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

'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y). — TonesInDeepFreeze

No, I believe it is a crucial point that is used in an important proof I gave you, which to put it in as short a manner as possible is: An item in a subset cannot be both a member of the subset and the set. If it is a member of the subset, it is a member of the subset. If it is a member of the set, it is a member of the set.

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". — TonesInDeepFreeze

If it's a member of other than itself, this means that it's not a member of itself. So my question to you is, what is the difference?

So that this point is not misunderstood:

An item in a subset cannot be both a member of the subset and the set. If it is a member of the subset, it is a member of the subset. If it is a member of the set, it is a member of the set. — Philosopher19

The following must be considered properly:

Something cannot be both a member of itself and a member of other than itself at the same time. For example, take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?) — Philosopher19

Then you're a dumbass for arguing with me when I was correct that Infinities do indeed have different sizes if that's your stance too. Either way, you're an "egregious" dipshit. "Oh, this guy is arguing the same thing as I am, and he's being comical, I should "correct," him about infinities not having sizes even though that's MY STANCE! Oh, wait, it's my stance, noone else can have it!" Eitherway, the fact is, you're a dumbass when it comes to communication. — Vaskane

I have never argued against the point that there are different sizes (cardinalities). And I have never said that other people may not also point out that there are different sizes (cardinalities).

An item in a subset cannot be both a member of the subset and the set. — Philosopher19

Do you mean: If S is a subset of some set T and x is member of S, then x cannot be a member of T ?

That's incorrect. By the definition of 'subset', if S is a subset of T, and x is a member of S, then x is a member T.

If it's a member of other than itself, this means that it's not a member of itself. — Philosopher19

Every set is a member of certain other sets. For example, every x is a member of {x}.

With the axiom of regularity, no set is a member of itself. So with the conjunction:

x is a member of x and x is a member of other sets

the first conjunct is false, therefore the conjunction is false even though the second conjunct is true.

Without the axiom of regularity, it is not precluded that there are sets that are members of themselves. Therefore, without the axiom regularity, we cannot infer that there is no set that is both a member of itself and a member of other sets too.

/

Or maybe your phrasing is not what you mean. When I take your phrasing literally, as best I can, I take you to be saying that a set cannot be a member of another set and also a member of itself.

But if all you mean is that a set cannot be both a member of itself and not a member of itself, then, of course, we have no disagreement.

I take you to be saying that a set cannot be a member of another set and also a member of itself. — TonesInDeepFreeze

Yes. The list of lists that list themselves is a member of itself in that list alone. Even though it is also a member of the list of all lists, it is not a member of itself in the list of all lists precisely because it is a member of the list of all lists and not the list of all lists that list themselves.

Note that the above shows the need to distinguish between "member of self" and "not member of self". To say no such distinction exists or is possible is to have an incomplete/contradictory theory in my opinion (contradictory because it argues the semantics of "member of self" and "not member of self" do not exist in Existence).

Do you mean: If S is a subset of some set T and x is member of S, then x cannot be a member of T ? — TonesInDeepFreeze

That's incorrect. By the definition of 'subset', if S is a subset of T, and x is a member of S, then x is a member T. — TonesInDeepFreeze

I get where you're coming from and I believe I completely get what you're saying. I don't deny the set of all natural numbers encompasses the set of all even numbers. Confusion occurs when one views sets that are not members of themselves (precisely because they are members of the set of all sets and not themselves) as being members of themselves.

To meaningfully talk about "member of self" and "not member of self", a set/context/reference is needed and adherence to it is necessary for the sake of consistency. The best way for me to convey to you what I'm saying in response to what you're saying, is the following:

Take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?) — Philosopher19

I'm thinking of starting a DIPSHIT PRIZE that we can nominate for and pass around here.

TDF really isn't that bad. He tries hard. You can be in charge.

Edit: To management...Not serious.

Thanks for your thoughtful and intelligent reply.

That's an interesting post. I've seen a few arguments that the success of eternalism in physics, to the extent that many popular physics texts openly suggest "eternalism is what physics says is true," largely flows from similar assumptions in mathematics. That is, it's a similar case of "this is how we think of mathematics, so this is how the world must be."

I am not sure if this is so much a problem with mathematics though as it is with how it gets applied to the sciences and philosophy. It seems to me that infinite divisibility might be worth investigating even if it doesn't accurately reflect "how things are."

distinguish between "member of self" and "not member of self" — Philosopher19

That is one thing you say that makes sense and is correct.

I am not sure if this is so much a problem with mathematics though as it is with how it gets applied to the sciences and philosophy. It seems to me that infinite divisibility might be worth investigating even if it doesn't accurately reflect "how things are." — Count Timothy von Icarus

I think there is a very close relationship between "mathematics" as the principles, rules etc., and the application of those principles. As Plato said, the people who use the tools ought to have a say in the design of the tool. And in reality they do, because the ones using the tools choose and buy the ones they like, therefore design and production is tailored for the market of application.

So in the case of "infinite" for example, the principles of calculus allow for the representation of an operation which is carried out without a limit. The limit is infinite, which essentially means there is no limit, and the operation proceeds endlessly. This representation proved to be very useful in application.

The issue we can look at, as philosophers, is what exactly is the effect of such an untruthful representation. First, we need to accept the fact that it is untruthful. To allow into any logical "conclusion", that an operation has been carried out without end is a false premise. In reality, the need to carry out the operation endlessly would deny the possibility of a conclusion.

The next step I believe, is to apprehend the level of ignorance which this untruthful representation propagates. There are some very specific problems produced from our conceptions of the continuity of space and time, which were demonstrated by Zeno. The mathematical representation (or more properly misrepresentation) as a premise in calculus, creates the illusion that these problems have been resolved, and so there is denial and ignorance concerning the reality of the problem amongst many people.

Finally, we can see how allowing this untruthful representation actually magnifies the problem rather than resolving it. When the usefulness of the misrepresentation is apprehended and recognized, it, and similar forms are allowed to pervade throughout the logical system (we can call this the propagation of self-deception). This creates the issue pointed to by the op, the need for different types of infinities, infinities of infinities, and the transfinite in general. The issue with the transfinite being, that some applications require the truth, a finite number, while others require the impossible, or false representation of a conclusion drawn from an endless operation, so some applications require a relationship between the two, hence the "transfinite". We can look at it as a bridge between the untruthful, and the truthful, a bridge which enables the self-deception.

I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one. I know there are a lot of physicists who claim that the universe must be computable, in which case it cannot truly require the reals to represent it, and space cannot be truly infinitely divisible. But from what I understand, experiments to support this idea have been wholly inconclusive. There are also folks like Gisin who argue that intuitionist mathematics is the better structure for representing physics, but they are a small minority (albeit seemingly a growing one). But there certainly do seem to be contrary voices who argue that mathematics based on true continuities, infinite points between any two points, works best for predicting empirical results precisely because it does represent the world. That is, perhaps not all elements of the world are infinitely divisible, but some, like space-time, would be.

I know Paul Davies claims to have an experiment that might settle this issue but it's currently impossible to actually perform, involving an insane number of beam splitters and accuracy. Other experiments looking at how light travels from distant stars have made predictions about how it must travel if space conforms to a finitist model, but the data ended up supporting a continuity, although from what I understand, these are no way definitive experiments. And then we can consider that certain discrete limits, like the Landauer Limit, appear in experimentation to turn out not to exist.

So, part of the problem might be that there seems to be informed disagreement on what represents truth here. Although, I do agree that it is problematic when a position becomes the "default" through inertia, despite not having strong evidence for it being the case over the contrary position. I would say reductionism is a strong example of this, where actual empirical support would seem to leave the question undecided, but it remains the default anyhow.

I find the intuitionist view fascinating because it would seem to allow for potential infinites of division, but not actual ones, a sort of near reintroduction of Aristotle.

I believe the solution to Russell's paradox is in here:
http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/ — Philosopher19

Honestly, I am having trouble dissecting the arguments used here.
Thoughts, @jgill ?

I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one. — Count Timothy von Icarus

It appears to me, like the quantum nature of energy demonstrates quite conclusively that the reality of space and time cannot be infinitely divisible. You see, the wave-function represents a continuity, but what it represents is not an observable aspect of reality. Observations indicate discrete occurrences of so-called particles (quanta), with not necessary continuity between the occurrences. If there is a true continuity, it is not represented by the wave-function, which represents possibilities. And, it is not the continuity of space-time, which fails at the quantum level. So it hasn't yet been determined.

I'm really sceptical of the idea that there is any one true math to decide these issues of infinity.

To me the best we can do is categorize models of infinity as conceptual mathematical objects.

As such the parameters are arbitrary and their usefulness is in a defined mathematical environment.

Under this categorization scheme, it can be possible that one model can be inconsistent with another and not be false.

Here is my example,
A smaller infinity can reach any finite number that a larger infinity can by freezing the larger infinity and letting the smaller one catch up.
I'm sure there are all kinds of problems with this in the standardized mathematics but in the sense of a conseptual mathematical object it is legitimate. I think I first said it as a bit of a joke but the idea is we can drive the math by abstractions.

I'm really sceptical of the idea that there is any one true math to decide these issues of infinity. — Mark Nyquist

We'd have to look at the arguments of people who have said that there is. Who do you have in mind? Naturally, we would look at realists such as Godel. And there are also cranks who at least present as if their own vague, undeveloped, impressionistic and incoherently suggested concept is the true concept, as meanwhile they do explicitly represent that classical mathematics is false.

it can be possible that one model can be inconsistent with another and not be false. — Mark Nyquist

I'd rather say 'theory' than 'model'.

But then we must ask what we mean by a theory being true or false. In a rigorous sense, a theory is true or false in a model for the language for the theory. A theory may be true in some models and false in others. And of course, if a theory T is inconsistent with a theory S, then there may be models in which T is true but other models in which S is true.

A smaller infinity can reach any finite number that a larger infinity can by freezing the larger infinity and letting the smaller one catch up. — Mark Nyquist

Define 'reach', 'freezing' and 'letting catch up'. Better yet, tell me your primitives and your sequence of definitions from the primitives.

What I've written is about as far as I've gotten on a 'theory'.

I was thinking there might be an application for this in central banking or distributing resources to competing unlimited wants. Maybe the math is out there in some form already. Wouldn't doubt it.

What about two infinity generating machines that spit out consecutive integers at variable speeds endlessly.
Set a dial and one or the other can go faster or slower or stop. If you have a system like that matched to physical systems that have finite limits it might be an interesting model

I'm in over my head but don't infinities have some rubber band like properties that can be set at will.

My interest is mostly going from brain state to doing the math as a basis for a philosophy of mathematics..... Real simple,. Brain; (math processes)

Don't expect everyone to do it perfectly and in learning math or new skills it's always a process of brain programing.

Fight the cranks all you like. Makes things interesting. It's just philosophy here not pure math.

It's just philosophy here not pure math. — Mark Nyquist

I have addressed that so many times in so many threads. Maybe earlier in this thread too.

I'm in over my head — Mark Nyquist

When in over one's head, it is recommended to keep one's mouth shut, and head for the shallows. People have drowned in these waters.

I've had college algebra, trig and calculus.
I can also design trusses and figure pressure loss in pipelines. Doesn't that sound exciting.

I can also design trusses and figure pressure loss in pipelines. Doesn't that sound exciting. — Mark Nyquist

No, I hate trusses. But hey, more power to civil engineers, though I would rather let the computer handle all those forces in different joints. Don't ask me about hyperstatic structures — I don't know.

I've had college algebra, trig and calculus.
I can also design trusses and figure pressure loss in pipelines. Doesn't that sound exciting. — Mark Nyquist

Well then, when in over your head, retreat to dry land and build a bridge.

I don't really design trusses but in addition to course work I made my own collection of scale model trusses of various designs. I still have them in a folder somewhere. Glue and cardboard.

I believe the solution to Russell's paradox is in here:
http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/ — Philosopher19

Honestly, I am having trouble dissecting the arguments used here.
Thoughts, jgill ? — Lionino

Russell's Paradox and infinity arguments hold no interest for me. After going round and round with the author on First Causes, I suspect I would learn little from this paper.

I can also design trusses and figure pressure loss in pipelines. Doesn't that sound exciting. — Mark Nyquist

I'm interested to know exactly how pressure is lost in pipelines, if there is no leaks. I've heard that in the USA a huge amount of natural gas just goes missing. Where does it go?

I've heard that in the USA a huge amount of natural gas just goes missing. Where does it go? — Metaphysician Undercover

Into your posts?

(Ok, but someone had to say it...)

Friction loss but it's way off topic.

Infinity is more of a process of continuing than a quantity?

I don't think the process of continuing forever amounts to anything infinite. I see infinity as the reason for the process of continuing forever as being possible/meaningful.

To me, Infinity and Existence denote the same.

I see Existence as the set of all trees, humans, numbers, existents/cardinalities. I see the set of all existents/cardinalities as Infinite. I'm not sure if I should describe Infinity as a quantity here or not. But I think something like 'the cardinality of absolutely all existents (so that's all numbers, letters, trees, semantics, hypothetical possibilities and so on), amounts to Infinity'. I don't see Existence as incomplete because such a view runs into contradictions, hence the need for Existence to equal Infinite (and possibly the need for Infinity to equal a quantity representative of the cardinality of absolutely all existents).