I am looking for proof that the set of natural numbers that each its member is finite has aleph_0 members.

So you are saying that aleph_0 is not a member of the set of natural numbers yet the number of its members is aleph_0. I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members.

It's not counterintuitive that there exist functions whose domain is the set of positive natural numbers. Any Calculus 1 textbook has such functions all through the book. Moreover, there are functions whose domain is the set of real numbers. Moreover, there are functions whose domain is a proper subset of the set of real numbers. Moreover, for any set whatsoever, there are functions whose domain is that set (except there is only one function whose domain is the empty set). — TonesInDeepFreeze

That I understand and that is not my problem.

To be clear, 'N' there does not stand for the set of natural numbers, so I'll use 'n'':

Yes, {1 ... n} is a set with n number of members and n is a member of {1 ... n}.

That doesn't vitiate that there are other sets with n number of members but such that n is not a member:

{0 1} has 2 members, but 2 is not a member.

{1 4 7} has 3 members, but 3 is not a member. — TonesInDeepFreeze

I understand there are sets with n members, but n is not a member of the sets. That was why I defined the domain D that has this specific property, the number of its members, n, is a member as well.

the set of natural numbers has aleph_0 members, but alelph_0 is not a member. — TonesInDeepFreeze

Let me ask you this question: Are all members of the natural number set finite? If yes, how the number of its members could be aleph_0?

Do you really not understand what a domain of a function is? Or are you trolling me? — TonesInDeepFreeze

I am so sorry that you feel that I am trolling. I respect your time and my time. I appreciate your effort in explaining things to me and I learned lots of things from you that I am grateful for.

This is counter-intuitive to me. Consider a function f with the domain D={1,2,...N} where N is a finite positive integer. The domain has N members and N is a member of the domain of f. Could you please explain what happens when N is aleph_0?

There are relative statements, like "I like cats". There are statements like, "man is mortal" which are objective. Therefore I think the conclusion is that the first category of statements, relative statements, are not part of truth whereas the second category of statements, objective statements, are part of truth.

I think I see where I'm going wrong. A relative truth would be that relative to a society of evangelical Christians, gay marriage is indeed wrong on the basis of their subjective belief that it is wrong. That goes further than just reporting a fact, which is what I did in my post. — ToothyMaw

:up: :100:

I think this post is getting at exactly that - is the property of being true based on facts that are verifiable independent of our feelings, or is the property of being true based on subjective experiences? Or at least, that's what I would like to think. — ToothyMaw

Given the definition of objective and subjective from here, the truth is objective if it is a set of statements that are true and independent of opinion, biase, conscious experience, and the like.

Why is f(aleph_0) meaningless when we know that natural numbers have aleph_0 members?

Have you studied the material of Calculus 1? At least the first week in which the definition of 'limit' is given? — TonesInDeepFreeze

I studied Calculus 1 40 years ago. I am familiar with this notation of the limit.

For the third time: The domain of f is the set of positive natural numbers, therefore, aleph_0 is not in the domain of f, therefore "f(aleph_0)" is meaningless. — TonesInDeepFreeze

What do you mean by infinity when you talk about the limit in this post?

How do you define truth?

There are limits. As an example, consider the sequence 1/2, 1/4, 1/8, 1/16, ...

We all know that the limit of this sequence is 0. You can certainly call this infinite division if you like, as long as you understand what limits are. There are indeed infinitely many elements of the sequence, and you CAN think of this as "infinite division."

You can even think of doing it "all at once" if you like

What it means formally is that the elements of the sequence get (and stay) arbitrarily close to 0.

What it does NOT mean is that there is some kind of magic number that the sequence attains that is a "distance of 0" from 0, but is not 0. That's a faulty intuition.

In fact the formal theory of limits, once one learns it, is the antidote to all our non-rigorous, faulty intuitions about infinite processes.

Does that help, or perhaps refresh your memory? I'm pretty sure that physicists must be exposed to the formal theory of limits at some point. — fishfry

I agree with what you stated.

Yes I understand and agree. But I am a little surprised that you seem to think that a sequence attains some kind of mysterious conclusion that lies at a distance of 0 from its limit, but is distinct from the limit. That's just not right.

Did my mention of limits ring a bell at all? Or raise any issues that we could clarify or focus on?

Because your idea of endless division is perfectly correct. But all that shows is that endlessly halving leads you to the limit of a sequence. But there's no "extra point" in there that's distinct from but at a distance of 0 from the limit.

Let me know if we're on the same page about this. — fishfry

Yes, we are on the same page and thank you very much for your contribution. I learned a lot of things and refreshed my memory. :)

Thank you very much for your contribution to my thread. I learned lots of things. You saved me a lot of time as well.

What do you mean by "infinity" used as a noun?

There is the adjective "is infinite": S is infinite if and only if S is not finite.

And there are various infinite sets, such as:

the least infinite ordinal = {n | n is a natural number} = w = aleph_0

the least infinite cardinal = aleph_0 = w = {n | n is a natural number}

the least infinite cardinal greater than aleph_0 = aleph_1

card(set of functions from w into 2) = 2^aleph_0 = card(the power set of aleph_0)

card(set of functions from aleph_1 into 2) = 2^aleph_1 = card(the power set of aleph_1)

There are various uses of "infinity" as a noun, including such things as infinity as a point in the extended reals or infinity as a hyperreal in nonstandard analysis. But you've not specified.

In any case, how can you ask your question when I explicitly defined the domain of f to be the set of positive natural numbers? (Whatever you mean by "infinity" used as a noun, it is not a member of the set of positive natural numbers.) — TonesInDeepFreeze

By infinity, I mean aleph_0. I thought that the value of f(alep_0)=0 which is why I asked for its value. But after some thinking, I realize that it is not. In fact, one could define a sequence g(n+1)=g(n)/10 where g(0)=1. It is easy to see that for any value of n g(n)<f(n) except n=0 if (f(0)=1. But g(aleph_0)=0.0...1 and we find g(aleph_0) >0 so f(aleph_0)>0 as well. I am sure you can define things better and provide a better argument.

Then just say "finite". But your particulars involve both countable and uncountable cardinals, all of which are greater than any finite cardinal but some of which are greater than any countable cardinal:

aleph_0 is countable

aleph_1 is uncountable

2^(aleph_1) is uncountable and greater than aleph_1 — TonesInDeepFreeze

Just out of curiosity, why aleph_1 is uncountable?

Greater than any countable number or greater than any finite number? — TonesInDeepFreeze

It is better to say greater than any finite number given the definition of a countable set in mathematics.

That's what I understand as a limit (process) - that can be approached but never reached, but if it could be reached, would yield whatever - the process of whatevering being nothing in itself, but useful as a tool. — tim wood

That is a correct interpretation if you divide the interval by two, then another time divide the result by two, ad infinitum. What I have in mind is that I simply divide the interval by 2^infinity in one step. This operation seems to be invalid though to mathematicians.

And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense. — TonesInDeepFreeze

Yes, I got that. Thank you very much for your explanation.

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.