## Proof that infinity does not come in different sizes

• 2.5k
I see no paradoxes or contradictions or foundational incompleteness in the beliefs that I uphold (mathematical or otherwise).

Which are?
• 3.2k
That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a triangle

He did it again! He completely skipped recognizing the refutation given him.
• 3k
We don't. He proved that they are the same size.
From the point of the set N, it looks like it is. But from the point of the set P, it looks like it is only a half set to N. What's going on?
• 276

If you look at the posts, I don't think I'm the one that has been showing the disrespect (if I have, it has been in response to disrespect). I wanted a discussion because I felt I had something to offer in response to something that I saw as contradictory. I don't think I entered the discussion closed-minded or dogmatic. And I think I tried to understand the other's point of view.

If someone is an "expert" in the field of something, but that something is evidently paradoxical or foundationally incomplete, it's absurd to treat them like an expert of anything useful. Some people are unreasonable/absurd. They want to hold on to their paradoxical or contradictory theory or belief at the cost of sincerity to Truth/Goodness/Existence/God

If people here witness that their beliefs or theories or axioms lead to no paradoxes or contradictions or foundational incompleteness, then I can't say to them they're misguided or lacking in knowledge.

You have those who recognise/witness that their theories are incomplete and act as such (there is honesty to them), and then you have those who recognise/witness this, but act as though they are the knowledgeable ones whilst all others are ignorant (which to me is the very definition of a "bad guy"). I believe spending hours or years or decades on something that is foundationally corrupt, does not make you an expert in anything other than something that is useless. What good is an expert in multishapism geometry that deals with the study of shapes such as round triangles and circular pentagons?

I don't feel like I have any contradictory or paradoxical theories or beliefs that I need to reconcile. I was trying to address what I saw as contradictory. If it's not contradictory, then it's not contradictory. But if it is contradictory/paradoxical and some are hardcore with regards to holding on to this, what can I say?
• 3.2k

The cardinality of N = the cardinality of P iff there is a bijection between N and P.

There is a bijection between N and P.

Therefore, the cardinality of N = the cardinality of P.

Meanwhile, there is no apparent meaning in "from the point of view".

Yes, P is a proper subset of N. Indeed the point is that it is a property of infinite sets that there are bijections between them and certain proper subsets of themselves.

The fallacy is in saying "half" in this context. For infinite sets, there is no division operation such that there is 1/2 the cardinality of an infinite set.
• 276
He did it again! He completely skipped recognizing the refutation given him.

Ok. Let me put it this way. I gave you a refutation with the z example. You started with insults, then you eventually said something like this:

By the way, we don't need to use temporal phrases such as "at the same time". Set theory does not mention temporality.

Then the rest of your z's and v's is irrelevant if it is supposed to refute the proofs I gave.

I decided discussing something with someone who seems to be emotional or biased is a waste of my time so I said I will stop, but I felt the need to add the following to the discussion:

I will just say this. That a set cannot be both a member of itself and a member of other than itself is the equivalent of saying that a shape cannot be both a square and a triangle (I have taken out the "at the same time" and the effect is still the same).

This dealt with your temporal phrases response.

Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?

And don't say to me something like "some set theories allow for this or that". I'm asking a basic logical question that has a basic and straight forward answer. There is no need to dance around anything. Just deal with the main issue at hand.
• 761

You are too pessimistic. You can have your view and they can have theirs.

You can always declare victory, plant your flag and call it a day. Really, say what you like. I agree there are contradictions and what I brought up about parameters that can be anything your brain can dream up.

Maybe there are real world applications to some of this as has been discussed by those who have actually done it. I assume they use what has proven to work. Math in practice has a precision component, not just theorizing.
• 3.2k
I don't think I'm the one that has been showing the disrespect

I don't care to say you are "disrespectful", but you are irrational and in bad faith when you skip refutations and explanations given you and instead just keep repeating your false and confused claims.

I don't think I entered the discussion closed-minded or dogmatic.

You are closed minded to the fact that you are close-minded and dogmatic. And you still won't face that your hyper-opinionating on a subject you know nothing about. If really were the fair minded person you claim to be, then you would get a book and find about the subject rather than posting misinformation and confusions about it.

"expert" in the field

Just for the record, I don't claim to be an expert in anything other than jazz, and even in that field I'm deficient in important ways.

There it is again! You say 'contradictory', again ignoring all the explanation given you about that.

incomplete

There is it is again! You say 'incomplete', again ignoring all the explanation given you about that.

act as though they are the knowledgeable ones whilst all others are ignorant

I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.

What good is an expert in multishapism geometry that deals with the study of shapes such as round triangles and circular pentagons?

Nope. Set theory doesn't do that.
• 276

I responded to you, you responded me with a refutation, I responded to your refutation with the following:

Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?

Where is my response? Is it me who ignores you or you who ignores me?
• 2.5k
All of them are here:

godisallthatmatters.com

Ok, so you are saying that your beliefs are not incomplete or contradictory in any way. That is not philosophy, that is religion, aka delusion.
• 276

I believe my beliefs are not foundationally incomplete or contradictory in any way from a rational/semantical point of view.
• 276
I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.

I didn't say all others are ignorant. I just said there are people who are like this. I did not specify who.
• 3.2k
I gave you a refutation. You started with insults

On some crucial points, you didn't even recognize them, let alone refute them. And when you did attempt to refute points, you failed, as your supposed refutations were false and confused.

You started with insults

That's a lie. I started with plain, cold information. And I did that for several posts. Eventually, it became clear that you are immune to rational discussion, and so I factually pointed out that you are confused, ignorant of the subject and in bad faith.

emotional or biased

As to bias, I have read a pretty good amount of the literature of this field with informed and responsible debates regarding classical mathematics. I am fascinated by and greatly enjoy informed and responsible critiques of classical mathematics. As to emotion, exasperation with cranks is natural.

Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?

I did answer it. Specifically and exactly.

And don't say to me something like "some set theories allow for this or that".

The relative consistency of those theories indicates that it is not contradictory that a set is a member of itself and also a member of other sets.

There is no need to dance around anything.

There's nothing terpsichorean about my reply. I gave you an exact refutation. The fact that you are ignorant of the context of set theory and alternative set theories is not my fault.
• 3.2k
Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?
— Philosopher19

Where is my response? Is it me who ignores you or you who ignores me?

My response is right where it was when I gave it.

And I also responded to your previous tu quoque, and you ignore that too.
• 276

The relative consistency of those theories indicates that it is not contradictory that a set is a member of itself and also a member of other sets.

It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself. Yet you want to persist by saying things like the above. Again, I asked:

Is it logically possible for a set to be both a member of itself and a member of other than itself? If it is a member of other than itself, then it is not a member of itself, is it? And if it is a member of itself, it is not a member of other than itself is it?

And don't say to me something like "some set theories allow for this or that". I'm asking a basic logical question that has a basic and straight forward answer. There is no need to dance around anything. Just deal with the main issue at hand.

It seems that what I added was ignored and what I asked was not answered. Until I see a good enough response, I'm done putting any more time into this. Once again:

It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

Who would reject this but the contradictory/unreasonable/irrational/absurd/insincere?
• 3.2k
I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.
— TonesInDeepFreeze

I didn't say all others are ignorant. I just said there are people who are like this. I did not specify who.

You are very confused. Yes, you didn't say all others are ignorant. And I didn't say that you said that all others are ignorant. Rather, as now you mentions again, you said that some people have regarded all others as ignorant.

You didn't specify anyone in particular. Good. Because there is no one who has even hinted at a suggestion that all others are ignorant. You take the sneaky road of impugning but leaving it open-ended who you are impugning though it is obvious who you mean. And my point stands: You are ignorant on the subject. That doesn't entail that others are ignorant on it.
• 2.5k

I skimmed through two pages. It seems to be a collection of semantic games. I am more concerned with what issues you solve with your beliefs. That your beliefs are not contradictory (big claim) is not a selling point for others to adopt it.
• 3.2k
It is blatantly contradictory for x to be both x and not x.

It's even contradictory just to say that x is not x.

And set theory does not say there is an x that is not x, nor that there is an x that is x and not x.

You ma
It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

Correct! Indeed that is a crucial point that is used in an important proof I gave you.

Yet you want to persist by saying things like the above.

That's a lie. Stop lying. I never said anything like that.
Once again:

It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

That is not "once again". Previously you said that "a set cannot be both a member of itself and a member of other than itself". That is different from "a set cannot be both a member of itself and not a member of itself".

I wondered a while ago whether you did not actually mean "a set cannot be both a member of itself and a member of other than itself" but actually meant " "a set cannot be both a member of itself and not a member of itself". But in my reply I addressed the former in such a way that if you hadn't meant it, then you could revise to what you did mean.

It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

Who would reject this but the contradictory/unreasonable/irrational/absurd/insincere?

Indeed. (Well, except for dialetheists and paraconsistent-ists.)
• 2.5k
I will say that for diagonal paradoxes, this thread is much better.

Relevant: Lawvere's fixed point theorem.
• 3.7k
is there real math behind the north pole of the riemann sphere?

Point at Infinity

:cool:
• 3.2k
Says the guy who tried arguing Cardinalities don't have size yet they do, as per the theorem I produced to prove you wrong. Since some Cardinalities are greater than others, we can say that some infinities are larger or even smaller than others. That you got your ass handed to you by someone suffering from "dunning-kruger"

You are egregiously and flagrantly putting words in my mouth.

I never said cardinalities don't have size.

But I'll say now that cardinalities are sizes.

Two sets are equinumerous iff there is a bijection between them.

The cardinality of a set is the cardinal number with which the set is equinumerous.

'the size of the set' and 'the cardinality of the set' are synonymous.

And we say that two sets have the same cardinality iff they are equinumerous.

And you have it backwards:

The original poster claims that it is contradictory to say that there are different infinite sizes. I have been saying that it is not contradictory to say that there are different infinite sizes. And I have been saying that in set theory it is easy to prove that there are different infinite sizes and indeed that some infinite sets are larger than other infinite sets.

It is amazing that you reversed it completely to characterize me as saying the opposite of what I have been saying.

/

There was no "ass handing" though you like the tough talk sound of that.

/

just goes to show you've got a lot to learn, but I'm happy to correct you any time pal.

I am continually overwhelmed by how much I don't know and could learn. But with you what I have learned is not about mathematics or philosophy.

I'm happy to correct you any time pal.

I'm happy to be corrected any time I am incorrect.
• 3.7k
Relevant: Lawvere's fixed point theorem.

Good for you. I flamed out at "epimorphism". (i.e., the beginning). And I have actually worked with fixed points in Banach spaces and specifically the complex plane.
• 1.5k

Nothing about what you said demonstrated my argument was circular. How was I begging the question?
• 3.2k
It could not be more clear.

You wrote:

"if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them."

But:

"there is an infinite [number of] natural numbers" is just another way of saying "[the] set [of natural numbers] is infinite".

So your argument is just that we prove the set of natural numbers is infinite because it is infinite (has infinitely many members).

Proving that a set is infinite is the same as proving that it has infinitely many members.

So it is question begging to assume the set of natural numbers has infinitely many members when that assumption is just another way of saying what you want to prove.

But you can look up actual proofs that the set of natural numbers is infinite.
• 12.8k
Here's actually some advice to all non-mathematicians (from a non-mathematician):

If you really can ask an interesting foundational question that isn't illogical or doesn't lacks basic understanding, you actually won't get an answer... because it really is an interesting foundational question!

Yet if the answer is, please start from reading "Elementary Set Theory" or something similar then yes, you do have faulty reasoning.
ssu

I disagree. The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles. In that case, the issue is not a matter of better learning the mathematical representation of the fundamental principles, and how to apply them mathematically, as a math text will demonstrate, it is a matter of disagreement with those mathematical representations. Therefore the reply of "please start from reading 'Elementary Set Theory' or something similar", is usually just a copout, a refusal to engage with the philosophical matter at hand as if further reading of the mathematics will change a person's mind, who already disagrees with it. That's like telling an atheist to go read some theology, as if this is the way to turn the person around.
• 3.2k
The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles.

Again, the intellectual dishonesty of the crank in action. In this case, blatant strawman by misrepresentation of what his interlocutors have said. And even more egregiously by dint of the fact that this strawman has been pointed out to him many many times.

It's not a matter of disagreement on principles, but rather ignorant and confused misrepresentation of the mathematics that is supposedly being discredited. It is fine and even essential that there be different points of view about foundations including critiques of classical mathematics. But it is pernicious against knowledge and understanding when the attacks on mathematics claim things about the mathematics that are crucially false, when the attacks are premised in an ignorant prejudice that the mathematics works in certain ways that it definitely does not. After the crank's error about this have been explained to him over and over and over and he still persists to spread the disinformation, then the best thing is to recommend that he get a basic textbook to inform himself in the subject that he has spent so much time already cultivating his self-imposed terrible misunderstandings.

That's like telling an atheist to go read some theology, as if this is the way to turn the person around.

It's nothing like that. It's the reverse. It's like telling the zealot denouncing scientific theories to get a textbook in biology.

I don't know anything about microbiology, so I don't spout a bunch of nonsense about. If I did, I should expect someone to kindly tell me to shut up about it and get an introductory text.
• 3.7k
The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles

You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better.

Are you working on a change in the fundamentals of math that might calm your concerns? I hope so, no one said math as it stands is perfect.
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. I am more concerned with what issues you solve with your beliefs.

I believe the solution to Russell's paradox is in here:

There are also other things on the website. I think they are appropriately titled with regards to what they try to do or highlight or solve or discuss.

It seems to be a collection of semantic games

A triangle is triangular (or the angles in a triangle add up to 180 degrees) is not a semantic game. It is use of semantics in a non-contradictory manner. Counting to infinity, or there being no set of all sets is use of semantics in a contradictory manner.
• 12.8k
You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better.

I think the obvious point to start with is divisibility. Generally, mathematics provides that a quantity, any quantity, can be divided in any way. We can call that "infinite divisibility". In reality, there is very clearly many division proposals which simply cannot be done. Because of this fact, that there are real restrictions on divisibility, there is a very big difference between dividing a group of things, and dividing a single object. Each of these two types of division projects has a different type of restrictions or limitations on it.

For example, to divide a group of seven human beings into two equal groups is a project that cannot be done, even though common math would say seven divided by two is three and a half. So we'd have to chop a person in half. But then we'd have eight objects instead of seven, because we'd have have two halves, which are two objects, but unequal to the other six objects. So we have to conclude that the way we divide a group, or quantity of things is seriously restricted.

Further, the way that we quantify something dictates the way that the quantity can be quantized. So if we use weight for example, to measure the volume of a group of grains of sand, we do not count the grains and divide the number of grains evenly, we look at the sand as one thing, with one weight, and divide that weight however we will. But there will still be a issue with precise division, when we get to the point of needing to divide individual grains of sand.

This leads into the problem of dividing single objects. An object is a unit, and this is fundamentally a unity of parts. If there is an object which is not composed of parts, like the ancient atomists proposed for the "atom", this object would be indivisible, and provide the basis for the rules of all division projects. However, such an object has not been found, so the guidelines for dividing a unit must follow the natural restrictions provided by the divisibility of the type of object. Different types require different rules, so mathematics provides for all possibilities (infinite divisibility). What physicists have found, is that the true restrictions to divisibility of all things, are based in mass and wave action, rather than composite "parts".

This means that in order to provide the proper rules or guidelines for the division of units, unities, we need to understand the real nature of space and time. Mass is a feature of temporal extension at a point in space, and waves are a feature of spatial extension at a point in time. Where the common principles of mathematics mislead us is the assumption of "continuity", and this is closely related to the simplistic notion of "infinite divisibility".

Now we have two closely related, but faulty principles of mathematics, infinite divisibility and continuity. They are applied by physicists, and people believe they provide a true representation of reality, when physicists know that the evidence indicates the presence of discrete quanta rather than an infinitely divisible continuity. Therefore our representations of spatial and temporal features need to be completely reworked. To begin with, as I've argued in other threads, representing space with distinct continuous dimensions (Euclidian geometry) is fundamentally flawed. The separations within space indicated by quantum physics, must indicate distinct incommensurable parts. These distinct parts are the parts which may be represented dimensionally, and the parts which cannot be represented that way. However, they must be incorporated together in a way which adequately represents what's real. At the current time, we have a dimensional, continuous line (numberline), with non-dimensional points (real numbers) which may divide the line infinitely, but this is just an unprincipled imaginary concept which in no way represents the real divisibility of space, and it becomes completely inapplicable when physicists approach the real divisibility of space.
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