicable to these realms either.

In nutshell I think that set theory involving "collections" "elements of collections" "sets" "members of sets" etc.. all of these can be well understood in terms of Mereo-topology in a fairly easy manner. The hierarchy of sets is I think very essential to understand higher kinds of mathematics. For who's to say even those can possibly find some application in the real world?!
• What is the difference between actual infinity and potential infinity?
The point is that your description of the distinction between "loose" and "tight" does not provide us with an indication as to what these terms really mean. The reality is that the parts of a collection are either loose or tight depending on what type of relation they have with each other. Therefore there are all sorts of different types of sets, which may be named dependent on the relations between the parts. We cannot just classify loose and tight sets, just like we cannot just class soft and hard physical objects, there are all sorts of different type of objects which we name, like 'cars' and "houses"

You Can! if your aim was to FOUND (i.e., lay the basis for) matters with. We leave "tight" and "loose" like blanks to be filled with the relevant application. So tight an loose are left as "primitive" concepts, those would take different meanings according to the working application. Of course collections would have different meaning across all applications, but they will have consistent meaning within the same application. Like how number 1 can have different meaning across applications.

I also want to note, actually an apology, that what I'm saying here about Mereological, well actually Mereo-topological, understanding of set theory is not the conventional line. It has been loosely suggested by David Lewis, but not with all such detail. So set theory is not dependent on those views I'm posting here. But to a great extend those views can make one understand what's going on with set theory as far as applying the rules and the logical flow within these set theories is concerned.

I find your idea that an object cannot have parts unless its subject to temporal separability as un-supported. Especially under imaginary grounds. We can fathom the imaginative line of having an eternal unicorn. Now this entity (which do not exist in the real world) does have parts, for instance the corn is a part of the unicorn, its head, tail, legs, etc.. all of these are parts of a unicorn, so the unicorn is not a mereological atom, even if it was an eternal being. So descriptive imaginative wise we can fathom what it does mean to have eternal objects with parts. But in the real world if one thinks that an object if composed of parts then those parts must have been in existence "before" the whole, i.e. every object must be "formed" after parts by some force connecting those parts that occurs in some moment of time of course after the existence of the parts, then according to this synthetic hypothesis (which holds mostly in the real physical world), then of course one would be bond to reject eternal objects being synthesized from parts, since there should have been a moment where those parts were separate and then after that another moment came where they'll possess a relationship to each other that caused the unit of that object.

In the mathematical realm, we don't adhere to such observations. The mathematical realm is changeless with time (unless time itself is adopted in some mathematical models where change is studied) and objects in that realm can be dealt with as having parts (proper parts I mean) without having to have a formalization (synthetic) moment. That applies to classes (i.e. collections) with many elements, where their singular parts can be understood as their elements, now those multipleton collections do have parts and they are supposed to be eternal in the platonic realm. Now that is obviously false in the physical realm. But this doesn't mean it is false of every realm! Platonists holds that the platonic realm exists, so it is a true realm, while fictionists think its false.

I'd say even if that platonic realm is FALSE (i.e.doesn't exist), still, the logical-mathematical rules displayed in them are not necessarily false. And they can hold of some real scenarios, and so can possibly find applications, and that what really matters!
• What is the difference between actual infinity and potential infinity?
So we need to distinguish temporal continuity from spatial continuity. If the parts are inseparable in the sense of temporal continuity, then the object is eternal, and it would appear contradictory to even talk about the object as being composed of parts. Such an object is the fundamental, or base "unit". It can have no parts because that would imply that the unit is separable in time. The spatial extension of such an object is dubious.

Ok, I agree it would be eternal since its not actually breakable. But why it can have no parts? Any object is itself a part of itself. Perhaps you mean it doesn't have "proper parts" [parts of an object other than itself]. I'm under the impression that you think that an object must be breakable to into parts in time in order for us to say that it has parts. But this is not correct. Even if we have an object that is eternally not breakable, still it can have many parts connected by tight connection in a manner that renders it a unit, it doesn't mean it doesn't have proper parts, it only means its no breakable to them, but it can have them always as parts of it. In real life having eternal objects is itself faulty. So the idea of eternal objects is hypothetical or actually imaginary, it suits framing mathematical objects, because usually we work with mathematical objects in some Platonic realm, and that realm appears to be time free, sometimes even disrupt spatial reasoning as well, anyhow.

Now if we work in an imaginary space in which time has no effect, i.e. doesn't change connection relations of objects to each other, still it is imaginable for those objects to have parts, so having parts is not a function of temporal separability as you hold. Not only that, still without time we can fathom of having objects that are composed of units that are loosely connected to each other. So we can have collections having many elements.

However, we may reach into a definition of "loose" and "tight" connections using this spatial x temporal distinction. For example you can say that:

x tightly connected to y if and only if at all times x is connected to y;

while

x is loosely connected to y if and only if sometimes x is connected to y and sometimes x is not connected to y.

According to this definition a "true unit" would be an object that is never breakable nor is continually in connection with an external object.

Of course in a mathematical realm in which time is not operable, like "most" of mathematical contexts, then all unit objects in that realm are true units.

Of course in an imaginary context one can incorporate time into the mathematical world and actually use that time-dependent definition of "true unit" and of loose and tight connection. So a collection would be either a true unit or a pseudo unit (a temporally appearing continuous object that is temporally separable from external objects), and being an element of a collection would be being a true unit part of that collection. Then we introduce naming of these collections with true units, and everything would run as I intended.

The problem is that this would add additional features to the picture, namely temporarily, which is not all that desirable in a mathematical realm. For the purpose of defining sets, we can simply hold the dichotomy of loose and tight connection as primitive concepts without relation to time. Our aim is largely descriptive. Since set theory serves as a foundation for mathematics then the particularities of what decides the "units" of a certain mathematical discipline is stuff related to the particularities of that discipline itself, so in Geometry units would be "points", in arithmetic units would be "numbers", in set theory units would be "sets", etc.... Here we are only concerned in introduced a general descriptive framework that can be applicable to diverse mathematical disciplines, and possibility even non-mathematical spheres of knowledge as well! For example the idea of having a "true unit" in time, might be useful in understanding the ontology of time and space?

I think that "collections" and "sets" are pervasive, so they can be used in any field of knowledge. That's why I'm dealing with that matter in a merely descriptive manner. So consider relations of "connection", "tight connection" to be primitive relations. Then using logic, we can define the terms "singular" (i.e. unit), collection, element, set, member. And that's all what we need. Then build the hierarchy in a gradual manner, and you are allowed to go as high as consistency permits. That set you get all the extensions of ZFC set theory, and thus encode any area of knowledge really, but specially mathematics.

So if we name "membership", what that name actually refers to is the relationship implied by "membership". Now we must guard against deception. We can name membership when no reasonable relationship has been identified. Therefore there is no point to naming "membership", unless to deceive. The relationship ought to be named directly, without the medium "membership". Furthermore, what follows from this, is that this "name", which is the name of the collection, but actually represents membership in the collection, which in itself represents a relation, is an even further layer of representation. So we have three levels of representation now, the name represents the collection, the collection represents membership, and membership represents a relationship. Plato warned us against such multi levels of representation, calling them "narrative". Any hierarchy produced in this manner would be extremely unreliable, as we ought to refer directly to "the good" to produce a hierarchy. Names are tools used for understanding, so the good here is understanding. Multi levels of representation are conducive to confusion rather than understanding.

I couldn't manage to follow that really. But in my usage when I used names, I used them to name "collections" which are "objects" and not relations. When I write "member" or "membership" this is a symbol to denote the membership relation, I don't mean by those symbols to be "names" those are not names, those are definable relation symbols. Names only are assigned to name collections which are objects. I don't think that a hierarchical build-up would be confusional, why? The definitions I gave were very strict. So I'm using "names" in a particular context, that is to name collections. symbols used to "name" relations are not called as "names" here.

ut where I find confusion is in your failure to recognize the distinction between naming an object, and naming a relation. These are two distinct uses of a name, like the difference between noun and verb.

You said I'm not discriminating between naming of objects and naming of relations. I need to see where exactly I made this confusion. I introduced names for the specific context of naming collections, and I defined collections as totalities of loosely connected singulars (units) and I justified my claim that such entities are totalities and such totalities are objects. So I was all the way speaking about naming objects. Where do you see me introducing names to name relations and confusing those relations as objects? I never said we can name for example the relation 'membership' and I've never introduced a name for it (in the particular context of name that I've used) where do you see me speaking about introducing a name for the relation "element-hood" and speaking about such naming? The relations that I've used are "connection" tight and loose, part, membership, element-hood, where do you see me speaking of attaching names to those? I never said that! I only spoke specifically about introducing singular names for collections, and I specifically defined what "collection" mean, and I justified that being an "object".
• What is the difference between actual infinity and potential infinity?
The hierarch is produced naturally by understanding the nature of continuity, and the strength of relationships. So we begin with some fundamental determinations concerning the nature of continuity; for example what is prescribed above concerning spatial and temporal existence, and we proceed to name relationships according to their strength, producing a hierarchy of relations. There is no need for a hierarchy of sets.

As attractive as it sounds, this proves to be extremely difficult. Experience along such lines are moot. Its hopeless. Without a hierarchy of sets, or similar structure, there is almost no hope to encode most of mathematics. You will only have the sketchy picture of prior to the twentieth century mathematics. But again this is one of the most useful kinds of mathematics.
• What is the difference between actual infinity and potential infinity?
Yet now you say that material connection does not imply "inseparability". I would agree with this if we clarify by distinguishing between actually separate, and separable.

I'm speaking within the confines of a mathematical realm, some platonic realm in which time doesn't cause any change to connection relations. So what is actually separate is always separate, so separable is separate, and so temporal x spatial connection is immaterial in this realm. We only have spatial connection and separation. That said we need to revert again to loose versus tight connections.

My account entails that the existence of connections between parts of an entity is what qualifies that entity to be an object. So having loose connection is fairly enough for that quest. You don't need tight connections between parts of an entity to qualify it for being an object! NO! loose connection can do the job, so an entity in which loose connections between its singular parts exist, is perfectly qualified of being an "object. However, you need tight connections to form units (singulars) but units are just special kinds of "objects", so an entity that has tight connections over its parts and it itself doesn't have that kind of connection to external objects, that would qualify it to be a unit object. But objects need not be units. They can be totalities of loosely connected units, or what I call as "collections". So as such collections qualifies for being "objects". I hope this resolves the confusion.

Now the loosest kind of a connection is the one made through descriptions. And so an entity of separate singular parts that are commonly describable in a manner that isolates them from objects not fulfilling that description, is the least needed condition to qualify it as an object. And we have this situation with collections as I defined.

The above coupled with a naming function that name collections by singular names, is enough to build up the required hierarchy in which almost all of mathematical objects can be carved!
• What is the difference between actual infinity and potential infinity?
But there is no need for a hierarchy of sets, that is a faulty premise.

Actually from experience with mathematics. We do need a hierarchy of sets. The other alternatives are not so promising.

I find temporal versus spatial separability not easy to fathom. I'll have a better try to fathom your notes about it.

I do admit that there is some ambiguity with my characterization of "loose" versus "tight" connection.

I view the connection between adjacent parts of an apple as being tight connections, while I view the descriptive joining of many apples satisfying some predicate as being a kind of loose connection. "Separability" in my sense means absence of tight connections, so loosely connected objects are separate, while tightly connected objects are not separate. I define a "unit" or a "singular" as an object K such that for any part x of K, the part of K that is the complementary part of x, denoted by x'^K, is in contact with x, i.e. x'^K is in contact with x. So when an object is a unit, then it is not the totality of non-tightly connected objects, and the other condition is that a unit must be not be in tight connection with an external object, i.e. it must be separate from external objects, so it can bear loose connections to external objects, but it cannot possess tight connections with them.

So a collection is a totality of one singular object, or a totality of many singular objects loosely connected to each other through descriptive joining of its singular parts through having a common description that isolates them from other singulars not fulfilling that description.

A name is a singular (a unit) that as you called it a "tool" that helps us understanding, here it helps us direct our attention to a specific collection, in such a manner that we can speak of multiplicity of collections in a hierarchical manner. Each collection has only one name, and each name only names one collection. Sets are singular names of collections. They are not names of relations. Naming of relations is a different subject, and I've never attempted to speak about it in any of my prior comments. I've been always speaking about naming collections, and so speaking about naming objects, and not relations.

When a set say set x names some collection C, then we call each "element" of C (i.e. each singular part of C) as a "member" of x. In some sense membership would copy element-hood but transfer it to an object external to the collection, that is to the name of the collection. But you need not confuse "membership" as a name for "element-hood", No! That is not the case. Membership is not a name, it is a relation, so it is not an object.

Now through membership relation and sets (i.e., names of collections), one can easily define a hierarchy of sets. And that build-up proves to be an extremely useful tool in our understanding of many mathematical entities. And the witness to that is SET THEORY. In particular ZFC set theory (Zermelo-Frankel set theory with Choice), which proves to be very powerful in understanding mathematical entities and rules, through the iterative buildup of a hierarchy of sets.

Of course for the development of set theory, all of our units are un-breakable over time, and they don't change their tight connections with time, so they are remotely different from natural objects which rut over time or combine with other objects to build bigger units, etc... Here in the platonic mathematical imaginary world, all individuals (units) have non-changeable tight connections over time. So they are as you said "eternal". Then we can freely form collections of them using the descriptive tool, and with the help of the naming relation, we can speak of a hierarchy of them, which helps us encode almost all of mathematical entities in it. Thus serving as a FOUNDATION for MATHEMATICs.

I'll try to keep track with your notions of temporal versus spatial continuity, and come back with comments about it.
• What is the difference between actual infinity and potential infinity?
Actually, I'm looking for compromise, by allowing that a descriptive rule implies a material connection.

Well I do agree that having a common description imply some material connection, but that connection is not the connection that imply inseparability. You can call these connections "loose" connections, as opposed to "tight" connection which is what causes continuity (inseparability), so if object K has tight connection to object L then they are in continuity, i.e. they are not separate, ie. they are in contact; while if object K has loose connection to object L then they are separate. Now what I call as an "individual" or a "unit" or sometimes I call as a "singular", this is an object that posses tight connections between parts of it in such a manner that it is not the totality of two parts that loosely connected to each other, and at the same manner the object itself must not possess tight connections to external objects. Now if we have many individuals such that there is a description that isolates them from others, i.e. there is a description common to all of them but not to other objects, then those individuals would be said to be LOOSELY connected by this descriptive joining, so they are still separate form each other. Now this would be an object! I call it a collection of objects satisfying this property, so it is a collection and its elements are the individuals that are loosely connected in it. So far for collections.

But this is not enough. You need representatives, or actually NAMEs, you can also call them tokens, or labels, those would be singular objects (units) that we arbitrarily assign to each collection, but provided that the assignment works along unique lines, I mean each collection is assigned only one name, and each name only names one collection. So although the choice of which object would name a collection is arbitrary, but once done naming of other collections cannot use that name, so the naming function is not totally arbitrary. Of course this is not Ontologically innocent, it involves adding unrelated material into the picture!

But why names? why should we assign an external object that is singular to act as a name to a collection that may have multiple elements, so why represent a multiplicity by a singular object? With external naming, there is no clear intimacy between the name and what is named, the assignment is arbitrary for that particular aspect. And this is what actually happens with naming generally, its artificial, for example the names used in language are all arbitrary, there is no special connection between the string of letters "horse" and the animal group it is used to represent. So that's the question: why we should bring an external object that doesn't bear a necessary relationship to a collection and make it act as a name, actually a "representative" for that collection?

The answer is to develop a hierarchical account about collections! This cannot be done in an efficient manner without the use of singular names. The idea is that through this artificially made unique naming process, we can define a new relation, called "membership", that act to copy the relation of element-hood in collections but raises this relation to the name of the collection, and since names are singular objects so they can be elements of collections (while collections when they are non-singular objects cannot be elements of collections, so we can't have a hierarchy of collections in collections using directly the "element-hood" relation!!!), so all elements of a collection wold be "members" of the name of that collection. The "name" of a collection, is what we call as "set" in set theory. So for example the set of objects k,l, denoted by {k,l}, is actually the name given to the collection whose only elements are k,l. so k,l would be "members" of that set, i.e. they bear the membership relation to the NAME of that collection, which is the set itself. Through this copying process of elements to members, one can speak of a hierarchy of sets that are members of sets and so on.... And so indirectly speak of collections of collection of...This would give the powerful mainframe needed to interpret almost all of mathematics.

Now you might be suspicious, and actually object, to such a buildup. Since its pivotal rule is built up through an intermediary that involves some arbitrariness, which is the choice of a name per particular collection of course. So its like building a big building that involves multiple big junks of tightly connected material put on top each other using light joining material, so the the whole buildup is bound to fail!

I'd say it would be extremely difficult to make that hierarchical buildup without using names, I tried myself to figure out these possibilities. For example you can use various grades of tightness of connections, like in saying we have: degree 0 loose connection (which are the tightest connections), degree 1 loose, degree 2 loose, etc.... where for each i the i+1 loose connection is looser than the i loose connection. We can do that and define elements of a collection by those bearing the loosest kind of connection between each other and internally of course they use harder degree of connections, and so on...

This can be done but largely on disjoint collections. When there are overlaps, for example like with the case of power-sets, then here it would become very bleak. And even worst in trying to capture non-well founded sets, that it becomes even impossible to use this method for that sake.

So we needs NAMES, to do the intermediary role in developing a hierarchy of sets of sets of..,etc.. It is the simplest way to do it! And this proves to be very powerful logically speaking, that almost all of mathematics can be encoded in it.

This build-up is not due to me. It is largely David Lewis's idea. But here I used 'sets' as names, while in his approach he use them as collection of names. The approaches are equivalent, but mine is more extensional than his. Also related work can be found in point-set topology although taking different technical paths.

According to this line of thought of mine, to me, sets being names, then they out to be assigned only to "definable" collections. Because with naming procedure, you need name something that you can describe first. And so to me all sets must be parameter free definable, i.e. they must name collections that are parameter free definable collections. And naming must proceed in a hierarchical build up from the simplest to the most complicated in a step-wise manner. So definable collections that are not the result of that buildup cannot have names, and this include the collection of all singular labeling objects (names) , and of all singular names that do are not part of the collections they label, etc.. those collections usually called as "big classes", are not reach-able by a hierarchical naming build-up from below, so they cannot be named, even though they are definable!

The nice corollary to this line of thought is that it proves the axiom of choice! and actually of much stronger form of choice, of a definable global choice!

So having a pertinent line of thought about what sets really are, can solve some technical problems, like with the famous problem of axiom of choice here.
• What is the difference between actual infinity and potential infinity?
This is evidence that your system is faulty. We need to recognize a collection as an entity itself, and not rely on a representative. A representative is often incapable of representing to us, the "thing" which is responsible for the real and valid existence of the collection. And this is proven by the fact that some valid collections have no representative.

My outcast is descriptive, while your's is largely etiologic!

Collections can be fairly described and recognized up to identity without resorting to any representative of them, yes I do agree with that. Representatives are neither essential for the existence nor for the characterization of a collection. However representatives of collections are essential for developing a hierarchical account about collections, i.e. when we want to speak about collections of collections of collections, etc...

I don't believe in random collections, yet I don't refute them. However the concept of "random" seem to be different to me than to what you mean by it. It seems from your accounts that you call a totality of unconnected parts as a random totality, because there is NO etiologic like connection between its parts in your sense, so you call such collections as arbitrary, random, etc.. While to me the concept of random only raise versus definable. To me a definable collection of separate unit objects, is itself an object, and it is not a random object because there is a strict "descriptive" rule that joins its separate unit parts. However that descriptive joining of its unit parts should NOT be understood as a kind of "connection" between its unit parts that renders them inseparable, otherwise those would seize to be units, the unit parts still remain "separated" since there is no material (or if you like call it substance) that joins them together, so they remain separate apart, even though they are descriptively linked in some manner. Now as long as there is a descriptive characterization of the collection in a unique manner, i.e. the collection is a definable entity in terms of its unit parts, i.e. like in saying it is the totality of all unit objects satisfying predicate \phi for example, then this definability is (to me) against saying that this collection is a random collection. To me randomness only arise if there is no such a definition, so we have an object that is the totality of unit objects and yet there is no description of those unit objects in our language. Those so called as indefinable collections are really what can be said to be random collections. So for example there is no clear etiologic connection between some particular tear drop and some particular orange, but we can descriptively define a collection of both of those individual objects. That collection is an artificial hybrid, a chimeras, still it is not random, since it has a unique description. Now if a collection is definable in terms of being a totality of unit objects satisfying some particular predicate, then we can assign a representative unit object that can serve to label it (i.e., represent it). However can we assign a label to an undefinable collection? My guess is NO, we CAN'T. Because intuitively we can only label what we can describe. That's why in my philosophic line of thought all sets in set theory are ought to be definable! That is they are names for definable collections!

However my account is different totality from your account. You refuse to admit a collection of "unconnected parts" being an object, to you there should be a kind of necessary relationship between the parts of an entity for it to be an object. That's why you call any try to describe a collection of unrelated objects, as an object, as being magical, since it brings to existence something out of nothing, to you it is some kind of fuzzy entity that doesn't qualify of being an object. While to me it qualifies as being an object no matter how much fuzzy it is, actually even if it is indefinable, still it is an object, so in principle I admit the possibility of the existence of fuzzy collections as well as indefinable collections. Yet I don't see such collections as useful, and I would be sympathetic with a line of argument rejecting their existence. But I don't admit fuzzy sets, since sets are labels, representatives, and it would be against the nature of naming to have them name fuzzy collections, this would be very confusional. And to say that we can name indefinable collections is even contradictory, it is like naming the unnameable. You see the difference of how I use the words "collection" and "set" above.
• What is the difference between actual infinity and potential infinity?
See, under this system, the collection, as an object, can only be apprehended as an object, to the extent provided by the representative.

No this is wrong. A collection can exist and be apprehended without having any representative, or even if it has a representative, the apprehension of the collection need not depend on it. Having representatives is and ADDITIONAL feature. It enables that collection to be a member of higher collections through the representation relation.In other words, representative singulars are only essential for having a hierarchical development of collections of collections of collections... etc.. It is not essential for our apprehension of the collection itself, which could be described in fairly specific manner without reliance on having a representative whatsoever.

What I'm trying to achieve is a hierarchical buildup like bringing separate bricks, define a collection of them, assign some brick (external to them) to act as a representative of them, actually just a label of the collection of those bricks, now there are other representative bricks representing other collections of bricks, now put those representative bricks into collections and also assign other bricks as representative of those collections, and so on... going up. Each brick is a unit, but a collection of separate bricks is not a unit. It is something like this envisioning that I want to construct.

OK, let me simplify this method. Lets use the concept of Mereological atom. An atom is an object that doesn't have parts other than themselves. Now a totality of atoms, is a collection. and an element of a collection is being an atom part of that collection.

Now the buildup I want to speak about is to have a Representation relation that assigns in a unique manner to each collection of atoms, some atom that act as a label (name) for that collection, I call this uniquely distinguishing label as a representative. Of course a collection of atoms might have a representative or might not have any one.

The buildup is to have collections of atoms, each of these collections is represented uniquely by a unique atom, now the next tier is to have collections of those representing atoms, and those collections would in turn also have representative atoms, and so on....

Set theory is about such a hierarchical build up.

Is this artificial. Yes it is! Not only that even the representation relation can be a fixed one. Much as naming symbols are arbitrary in nature.

You can call this hierarchy of names, or Naming Hierarchy. Each name can be understood as a mereological atom. Now we have collections of names, those collections themselves have names that names them, then we have collection of names of collections of names, then we have names of those, then collection of those, then names of those..etc... I'm claiming that Set theory of mathematics thrives in such a naming hierarchy which I happen to call the representative hierarchy.

One need to completely disentangle the concept of representation (unique naming) from that of collection, that's the point that I'm trying to insist on here. A collection of mereological atoms satisfying a predicate \phi, lets denote it as C^phi, is the totality of all of those atoms, i.e. C^phi is an object that has each of those atoms (satisfying phi) as a part of, and such that any object that has each of those atoms as a part of, would have C^phi as a part of! This is substantially different from the *SET* of all \phi atoms, lets denote it by S^phi, here S^phi is the atom that names the collection C^\phi. Now being an *element* of the collection C^phi means being an atom that satisfy phi, that is a part of C^phi, while being a *member* of set S^phi means being an atom that is part of the collection named by S^phi, so it doesn't necessarily mean being an atom that is a part of S^phi. In some sense a set is one step higher than its members, while a collection is not higher than its elements.
• What is the difference between actual infinity and potential infinity?
Now you claim that any random collection of elements is a "unit"

I never said that, nor did I claim it. Actually what I said refutes that!

If I correctly understand what you are saying here, a "set" is a collection. As a "unit" the set is complete, a totality, or whole

You didn't correctly understand what I was saying!

to say that any random collection of elements is a unity is to utter nonsense.

I cannot agree more! Of course, and that's what I was saying. But you totally misread what I was writing. I think because of you "apparently" not having experience with the topic of Mereo-topology.

What I'm saying is a little bit complicated. Seeing your comments, I realize that you completely mis-understood me. But I do concede that what I wrote was too compact.

Lets come to what I meant by "UNIT", I mean by that an individual. For example an apple is a unit, while the collection of two separate apples is not a unit. Now I envision a unit as an object that is not the whole of two separate objects, that is at the same time separate form other objects. This has something to do with separateness and contact. So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. While the collection of some two separate apples is not like that, you have one apple being a part of that collection and the other apple also being a part of that collection but you have a breach of material between them, i.e. the two apples are separate, i.e. not in contact with each other and no part of that collection is in contact with these two parts, such collections are NOT units, they are collections of separate units. The only collection that is a unit is a collection that have one individual, like the collection of one apple, like the collection of one bird, etc.. those are unit collections. You need some experience with Mereology (Part-whole formal study) and connectedness (Separate-contact) formal study, joining both fields you have what is known as Mereo-topology. You need to be familiar with the axiomatization of Mereo-topology, in order to get the grasp of what I'm writing here. These are particular concepts, they are not that philosophical, but of course they can be realized on philosophical grounds.

I define "collection" as a totality of units, of course that totality itself may be a unit (in the case the collection has only one unit part of it), or might not be a unit (like a collection of multiple units: like of two apples, 10 cats, etc...). I need to stress here that "being a unit" or not, has nothing to do with the collection being definable or not, even if it is definable after some predicate still the collection if it contains many units, still it is NOT a unit. Being a unit depends on the continuity of the material in the collection, and not on definability issues or the alike. The only collection that is at the same time a unit, is the singular collection, i.e. the collection having one element, i.e. has one unit part. Otherwise collections having multiple elements whether definable or not, are always not units.

A set (as that term is used in set theory) is a unit object that represent a collection of units, like in how a lawyer represent a collection of many accused persons. Each accused person is a unit object (because its material is in continuity, and it itself is separate form other material) and the lawyer is also a unit object, so here you have an example of some Representation relation where a collection of unit objects (that is itself (i.e. the collection) not a unit since there are many accused person in that collection of our example) that is represented by a unit object (the lawyer). That was an example of EXTERNAL REPRESENTATION. On the other hand there is INTERNAL REPRESENTATION where a single unit in the collection can stand to represent the total collection, like for example when the HEAD of some tribe represents the whole of its tribe in some meeting of head of tribes. The head of a tribe is a unit part of that tribe, and yet it can represent the whole tribe. Any group (collection) of people can always chose one among them that can stand to represent the whole group. This is internal representation.

The usual set theory with well founded sets is a theory of external representation of collections of representatives of collections of representatives of..... It is about tiers of representation of collections.

The empty set can be ANY non-representing individual object. For example take any particular apple. This can serve as the empty set, since apples are not representatives of collections of representatives..

Now take some unit object that serves to represent the chosen apple above (the one we called the empty set). This must be different form that apple, because the apple is not a representative of anything, while that object is representing that apple itself. This latter object would act as the singleton set of the empty set, denoted by {{}}. Now you can take a third object that act as a representative of the collection of the apple (the empty set) and the object that represents that apple (the singleton of the empty set), now this representative object would be the set of the empty set and the singleton of the empty set, denoted by { {}, {{}} }. And so on....

So one need to discriminate sets (which are unit collections that act as representatives of collections) from collections (which are totalities of unit objects). If one manage to fathom that discrimination, then one can of course understand the difference between being an element of a collection, which is being an individual (i.e. a unit) part of that collection, and between being a "member" of a set, here a set is a representative of a collection, and with well founded sets, they are always external representatives of collections (like in the lawyer, accused example), now being a member of a set is actually to be an element (i.e. a unit part) of the collection represented by that set. Membership of sets is a representational act, it is a kind of a singular representational act. Discrimination between the concepts of Collections and their elements, from Sets and their members, is vital for a proper understanding of the subject of sets and classes, and it is something often misunderstood, and misrepresented even at official text-books unfortunately.

Actually If I was to rename matters, I'd call collections as sets, and what is termed as "sets" in set theory I'll call as representatives, and epsilon membership, I'll re-name as "representation step". Anyhow

One needs to be careful! Not every collection has a representative! Even some well definable collections might not have representatives. Although this largely depends on what is meant by "well definable".

I hope you can re-read my prior posting with this clarification.

As about the question of random collections and defined ones, this is another matter, I didn't allude to those yet. I want to define the basic terms, and then if we have some agreement over those, we can go to those issues. But basically I do agree with the sentiment that ALL collections are aught to be definable!
• What is the difference between actual infinity and potential infinity?
You are assuming that two apples is a whole, without any reasons why two apples may be a whole, and why it doesn't take three, four, five, or the proper totality of all apples, to make a whole. Surely there are more apples than two, so by what principle do you apprehend two as a whole?

All of those have their wholes, For any predicate that hold of apples there is a totality of all apples fulfilling that predicate. And those totalities would be different totalities if the apples constituting them are different. But I've just presented to you a particular case. There is nothing special about two here or three or any number.

I like to present matters in a Mereo-topological manner. Now a unit is an object that is not a whole of two separate (not in contact) parts, and at the same time it is separate from any other object. A totality of units is a collection. The smallest collection is a unit. An element of a collection is a unit part of that collection. So the unit collection is the sole element of itself. Multipleton collections are those that are constituted of many units. So they are not the elements of themselves. Now for the sake of simplicity let's assume the ideal condition of all units being unbreakable and actually in-changeable over time, and they won't be in contact with other objects at other moments of time. So no unit object can be split into two separate objects at some other moment of time, nor it would be a part of another unit object at other moment of time. Of course this is an ideal condition. Under that assumption we can have stable totalities and thus I can extend any predicate in the object world as far as that predicate only hold of unchangeable unit objects. If the units are breakable (as it is the case with the real object world) or can come in contact with other units to form bigger units (as it is the case with the real object world) then this method fail, or at least becomes very extremely complex.

Set theory can be explained as an imaginary try to REPRESENT stable collections of units, by stable units. So any two stable collections (i.e. their units are unchangeable over time) would have distinct representative units (whether those representative units are part of those collections or external to them) as long as they are not the same, and each collection is only represented by one unit. This theory of representation of collections by units, is the essence of Set theory. Of course the representative units are ideal, i.e. unchangeable over time. Now while element-hood of collections are being unit parts of those collections, yet "membership" in a set is another matter. Membership in sets can be defined in two ways, l personally like the definition of them being elements of collections represented by the unit, i.e. every set is actually a unit object that represents a collection of units, now those units of the represented collection are the members of that set. Let me put it formally:

x is member of y if and only if there exists a collection z such that y is the representative of z and x is an element of z.

We start with the non representative unit, i.e. a unit object that do not represent any collection of units, this would stand for the empty set. then go upwards in a hierarchy. What I call as the representational hierarchy, where collections are represented by sets (units) and sets themselves are collected into collections, which are represented by sets, etc..... This step-wise hierarchical approach enables a gradual transition from the less complex to the next more complex to the next, and so on... So a nice way would be to start with the empty object (the non representing unit), then to the collection of all sets (i.e. units) representing parts of that empty object, then to the collection of all sets (i.e. units) representing parts of the resulting objects, etc... According to this view a set is always a unit, and that unit act to represent a collection of units.

We can extend the representational hierarchy as long as we don't have a clear inconsistency with it. This way we can encode almost all of mathematical objects in that hierarchy.

Sets not only can represent finished collections, it can also represent unfinished collection, as long as the process of producing the elements of that collection is well defined, like the process of making the naturals by succession from prior naturals and so on.. we can have a set that would represent the process of that natural production. And those are the infinite sets of naturals.

So set theory of mathematics like in ZFC are just a theory about representation of actually finished collections and of potentially non-finishing collections.

I just wanted to put you in the picture, that sets (as used in mathematics) are different from the collections I've spoken about. While the genre of collections is the same as the genre of their elements, sets on the other hand can be totally external to the collections they represent and can indeed be of a different nature. There is a lot of confusion between collections and sets, even in standard text-books of mathematics, and especially there is the confusion between element-hood of collections and membership in sets, that many mathematical textbooks on set theory introduce sets in terms of collections and set membership in terms of element-hood of collections, and this is a great confusion. Sets do not function as collections, no they function as unit representatives of collections, thereby enabling us to speak of a hierarchy of multiplicities within multiplicities and so on... So the set concept is a stronger concept than the collective concept. The former is representational and latter is mereological.
• The bijection problem the natural numbers and the even numbers
However, let's do something different. We take the same sets N and E. We know that N has the even numbers. So we pair the members of E with the even numbers in N. We can do that perfectly and with each member of E in bijection with the even number members of N. What now of the odd numbers in N? They have no matching counterpart in E.

Doesn't this mean N > E?

Not it doesn't. Because it doesn't meet the definition of ">", here is the definition:

X > Y if and only if there is an injection from Y to X, but there is no injection from X to Y.

#### Zuhair

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