So we have division, which is a binary operation. But we also prove that for every positive real number r, there exists a function f whose domain is the set of positive natural numbers and such that, for every positive natural number n:

f(1) = r
f(n+1) = f(n)/2

f is a function from the set of positive natural numbers into the set of real numbers. — TonesInDeepFreeze

Is f(infinity) a member of the above sequence? If yes, what is its value? If not, how could the sequence be an infinite one?

Notice that it is trivial to prove that for no n is it the case that f(n) = 0. — TonesInDeepFreeze

Let's wait for your answer to the previous questions.

Quantum field theory exists. The theory combines classical field theory, special relativity, and quantum mechanics (from Wiki). People are however troubled to find a theory that combines general relativity and quantum mechanics, such as string theory and loop quantum gravity, each has its problems.

Just so, so when you talk about something that "strictly" can be, but which cannot itself be, then you tell me what the sense is. — tim wood

I don't understand your objection to my comments. I used "strictly" when I wanted to see what is the result of an interval divided infinite number of times by two. Mathematicians think that there is no operation of infinite division in the real number system. I don't know why and I asked @TonesInDeepFreeze for an explanation. Perhaps he can answer my question in a simple term so you can understand as well.

I have already qualified myself as a high-school "mathematician" - that being why I try to make sense in English. — tim wood

Ok, I try my best to answer your questions. Perhaps, others (mathematicians @TonesInDeepFreeze, @fishery, and @jgill) would participate and answer your questions in a simple manner.

Let's try this: it seems to me you are confusing ideas of number with limit. — tim wood

No, I am not confusing the ideas of the number and the limit.

For example, the decimal expansion of the square root of two goes on forever, is infinite, so what is the last digit? — tim wood

There is no last digit. The square root of two is an irrational number. The Irrational set, the set of all irrational numbers, is a subset of the real number set. Almost most of the real numbers do not have the last digit (the number of digits is infinite).

You want to divide something a "strictly" infinite number of times: great, how many times is that? — tim wood

Bigger than any countable number.

If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? — Gregory

The top of the cake looks like the top of the cake no matter how many times you divide the cake horizontally.

Isn't it indefinite? — Gregory

It is not.

But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? — Gregory

I don't understand what you are talking about.

I think this is what must be asked about the continuum. Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry? — Gregory

We are not talking about space and time here. Whether space and time are continuous or not is the subject of other threads.

So if no real number is an infinitesimal, numbers are then what is relation to geometry. Is 2 then 2 points, or are all numbers a point? — Gregory

Yes, there is no infinitesimal in the real number system. I don't understand the rest of your comment.

According to Wki both Cauchy (in Cours d'Analyse) and Edwars Nelson also compared infinite points to the numberline. Long before hyperreals i believe. The great writer and philosopher George Berkeley rejected infinitesimals on both mathematical and philosophical grounds — Gregory

I didn't read Berkeley at all so I don't know what he is arguing about. Is he arguing that there is no infinitesimal in the real number system or he is arguing that there is no infinitesimal in any mathematical system? According to @TonesInDeepFreeze there are mathematical systems with infinitesimal.

What about imaginary numbers, however? Stephen Hawking, in his attempt at find the wave function of the universe, proposed his (yep) No Boundary Proposal in 1983. I like to apply this "theorem" to consciousness. Hawking uses imaginary numbers to describe time as it goes backwards, behind the Big Bang. How are we to understand mathematically a state not having any boundaries? There is always a "here" and "there" in our experience. That is, except in consciousness wherein we can go deeper and deeper and we find no edge. The "limit" seems to be death, but in our experience we are infinite. Hence we can think about infinities.. — Gregory

I don't understand what you are trying to argue here.

The kalam cosmological argument gives a great example of infinities embedded in another. The argument fails in its purpose because eternity, an infinity, contains all steps of infinity. There can be that infinity if there is the eternity. QED? — Gregory

The Kalam cosmological argument states that there cannot be an infinite number of past events therefore there is a beginning. The rest of the argument is about proof of the existence of God which I don't agree with.

Don't feel bad. I"m a very old retired mathematician and have had to look up filter trying to understand sime 's comments. — jgill

I am not feeling bad. At worst I am wrong and learn a new thing. At best I am right so others learn a new thing. Thank you for your support anyway. :)

This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third between them, by claiming there are two points at a distance of zero. Can you see the circularity of your argument?

I explained this to you at length in a post you didn't bother to engage with. — fishfry

It seems that there is no operation of infinite division in the real number system. That was something I didn't know.

I believe you. Physicists attempting to do math are often a source of humor and/or horror to mathematicians. But I'm sure you know that :-) — fishfry

Oh yeah, I can guess that. We, physicists, work with the infinities all the time. Of course, mathematicians do not agree with how we deal with infinities but strangely physics works. :)

What Tones said. — fishfry

I am not confusing the two.

You need it spelled out for you again?

Definition: x is an infinitesimal if and only if, for every positive real number y, |x| < y.

Theorem: No non-negative real number is an infinitesimal.
Proof: Suppose x is a non-negative real number. Since x is a non-negative real number, |x|/2 is a positive real number and |x|/2| < |x|. So it is not the case that |x| < |x|/2|. So it is not the case that for every positive real number y, |x| < y. So x is not an infinitesimal. So no non-negative real number is an infinitesimal. — TonesInDeepFreeze

You didn't provide this argument before. Did you? You just defined infinitesimal!

You skip that I addressed that. There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0. — TonesInDeepFreeze

How could you have an infinite sequence of divided results without infinite division operations?

It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. — TonesInDeepFreeze

I was not confusing these. In fact, I mentioned the number of division operations to be strictly infinite.

And the fact that the limit of the sequence is 0 does not refute that between any two distinct real numbers there is real number strictly between them. Indeed, the convergence to 0 itself depends on the fact that between any two distinct real numbers there is a real number strictly between them. — TonesInDeepFreeze

That is understandable but I was not arguing against that. I just argued that if the number of divisions is strictly infinite then you cannot get anything new by dividing the result further since the result is zero. I now know that the number of division operations cannot be infinite in the real number system. I don't know why!

Sure, gets smaller. Which in the very expression of which says that there is again a smaller - always. I suppose you can introduce a rule or limit that says at that limit the distance is zero, but then the point is identical with itself. I don't see how you get out of or around this. — tim wood

I consider the number of the operation (by operation I mean dividing by two) to be strictly infinite. @TonesInDeepFreeze however claims that such an operation does not exist in the real number system: "There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0."

I am however confused by this statement. How could you have an infinite sequence of divided results without infinite division!?

Doubts may be experienced so often by an individual. I certainly feel in a maze, or even a fog of confusion of possibilities on a frequent basis. That is often because it is difficult to see the larger picture, especially of the unknown future. What I like about Watson and Skinner's picture of rats iand mouses n mazes isn't the actual deterministic picture of behaviorism but the metaphor of the creatures within the maze.

Behaviorism certainly paints a picture of determinism. However, the later development of cognitive behavioral approaches may alter this. That is cognition plays a part in making sense of it all, including the mazes, even if there are not any easy solutions. — Jack Cummins

I don't think that doubts can be resolved by a deterministic entity such as the brain since doubts are not deterministic states. Hence, I think doubts are resolved by the mind that has the ability to freely decide.

It is an odd thought that all the movements of particles/energy in our brains could cause feelings of doubt about the resolution as they all resolve into the only brain state into which they could possibly resolve. — Patterner

Are you questioning how doubt could arise due to neurobiological processes or how they could be resolved? We don't know how neurobiological processes could cause all sorts of brain states, such as thoughts, feelings, etc. So the answer to the first question is that we don't know. The answer to the second question is, that although we know that doubts are caused by neurobiological processes in the brain, the brain cannot possibly resolve doubts since the brain is a deterministic entity. Therefore, I think that doubts are resolved by the mind which has the ability to freely decide.

That is, two points on the number line, zero distance between them: how can they not be the same point, or number, and if not the same, how can there be zero distance between them? — tim wood

You are correct in your observation. If the distance between two points is zero then we are dealing with the same point. This means that there cannot be a point between points since they are the same point. This is against the argument that there is always a point between two points on the real number. What I showed is that the distance between consecutive "means" (by "mean" I mean the point between two points) tends to zero. I hope things are clear now. If not please let me know so I would elaborate.

Perhaps this post help you. What I showed in that post is that the distance between consecutive means tends to zero for large $i$. The last step of my argument seems to be problematic because of the way the division is defined between cardinal numbers. You don't need to worry about the division of the cardinal numbers though since I don't need that step. All I need is to show that the distance between consecutive means is $d_i=\frac{(b-a)}{2^i}$. This distance is zero when $i$ is strictly infinite. I am currently trying to make sense of this post though and I have to say it is very technical.

I am a retired physicist and my knowledge of mathematics is very rusty due to my age. I however understand what you are saying. Thanks for helping me to learn new things and refresh what I learned a long time ago.

For a mathematics for the sciences, ordinarily we use a complete ordered field. That requires having a non-empty set, a 2-place relation (<) on the set and two 2-place operations (+ *) on the set such that for all x, y and z:

ORDERED FIELD
x+(y+z) = (x+y)+z (associativity of addition)
x+y = y+x (commutativity of addition)
EyAx x+y = x (additive identity element)
Theorem: E!yAx x+y = x
Definition: 0 = the unique y such Ax x+y = x
Ey x+y = 0 (additive inverse)
EyAx x*y = x (multiplicative identity element)
Theorem: E!yAx x*y = x
Definition: 1 = the unique y such that Ax x*y = x
0 not= 1
x*y = y*x (commutativity of multiplication)
x*(y*z) = (x*y)*z (associativity of multiplication)
x*(y+z) = (x*y)+(x*z) (distributivity)
x not= 0 -> Ey x*y = 1 (multiplicative inverse)
(x < y & y < z) -> x < z (transitivity)
exactly one: x < y, y < x, x = y (trichotomy)
x < y -> x+z < y+z (monotonicity of addition)
(0 < z & x < y) -> x*z < y*z (monotonicity of multiplication)

COMPLETE ORDERED FIELD
In set theory, we prove that there is a carrier set (called 'R') for such a system and such that, for any upper bounded non-empty subset of S of R, S has a least upper bound. With that and the rest of the set theory axioms we can do the mathematics of derivatives and integrals for the sciences. — TonesInDeepFreeze

Ok, thanks for the elaboration. I got that.

An alternative is to have a system with infinitesimals. But still, ordinarily, we need to define <, + and * and to prove whatever theorems are needed for the machinery of mathematics.

To just wave a hand and say "Voila, this is my infinitesimal" does not provide the needed definitions of < + and * with infinitesimals nor the needed proofs.

So how do we go about proving the existence of a system with infinitesimals? For your purposes, it would help to first define 'is an infinitesimal'. I provided a definition previously, but I notice that many authors include 0 as an infinitesimal. So perhaps use this definition:

x is an infinitesimal if and only if, for every positive real number y, |x| < y.

It has been proven for you that for every real number x there is a positive real number y such that y < |x|.

So no non-zero real number is an infinitesimal.

One more time: No non-zero real number is an infinitesimal. The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every real number x there is a positive real number y such that y < |x|. We don't need to keep going over this over and over. — TonesInDeepFreeze

I looked at all your posts and didn't find the proof that no non-zero real number is an infinitesimal. Could you please provide the proof?

The sequence of half distances converges to 0. So what? That doesn't prove that it's not the case that between any two different real numbers there is another different real number. — TonesInDeepFreeze

Between any two distinct real numbers, there is always another one strictly between them. — fishfry

This was a reply to the above comment from @fishfry who claimed between any two distinct real numbers, there is always another one strictly between. The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case.

Our thoughts, feelings, desires, and the like help us to decide in most situations. There are situations in which thoughts, feelings, desires, and the like cannot help us to decide for example when we have doubts. The existence of doubt together with our ability to decide when we have doubt means that we, whether a mouse in a maze, a human who wants to invest in the market, etc. are not deterministic agents.

If you approach religion like that, you will find no solution to the question at all. — Constance

Yes, I know. The problem is if there is one God then why are religions so diverse and inconsistent?

Ask, why doesn't science have this problem? It is the consistency of results: put nitroglycerin in the same experimental context, the results will be the same. — Constance

Yes, science is consistent, religions are not.

If you treat religion like a culture, like you seem to be doing, then all you get is cultural differences, but if you look for the essence of religion to see if there is something just as unwavering, and you look "through" the narratives, the churchy fetishes, the bad metaphysics, and so forth, to what survives after all of these contingencies are suspended, and you find the metaethical indeterminacy of our existence. This is what religion is all about. — Constance

There are many reasons why people believe in religion, such as fear of death, fear of punishment, the promised rewards, and the like. Why do religions survive? Because of the mentioned reasons. Because people do not realize the conflict between religions and the conflict within a single religion.

Very long story short: a determinate ethics is simple to understand. We see it in our laws, rules, principles, explicit or implicit, and so on. The ethical normativity of our existence. Indeterminacy is what we run into when we ask for basic rationality on which these are founded: why pay taxes? Because we need money to run a society. What is the point of that? See contract theory: it's better than the state of nature; much better, because people are safer from harm. What is wrong with harm? Errrr, What do you mean? This is an indeterminacy that runs through all of our affairs, hidden beneath the veneer of conversation. The prima facie moral call not to cause harm really has NO justification beyond it being stand alone bad, which is weird for anyone who likes explanations.

But take those ethical complaints that intrinsically deal with harm, and there you are stricken with plague or burning to death in a car somewhere, and there are no laws to protect you, no authority to redress the wrongs, that is, the intrinsic wrong of it being there AT ALL. Take the broad context of our ethical issues in the world, and see that ultimately, no redress is forthcoming at the foundational level! THIS is where religion has its essence, why, that is, societies "came up with" religion, and why religion is in all cultures. We are all "thrown into" a world of unredeemed suffering and unconsummated desire. This is the essence of religion: to bring these to their completion. — Constance

We have a common conscience and we can establish a stable society based on that. Moreover, harming others is a very common concept within different religions, like stoning to death, cutting hands or fingers, and killing those who do not believe in God.

Thank you, I try. — Philosophim

Thank you. Before going further I would like to define other terms that I used in my previous post to help both of us understand each other better and communicate easier. I already define good and evil. I however use two other terms namely right and wrong which I haven't defined yet. Right is something we ought to do and wrong is something we ought not to do. As an example, think of the nasty kid. The punishment is evil given the definition of evil but it is right in this case. The reward is good given the definition of good but it is wrong in this case.

Taken in that limited context, is that really evil then? Preferably, we would like there to be infinite resources. Then there would be no need for evolution. But if there are finite resources, and also threats that could potentially prevent beings from getting them, isn't evolution the best to handle a situation? — Philosophim

Evolution is evil since the weak species suffer and eventually die out. Evolution is however positive.

Because if there wasn't evolution, wouldn't it all just die out? — Philosophim

We just couldn't have different sorts of species that fit very well with the different environments and hazards.

Evil is not, "What is inconvenient". — Philosophim

It is given my definition of evil.

What is preferable, having a world with evolution, or no world at all? — Philosophim

We cannot avoid evolution given the fact that the resources are finite.

What should be is what is good, and what should not be is what is bad. — Philosophim

I use the terms positive and negative instead of good and bad when it comes to evolution. Evolution is positive and it is not negative. I use these terms to avoid the confusion of using terms good and evil when it comes to morality.

Sometimes we might want something, but its not possible to obtain. We all want a world with no sickness or death. That would be a better world if it were possible. But since its not, does that automatically make our world evil? — Philosophim

Sickness and death are natural evil.

Are those things that we do not want in excess, or are they evil innately?

If someone comes into your home to murder you and your family, hate can be the motivation that lets you fight them off. — Philosophim

Hate is evil and in this case, is right.

Pain lets you know when your body is injured. There are people who can't feel pain, and they often die young. Here's an article to ease into the concept. — Philosophim

Yes, the pain is evil and it is necessary for the reason you mentioned.

So you see where I'm going with this. My goal here is to get to the very foundation of the words. At its very foundation I see good as "What should be" and evil as "What should not be". — Philosophim

Evil as I mentioned is a psychological state and it is necessary. It is not what should not be.

It keeps it clear, distinct, and allows clear identification. Because as you've noted, things that seem 'evil' in some circumstances, aren't. — Philosophim

Evil is involved in things like body and mental exercise. But the body and mental exercise are positive.

It is tricky. And all of your examples I would intuitively think are examples of good. — Philosophim

The act of punishing the nasty kid or killing the person who is terminally ill is evil but it is right. These acts are not good. I already make a distinction between good and right to avoid confusion.

Good and evil are both about intention and outcome. Punishments done to teach and discipline are good. — Philosophim

Punishment to teach and discipline is right. It is not good given the definition of right and good.

Punishments done as revenge and to simply cause hurt are evil. — Philosophim

Punishment is generally evil. Punishment could be right or wrong though given the circumstances. Punishment for simply causing hurt is wrong.

But of course religion "provides reasons why an act, good or evil, is right or wrong." Religion tells us that God is moral foundation of such "reasons". — Constance

Well, if they say so. But that does not make God a moral foundation. The reason for that is the very diverse range of religions with different teachings. Most religions give teachings that contradict the teachings of others. There are even contradictions within a single religion. Not all religions are the same and all of them could not be possibly true. So even if accept the premise that God is the moral foundation then we still face a problem: Which religion is true?

I think the proper adjective for existence is positive rather than good. By positive I mean consisting in or characterized by the presence rather than the absence of distinguishing features. This way, we avoid the confusion of the term good used in morality with positive which is related to existence.

It is fine to disagree. — Philosophim

I am glad to discuss things with an open-minded person like you.

But I'm going to ask, "Is it better to have good states of reality or evil states of reality?" — Philosophim

No. Good and evil are fundamental and they are both necessary. Think of evolution for example. The weak agents are eliminated in the process of evolution so room is left for the stronger to survive since the resources are finite. Evolution is evil since weaker agents are eliminated for the sake of stronger ones.

I do agree that we can also use morality in a sense that we have already determined what is good or evil. But this is the conclusion after evaluation. I do not mind either use. — Philosophim

I have to first answer what good and evil are before discussing morality. Good and evil as I mentioned are two categories of psychological states. I cannot define good and evil but I can give examples
of psychological states in which a set of psychological states are good and others are evil. Good like love, happiness, pleasure, and the like. Evil like hate, sadness, pain, and the like.

Can you escape the notion that good is what should be, while evil is what should not be? — Philosophim

Apparently, we cannot. We have to accept the reality as it is. Think of mental or physical exercises for a moment. Without physical activity which is tiresome and painful, therefore evil, you cannot have a body in good shape. The same applies to mental exercise. You must read, think, memorize, and discuss things to become mentally strong. This is also tiresome and painful, so we cannot avoid evil when it comes to mental exercise.

Doesn't that require us to evaluate the situation? — Philosophim

Sure, we need to evaluate the situation before deciding whether we should do good or evil.

And how do we know what is a right action? — Philosophim

This is a tricky part so I have to give examples of a few situations to make things clear. Think of a situation in which you have a nasty kid who breaks things and messes up your house. You don't reward him for what he does instead you punish him. The first act, rewarding, is good and the second act, punishing, is evil. Therefore, evil is right depending on the situation. Think of a person who is terminally ill. The act of killing any person is evil since it causes sadness to friends or relatives. But the act of killing a person who is terminally ill is right if she or he wants it. Here, I just gave a couple of examples of the situations in which evil acts are right. I am sure you can come up with situations in which a good act is the right choice.

What do you think about the logic of the rest of the post? — Philosophim

I read your entire OP once but I have to read it a couple of more times before I become ready to discuss it in depth. For now, let's see if we agree on the definition.

Thank you very much for your post. It made many things clear to me after I read it a few times. I am still not convinced that infinitesimal does not exist though. Can you prove it? I however can show that the distance between consecutive means tends to zero. You can find my argument here. Please let me know what do you think.

I googled and I found two references about the division of cardinal numbers. You can find the references here.

It appears all you have shown is the distance between consecutive means tends to zero. — jgill

Well, I showed that the distance between consecutive means is zero if the number of divisions is $\aleph_1$.

The last sentence is a little weird. — jgill

Well, that is the division of two cardinal numbers. I googled about the division of cardinal numbers and I found two references here.

The previous sentence says it all if one takes a limit. — jgill

Ok, I see what you mean.

Good - what should be — Philosophim

I'm afraid I have to disagree. Good and evil are psychological states of affairs and are features of reality.

Morality - a method of evaluating what is good — Philosophim

Morality is about releasing what is a right action, good or evil, in a situation.

Between any two distinct real numbers, there is always another one strictly between them. — fishfry

That is not correct: Consider two numbers on the real number such as $a$ and $b$. Let's define the mean as $c_1=\frac{a+b}{2}$. We can determine the next mean as either $c_2=\frac{a+c_1}{2}$ or $c_2=\frac{c_1+b}{2}$ in which in the first case we approach to $a$ from the right and in the second case we approach to $b$ from the left. Let's work with the first approach: $c_2=\frac{a+c_1}{2}$. The next mean can then be determined by $c_3=\frac{a+c_2}{2}$. We can write $c_{i+1}=\frac{a+c_i}{2}$. The distance between $c_{i+1}$ and $a$ is $d_{i+1}=c_{i+1}-a=c_i-c_{i+1}=\frac{(2^i-1)*a+b}{2^i}-\frac{(2^{i+1}-1)*a+b}{2^{i+1}}$. Therefore, we have $d_{i+1}=\frac{2*((2^i-1)*a+b)-((2^{i+1}-1)*a+b)}{2^{i+1}}=\frac{b-a}{2^{i+1}}$. So, $d_{\text{Aleph}_1}=\frac{b-a}{2^{\text{Aleph}_1}}\lt\frac{\text{Aleph}_1}{2^{\text{Aleph}_1}}=0$. Therefore, your statement does not follow.

No. What is defended and discussed here is an analytic of religion, not religion as it is taken up in regular affairs. Read the OP. — Constance

Ok, my apology. I read your OP a couple of times and now I know what you are arguing about. To me, the essence of religion is not about ethics at all but about spiritual and mystical experiences. Although there are religions with a set of commands, what we ought to do and what we ought not to do, but to my understanding there is no religion that provides reasons why an act, good or evil, is right or wrong. Therefore, religion is not about ethics.

God is a term that issues from the basic religious situation. It belongs to a basket of terms that create issues rather than resolve them. Say God is the greatest possible being, then I will give a hundred ways to entangle this into an entirely contrived issue generating concept. Wittgenstein was right: leave such things alone for, putting is simply, lack of grounding in the world. In other words, the world shows itself to us and our job in philosophy is to say what is there at the most basic level. — Constance

I don't think that philosophy can resolve the problems regarding spirituality hence religion. You either have spiritual experience or not. You cannot tell whether a spiritual experience is an illusion created by the brain or it is real (by real I mean that there are spiritual agents in charge of causing the experience).

But you don't give the idea its due: take someone's masochism regards beatings as somehow delightful. The pain of the beatings is no longer, therefore bad, or another way to go would be to say that what is bad in the pain is entangled with something that makes if good (a fetish's very definition) and familiarity makes for a settled matter, psychologically. On this point I don't care about the variability of the way we experience the world. All that matters is the value in play when value is entangled (it almost always is). It can be clear as a bell, as when the flame is put beneath the palm of my hand--hard to fetishize this one. Not impossible, but then...well, I hard to even imagine. I can imagine Thích Quảng Đức did; he was the Buddhist monk who set himself ablaze in protest. But this is a different matter as he had trained himself to ignore the pain, not enjoy it. But the source of enjoyment is just not at issue. What is at issue is the nature of pain when one is feeling pain. Just that. You have a fetish such that burns and beatings are a good time, then I do not classify your beatings as painful, but delightful. — Constance

Glad to see that you agree that the pain is not bad for all agents.

I am of the school that says if something hasn't been through the analytic grinder, then it is not worthy of belief. — Constance

Well, that is quite the opposite of what you stated regarding religion. To you: "But religions are about a dogmatic authority, and so the analytic of good and bad has no place.".

I no more take religion in any popular sense seriously at all. Such a thing is no longer a a living possibility. — Constance

Do you believe in God? If yes which kind of God It is?

If a masochist likes X, then X isn't pain to the masochist. I take this as both analytically true as well as experientially. — Constance

I am a masochist myself so I can tell you that is the pain that I like.

:up: :100:

But religions are about a dogmatic authority, and so the analytic of good and bad has no place. As for a description, this is what observation does. So what is there to observe? Just the arbitrary command (which may be a good idea or not. The point is that the determination about its goodness or badness is not based on justification and merit). — Constance

What is your religion and why did you choose it?

Pain is apodictically "bad". — Constance

Not to a masochist.

Well, it tries to. — Constance

To my knowledge, no religion describes good, evil, right, and wrong. It just gives a set of commands: what we ought to do (considered as good) and what we ought not to do (considered as evil).

But the point here is that when we are trying to understand something in the world, we look to a description of how that thing appears. So we "observe" religion much as we would, say, the law, or geology or anything we want to understand. I am saying religion is what we encounter when ethics meets metaphysics. So what is ethics and what is metaphysics? In ethics, there turns up something apodictic, which is really not the way philosophers prefer to think about ethics, because apodicticity is irreducible. I.e., nothing to talk about.
So what to do now? What if ethics were apodictic? I am claiming it is. — Constance

There are two problems here even if we accept that ethics is apodictic: (1) Which religion is the correct one? and (2) What is the reason for religion being the only reliable source when it comes to ethics?

(1) is important since there are conflicts in many religions and even there are conflicts within a single religion. (2) What if someone comes up with an apodictic idea regarding ethics such as each human has all rights when it comes to his or her life but she or he does not have any right when it comes to the life of others unless both individuals agree on terms and conditions?

Religion IS metaethics... — Constance

I cannot see how that could be true. Religion does not tell us what good, evil, right, and wrong are. Does it?

Thanks for the elaboration.

Thank you very much for your reply and fruitful input.

Let's start the discussion by defining the different gaps. Given a set of points, points of real number for example, I define G1 as an interval between two distinct points. I define G2 as an interval between two immediate points with no point between them (what I call an abrupt change in OP). I am interested in understanding whether there are gaps of type G2 in the set of real number given the definition of the set of real number which can be found here. The set of real can be defined as a set of numbers each number has an integer and decimal fraction. The decimal fraction contains an infinite number of digits, to be precise Aleph_0, where Aleph_0 is the cardinality of the natural number. To show that G2 exists in the set of real number I simply construct an infinitesimal_1 as follows: infinitesimal_1 = 0.0...01 where by "..." I mean an infinite number of zero to be more precise by the infinity in here I mean Aleph_0. I can show the second smallest number is 0.0...02 with no number between. Think of a number between, 0.0...011. This number can be divided into two parts 0.0...01 and 0.0...001. 0.0...001 however is 0.0...01 given the property of Aleph_0 in which Aleph_0+1=Aleph_0 therefore 0.0...011 is equal to 0.0...02.

I am sure that mathematicians define things in a more rigorous way when it comes to infinitesimal. I am interested in whether you can show that there is an infinitesimal in the set of real number given your definition.

Cantor's generalized theorem says that there is no onto mapping possible between a set and its power set, even when such set has an infinite cardinality. — Tarskian

I don't understand how one can disprove Laplace's Demon using Cantor's theorem. Do you mind elaborating?

What do you mean by 'a gap'? If you mean that the two distinct points are not the same point, then yes, by definition. There's a gap between 4 and 13. — noAxioms

By a gap, I mean an interval.

Without a definition of a gap, P1 is ambiguous. It states that either G or ~G, which is tautologically true, making P1 empty. The word 'distinct' is not part of P1. — noAxioms

It shouldn't be.

Show it then. What about the number that is halfway between this smallest positive number and zero? You've shown that it doesn't exist? — noAxioms

Well, I construct the infinitesimal in this way: 0.0...01. By "..." I mean Aleph_0 zero. The next number is then 0.0...02 therefore there is a gap, 0.0...01 between these two numbers. One could say how about 0.0...011? It can be shown that 0.0...011 is 0.0...02 by simple math. 0.0...011=0.0...01+0.0....01. By "...." (where dots appears four times) I mean Aleph_0+1. But Aleph_0+1=Aleph_0 therefore 0.0...01+0.0....01=0.0...01+0.0...01=0.0...02.

Define it then, without making classical assumptions (like a particle having a location, or some counterfactual property. — noAxioms

This is off-topic but I give it a try. Consider a hydrogen atom for example. R is the center of mass position operator of the atom that is related to the position operator of the nucleus (r_n) and the position operator of the electron (r_e). The relation is R=(m_n*r_n+m_e*r_e)/(m_n+m_e) where m_n and m_e are the mass of the nucleus and electron respectively. The center of mass therefore can be calculated as <Psi(R,t)|R|Psi(R,t)>.

Also, why do you think that death is impossible? — boundless

I didn't say that death is impossible.

C2 doesn't follow at all. In the real numbers, there being a gap between 4 and 13 does not imply that the real numbers (or even the rationals) is not a continuum. — noAxioms

C1 states that there is a gap between all pairs of distinct points of the continuum.

You need to demonstrate that there is nothing between some pairs of points that are not the same point. Then you've falsified the continuum premise. — noAxioms

Are you challenging (P1)? If yes, I already illustrated that given the definition of the real number one can construct the smallest number or the smallest interval so-called infinitesimal.

Only in classical physics, and our universe isn't classical. But I accept your refutation of the rebuttal to the OP. Do you accept my rebuttal? — noAxioms

You can define it in quantum physics as well. Of course, you cannot measure it.