What is this supposed to mean? — Michael

I think it's clear enough, therefore, I don't want to clarify further.

The angles in a true triangle add up to 180 degrees because that is the nature of Existence. It is not because someone said it or highlighted it.

What do you mean by an "imperfect" triangle? — Michael

What you may call a non-euclidian triangle, I call an imperfect triangle. A perfect triangle has perfectly straight lines and its angles add up to 180 degrees. Another shape may resemble this without actually perfectly being this (like an imperfect triangle whose angles don't add up to 180 degrees but is near)

Suppose someone produces an axiom. Will it not be the case that that axiom will either be contradictory in relation to certain truths or consistent in relation to certain truths? Existence determines what is true and what is false. Whether any belief or axiom highlights truths or is contradictory to truth is determined by Existence/Truth. If not, there is no truth or semantics to work with to deduce further truths.

You should see non-Euclidean geometry where the angles in a triangle can be more or less than 180 degrees. — Michael

Imperfect triangles are imperfect by definition. I'm focused on absolutes.

where there are an infinite number of elements (including fractions and irrationals) between 0 and 1 — punos

Compare the following:

1) There are an infinite number of elements between 0 and 1
2) There is no end to the number of elements between 0 and 1

If there is no end to something, how can another thing with no end be twice as large as it? Don't they both have no ends?

This is why there is a distinction between something that can go on forever and something that is infinite. Infinity allows for things like a number sequence to go on without end, but the thing that goes on without end is not infinite, it just goes on without end without actually reaching infinity just as one cannot count to infinity and reach infinity even if one was projected to count forever (so the number sequence is not infinite).

The answer to your problem is quite simple. In mathematics things are done by axiom. If you want to count to infinity and beyond, simply produce an axiom which allows you to do that, and bingo the infinite is countable, and you're ready to go beyond. Look closely at the following: — Metaphysician Undercover

My belief is that we can't just produce axioms. We can only recognise truths about Existence such as 1 add 1 equals 2 or the angles in a triangle add up to 180 degrees or one cannot count to infinity.

But having a different concept and definition of infinitude doesn't thereby entail that there is a contradiction in set theory or mathematics. — TonesInDeepFreeze

You see, my position is that there is only one semantic/definition for the label "infinity". If we are focused on different semantics, we are not talking about the same thing. I think it would then help if we don't use the same label for that thing so as to make it clear that we are talking about different semantics.

Again, yes, there may be a contradiction between set theory and certain other formulations. But that does not entail that there is a contradiction within set theory. — TonesInDeepFreeze

I think a belief or theory has to be consistent with Existence as a whole, and not just consistent in isolation. To me, by definition, any theory or belief that encompasses the following belief "the set of all sets is contradictory" is a contradictory belief. It would be like any theory or belief encompassing the belief that "triangles are not triangular", which is contradictory belief to encompass.

set theory does not refer to an object named 'infinity' but rather to the property of being infinite, which is a crucial distinction. — TonesInDeepFreeze

Again, I think you are focused on a different semantic to me. To me, the semantic of infinity is one quantity or quality. It is a quantity or measure or quality that can never be reached. My position is best summed up with the following:

The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite. The only thing I view as Infinite, is Existence.

I don't know what more to say. When I use the label/word "infinity", I'm not sure you're focused on the same semantic that I'm focused on.

Your posting is an absurd loop — TonesInDeepFreeze

I'm sorry if discussing with me has been a negative experience for you.

I am addressing your point. I believe you are not reading all of it. See my last post to you.

I also want to add one more thing to the following

But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers. — TonesInDeepFreeze

Suppose something goes on forever such that it covers more and more distance as it goes on. So it covers 5km, 10km, 15km ad infinitum. I can't say {5km, 15km, 20km, ...km} is an informal notation to stand for all the distance it covered and that that distance is infinite. Do you see what I'm saying? You can't just say {1,2,3,4,...} is an informal notation to stand for the set of all and only the natural numbers and that the total number of natural numbers in that set is infinity.

I believe in the same way that I can't say the total distance covered is infinite, you can't say the total number of natural numbers in that set is infinity.

But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers. — TonesInDeepFreeze

I don't deny that there is such a set, but I deny that the total number of natural numbers in this set reaches infinity. Imagine you have all the natural numbers in {1,2,3,4,...}. Can you show what number comes before infinity to be able to meaningfully assert something like {1,2,3,4,...} consists of an infinite number of natural numbers?

You can't just say it has all the numbers and all the numbers amount to infinity

There is least infinite cardinal, which is the cardinality of the set of the infinite set of natural numbers. And there are cardinals greater than the least infinite cardinal. Moreover, for each cardinal, whether it is a finite cardinal or infinite cardinal, there is a greater cardinal. — TonesInDeepFreeze

We are in disagreement right there. You say you can have an infinite number of natural numbers. I say this statement will lead to a contradiction. That contradiction being "this infinite set is bigger than that infinite set". This is a contradiction because infinity is that which you cannot add to or have more than of. If you don't believe in this then how can we possibly agree?

This is why I said:

The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite. — Philosopher19

My position is now probably best represented by the following:

The only reason something like a sequence of numbers can go on forever, is because of Infinity. It is not because the sequence of numbers are Infinite.

Two different things can go on forever at different speeds, but this does not mean that one will go farther than the other when both are set to go on forever. It may look that way if you were to try and "map the distance covered by one to the other", but neither will ever cover an Infinite amount of distance for one to be able to conclude something like "this Infinite distance covered is greater than that Infinite distance covered". Of course, this is not the same as saying something like "this amount of distance covered in Infinity is greater than that amount of distance covered in Infinity".

It seems to me that you think I'm not paying attention to what you're saying and I think you're not paying attention to what I'm saying. I think we should end our discussion.

Peace

A quick look will tell you that there are twice as many feet as there are people. You do not need to count the number of people to know this to be true; just check for amputees... — Banno

The benefit to me of what you've posted here is that I now reject the following from the OP and would change the last part of it in the link I provided to my post:

Perhaps one might argue that there is no count involved with regards to the latter and that it's just a fact that Infinity encompasses an infinite number of natural numbers. But if that's the case, then Infinity also encompasses an infinite number of possible real numbers and possible letters or possible x. But where there is no counting involved, all infinites are of the same size/quantity (or rather, infinity is one quantity as opposed to different quantities). — Philosopher19

I still hold the belief that saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers. One has essentially suggested a number sequence goes on forever. But since one cannot count to infinity, it is the case that the total number of items in that sequence will not be infinity. If I do not do this, I will hit contradictions. If I do this, I will avoid contradictions.

Seeing as your post benefited me, I should thank you. So thank you.

for any infinite cardinality there is a greater infinite cardinality — TonesInDeepFreeze

Which is the equivalent of saying beyond the quantity of infinity, there is a greater quantity of infinity (which is contradictory to the semantic of infinity). Again, you can add one to any quantity except of course the quantity of infinity.

Time well spent would be to learn some mathematics rather than claiming untrue things about it. — TonesInDeepFreeze

I don't believe I'm the one saying untrue things.

Is it that Philosopher19 has a picture of infinity such that, since one cannot count to infinity, one cannot have a grasp of infinity? — Banno

No I think we have a grasp of infinity or an awareness of the semantic. Some are more focused on this awareness than others. Some are more sincere to this awareness than others. Part of that awareness entails one cannot count to infinity. You can add one to any quantity, except of course infinity. Such is the nature of infinity. Yet, it seems to me that some seem to believe "beyond infinity" is meaningful.
What more can I say?

I believe I've said enough in this discussion and that beyond this is time not well spent.

Peace

I just want to say that I've had a look at the OP and I believe it to be very outdated. I believe that instead of reading the OP one is much better off reading the following (if one was to read my work):

http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/

Peace

By bijection. See Open Logic Ch.4. — Banno

A one to one to correspondence implies a count of one side compared to the other. But infinity is not reached or exhausted and cannot be counted to

"Counting", and ill-defined notion, is not involved in bijection, although "enumeration", a well-defined notion, is. — Banno

Is it not? Do you not count how many maps onto how many?

See Cantor's diagonal argument. — Banno

I have already seen. Tell me what about it suggests that infinity is more than one possible quantity despite it being the case that infinity is one semantic as opposed to two. Note that 5 is one semantic as opposed to two.

Two questions were asked, no answers were given:

How would a difference in size be established between two sets when there is no counting of the number of items in the sets involved?

If there is counting involved, how has one reached an infinite number of items?

I also asked an additional question:

If infinity is a quantity, how is it more than one different quantity?

If I ask how many items in that set and the answer is infinite and I ask how many items in that other set, it is surely contradictory for someone to say to me and even bigger infinity. There is no beyond one infinity for there to be the possibility of a bigger infinity.

Because the choice is between the whole of the remainder of that project being founded on an error unnoticed by more than a century of study by logicians world wide; and your being mistaken. — Banno

That sounds like bias and dogma to me as opposed to actual discussing of the argument with attention to detail.

And this shows that you have not understood R = {x : x ∉ x}: — Banno

To me it looks like no meaningful answer has been given to me and that what I have posted has not been paid attention to.

I think it's clear that one cannot count to infinity So one cannot say that x is an infinite sequence of numbers just because it goes on forever. If I count forever I will not reach infinity. I cannot say assume I completed my count with this set of numbers and that set of numbers and then argue that that set is a bigger infinite set than the other.

Just think: how can one infinite quantity be bigger than another when the quantity of infinity is one quantity?

Again, I have said this multiple times. I recognise and acknowledge the following:

Proof. If R = {x : x ∉ x} exists, then R ∈ R iff R ∉ R, which is a contradiction. — Open Logic: Complete build

What has not been shown to me is how this logically obliges us to view the set of all sets as contradictory.

Again, it is clearly contradictory for any set to contain all sets that are not members of themselves and no other set. But this does not mean that it is contradictory for the set of all sets to contain all sets that are not members of themselves. The set of all sets contains absolutely all sets that are not members of themselves and it is a member of itself.
Just because one set is contradictory (In the above case R), doesn't mean another set is also contradictory when they are not the same sets.

The set of all sets and R are not the same sets.

I can't give an opinion but you have put some work into it to your credit. — Mark Nyquist

Thanks Mark

My work on infinity and the universal set if anyone is interested:

http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/

Given your responses in the other discussion (the infinity one). I see no point in continuing this discussion.

Peace

When you say the axioms of naive set theory, are you referring to those notations that I asked you to put in clear language. If so, it seems to me you left half way through trying to clarity on it.

Yes, we can establish set X as being "bigger" than set Y without counting the number of items in X and Y. We can establish this by using Cantor's diagonal argument. If you were a mathematician you would understand it. — Michael

If you used reason you'd know that you cannot count to infinity and that you cannot say x is bigger than y without some measurement/count involved to compare the sizes of the two.

I leave the following as an open question to anyone who believes in infinite sets of varying sizes:

Can we establish set x as being bigger than set y without counting the number of items in sets x and y? If yes, how? If no, what do we do with the problem of "one cannot count to infinity"?

Peace

Again:

I've seen Cantor's diagonal argument. It does not answer the questions I asked you in my last post to you. — Philosopher19

That is not an answer.

It's like me asking "can you count to infinity?" where the answer should be no, but someone responding with "Jack's diagonal argument" implying you can without actually showing how.
Again, I've seen Cantor's diagonal argument. It does not answer the questions I asked you in my last post to you.

Can you answer the following:

Can we establish set x as being bigger than set y without counting the number of items in x and y? If yes, how? If no, what do we do with the problem of "one cannot count to infinity"?

Russell proved that these two axioms entail that there is a set that only contains all sets that are not members of themselves (the Russell set). — Michael

I followed your original notation and tried to get clarity on it. We came to the following:

So when you say:
There exists a set B whose members are precisely those objects that satisfy the predicate
— Michael
are you essentially saying "there is a set that contains all sets that are not members of themselves"? If not, can you clarify? — Philosopher19

To which you answered "yes". To which I highlighted to you a clear difference:

1) There exists a set whose members are sets that are not members of themselves
2) There exists a set that contains all sets that are not members of themselves — Philosopher19

Again, 1 is contradictory. Put it in clear language as to why the contradictoriness of 1 obliges us to reject 2 or to view the set of all sets as contradictory.

I've seen cantor's diagonal argument and the following objection applies to it:

How would a difference in size be established between two infinite sets when there is no counting involved? And if there is counting involved, how would infinity be reached given that one cannot count to infinity?

You've already admitted that you're not a mathematician, so it's strange that you think you know mathematics better than Cantor (and Russell). — Michael

Reason is accessible to everyone (not just mathematicians). I try to focus on the argument at hand as opposed to who is doing the arguing.

But you haven't addressed my point.

We are in agreement that p) you cannot have a set that only contains all sets that are not members of themselves. I am saying this has nothing to do with the fact that you can have q) a set that contains all sets that are not members of themselves. The set of all sets contains all sets that are not members of themselves (plus itself). Again, where is the contradiction in this?

It is clear that p is contradictory. You have not shown how this logically obliges us to view the set of all sets as contradictory (you have only shown how p is contradictory. See your post). And you have also not shown any contradiction in q.

↪Philosopher19 Yes. Given the axiom schema of unrestricted comprehension there exists a set B whose members are sets that are not members of themselves. This leads to a contadiction. If B is not a member of itself then it should be a member of itself. — Michael

There is a difference between:

1) There exists a set whose members are sets that are not members of themselves
2) There exists a set that contains all sets that are not members of themselves

2 is not contradictory at all whereas 1 could be contradictory.

2 is not contradictory because by definition, the set of all sets contains all sets that are not members of themselves and it is a member of itself. Where is the contradiction here?

1 is contradictory if you say set B only contains all sets that are not members of themselves. Again, the only set that by definition can contain all sets that are not members of themselves, is the set of all sets and it does not just contain all sets that are not members of themselves, it contains itself too.

All sets that are not members of themselves have to be a member of something don't they? It's like saying all existing non-self-contingent things have to be contingent on something don't they?

In Russell's paradox, φ is "sets that are not members of themselves". — Michael

Thanks.

So when you say:

There exists a set B whose members are precisely those objects that satisfy the predicate — Michael

are you essentially saying "there is a set that contains all sets that are not members of themselves"? If not, can you clarify?

It says that A and B are equal if every member of A is a member of B and every member of B is a member of A. — Michael

Ok

There exists a set B whose members are precisely those objects that satisfy the predicate — Michael

is predicate φ "A and B are equal if every member of A is a member of B and every member of B is a member of A"? If not, what is it?

Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B (the of axiom of extensionality) — Michael

Are you saying that A and B only contain X as members of themselves and they contain nothing other than X as members of themselves? If you are, I don't see it included in the above. If you are not, then either A or B can contain members other than X such that A is not equal to B despite both A and B have all Xs as members of themselves