No!

The paradox asks if X is a member of X. — SophistiCat

Let 'All sets that do not contain themselves as members' be

a = {x}
b = {y}
c = {z}
d = ... and these sets go on for as long as is necessary, e, f, g, h,...

Set X = {{x}, {y}, {z},...}

Suppose for some set h, h = {X}

I am saying X = {{x}, {y}, {z},...}\h

That is, X = {{x}, {y}, {z},...}\{X}

There may be h such that h = {X} or there may not.

I am saying X\h regardless and this is the definition of X.

In simple language X = "All sets that do not include themselves as members, except {X}"

You seem to be assuming that {X} is included in X but by definition it is not.

Or suppose Set V = {{x}, {y}, {z},...{X}}

Set X = V\{X} and there you have it.

If we define things as follows it might make it clearer-

a = {x}
b = {y}
c = {z}

Set X is the set of sets a, b, c so

Set X = {{x}, {y}, {z}}

If X is included

X = {{x}, {y}, {z}, {{x}, {y}, {z}}}

If X is not included

X = {{x}, {y}, {z}}

So X\X is {{x}, {y}, {z}} which is what I originally meant by X\X or X\{X}

Your notation is confusing. If you want to say that a is a member of X (a ∈ X), you would write that as

X = {a, ...}

which is not the same as

X = {{a}, ...}

{a} is a singleton set with a as the sole member. — SophistiCat

Yes, but X is a set of sets so X = {{a}, {b}, {c},...} but {a, b, c, ...} might be correct too as long as the logic of what I'm saying holds up. Link: https://truebeautyofmath.com/lesson-4-sets-of-sets/

I'm not seeing how you can "without X" and still have any X left - in terms of the notation. — tim wood

It is not 'without X' it is 'without {X}' as a set. {X} is not the same as X, my bad notation in the beginning notwithstanding. X\{X} is every set in X but not the set {X} itself.

X\{X} = {{a}, {b}, {c},...} but not {X}, regardless of whether {X} can be a member of X.

Excluding {X} is not the same as excluding X.

The paradox asks if {X} is a member of X but I am disposing of the paradox by defining X as X\{X} so there is no contradiction.

I get, "the set of all sets that do not contain themselves as subsets" = "the set of all sets that do not contain themselves as subsets" and/but excluding "the set of all sets that do not contain themselves as subsets." And that looks like the empty set. — tim wood

Yes, you are correct. Since we are talking about sets of sets, a better notation would be-

Set X = X\{X}

Set X₂ = (X U {X})\{X₂}

Set X₃ = (X₂ U {X₂})\{X₃} and so on.

Apologies for the sloppy notation.

There is no need to redefine the set. — EnPassant
But that's what you did. — Banno

No, I am saying there are infinite collections of things that are not a set.
See this link https://math.stackexchange.com/questions/24507/why-did-mathematicians-take-russells-paradox-seriously

Is it correct to rewrite this as X = X\X ? Can you translate into English? — tim wood

The paradox asks the question "Is X a member of itself?"

Let's say Set X = {{a}, {b}, {c},....}

If {X} is a member of X then

Set X = {{a}, {b}, {c},....{X}}

If {X} is a not member of X then

Set X = {{a}, {b}, {c},....}

But I am excluding {X} regardless of whether it can be a member of itself.

So Set X = {{a}, {b}, {c},....}

{X} can only be a member of itself according to the definition of X. I am explicitly excluding {X} from being a member of itself by definition. X\{X} excludes {X} as a member, not as the entire set.

Then I unite Set X with {X} in

X₂ = (X U {X})\{X₂} (See my next post)

Since the paradox shows that X is a kind of 'pathological' set we don't know where to put it.
I am creating X₂ and putting it in there. Then the process is repeated infinitely so that all relevant sets can be contained. The result is an infinite progression of sets that contain "All sets that are not members of themselves" And this entity turns out to be an infinity of sets, each nested within the other.

(It may also be that every X_i contains every X_j but not {X_i} but I have not got this far with it yet.)

{a} is a subset of A, B and C, but not a subset of X. — SophistiCat

Subset is transitive: If A is a subset of B and B is a subset of C, A is a subset of C.

{a} is a subset of {A} and {A} is a subset of {X} ---> {a} is a subset of {X}

No, {a} is not "in" A,B,C,... — jgill

{a} is a subset of A and A is a subset of X therefore {a} is a subset of X

Also, when it comes to set of sets, {a} can be an element.

Ok, change it to 'subset'. Post edited.

"I am having some trouble thinking of any well-defined set that does contain itself. Help?" — jgill

Set A = {a, w}
Set B = {a, x}
Set C = {a, y}

Set X = the set of sets that have {a} as an subset.

Set X = {A, B, C,...}

{a} is in X (because {a} is in A, B, C,...)

therefore X contains X

Well, a set is an unordered collection of individuals. The unordered collections of individuals that do not contain themselves is an unordered collection of individuals; therefor it is a set. — Banno

There is no need to redefine the set. All that is needed is to see that there are collections that are not single sets - as the paradox implies. I think Russel's Paradox is superficial and I never believed it "undermines mathematics" which strikes me as an unjustifiably dramatic statement.

In fact it is a trick question because of the way it is stated: "The set of all sets that do not contain themselves as subsets." Why are they calling it a set? I don't think something can be called a set unless it can be demonstrated to be such. And the paradox shows that it cannot be a set. The entity should be defined as "All sets that do not contain themselves as subsets" There is no ambiguity in stating it this way and intuitively I feel that such an entity is possible. But what is it if it is not a set?

If the logic of my first post is correct it is an infinite collection of sets, each nested within another.
What seems to be the case is that there can be infinite sets that are not simply a set. I say this, not because of my reasoning but also because Russel's solution involves something similar: an infinity of "types", each nested one within the other. But it may be possible to resolve the issue with sets alone (Russel's solution seems very artificial and contrived)

But the real question I am asking in my first post is: Is the logic I am using coherent? I don't see anything wrong with it, unless you can.

My largest uncertainty is not understanding what space-time is. If I think about it, it rather sounds like we are describing trajectories. Space... doesn't seem to exist. Does it? What is space? It's simply a dimension as far as I can tell (ie distance, relationship etc). Time is a measure of change. — Graeme M

Time is often defined as change but this is a weak definition; change is evidence of time but not a definition of it. (Space)time is a geometry according to which change happens. That is, time is the way change happens; it is the order according to which change happens. In physical spacetime general relativity describes how change happens. G.R. describes the geometry of time and how things happen in it. That geometry is time.

You say there is an object in universe A. But if an object is to exist space must exist and space must be coherent/mathematical. A non mathematical universe would be chaos. It would not be possible to move from a to b firstly because there would be no coherent space between a and b and secondly, there would be no space for an object to occupy. Chaos is chaotic beyond imagining

If the objects you are talking about are coherent objects the space within and around them must be coherent and laws would simply be a description of the shape of this space. Once you have space you have law/mathematics.

I don't know, but it's difficult to say that math is entirely made-up when it's so useful in scientific theories. Quantities of things exist, so does topography and function. — Marchesk

Here is a thought. Write the squares of numbers like this-

1 squared = 1
2 squared = 4
3 squared = 9 etc.

Now, you can plot this sequence of squares on a graph as a quadratic curve, the curve of x^2.

The question is, how can a flat piece of paper receive this concept of squared numbers so faithfully? How is it that it is possible to translate a thought about numbers onto a graph in flat space?

This can only be possible if there is a natural correspondence between mind and space. If mind and space were utterly different it would not be possible to create an image of mathematical ideas on a flat space. But if there is a natural correspondence between mind and space what is it? The only common factor I can think of is mathematics. That is, mind and space must be intrinsically mathematical.

More or less true in set theory, a particular branch of mathematics. My area was complex analysis and when I deal with the concept of infinity it is in the sense of unboundedness of sequences or processes. — jgill

Yes, but the limit can be defined independently of time.

Whether being and conscious awareness ("thinking") are the same is an interesting question. Again I find Heidegger a very interesting resource on these issues. I don't want to make this about Heidegger -- I have another thread for that -- but needless to say your question is a good one. — Xtrix

Suppose we define philosophy as 'knowing the world'. Then a cat is a philosopher because the cat, through consciousness, knows the world. And knows it in ways we cannot easily imagine. I don't want to be facetious but extend this to human consciousness; don't we know the world through consciousness? If you eat an apple you know what an apple is in a way that the intellect will never explain to you.

But we 'sophisticated' people in the 21st century are addicted to 'reason' and are conceited about any kind of knowledge that does not come from 'reason'. Reason is abstract, consciousness is concrete. Which is more truthful about the world?

I think it's a very weak relationship. That way you can equate St. Teresa of Jesus with Albert Einstein. It seems to me much more what separates them. — David Mo

Only, perhaps, in the way that the elephant's foot is very unlike its ears. They are both 'elephant stuff'. The non physical 'world beyond' is equally a quantum world and a divine world. The universe is immense and looks different from different angles. The Platonic realm and Teresa's world and quantum energy fields my well be the same world.

Setting side those never ending debates, what does it mean for a constructionist to be able to offer a proof for any conjecture involving an infinite sequence, such as any number greater than two is the sum of two primes? — Marchesk

It seems to me that a pure constructionist cannot even admit that there are an infinity of natural numbers. Induction proves that there are and many mathematical proofs rely on induction. But this comes back to what we mean by 'exist' in relation to numbers in a Platonic sense. What does 'exist' mean?

One unresolved question in philosophy is why there is something rather than nothing. We don't know but we know there is something. This necessary something that is, before all created things, is what is, eternally. This eternal substance is existence. It is not that this necessary something has the property 'existence' it is existence because existence cannot be a property. So, if numbers exist, they must be intrinsic to existence. And since it takes Mind for numbers to exist, existence must be Mind, if numbers are in existence. The only eternal mind in which numbers can exist is God's Mind.

What all this means is that existence, mind, and God are three names for the same thing.
In this context I am using the word 'existence' to mean that which necessarily is.

So how does a constructionist handle such a number? Do they deny that the set of all numbers is properly mathematical? — Marchesk

Kummer, Cantor's arch enemy, was a kind of constructionist and denied the reality of real numbers. I guess they just don't agree. The question here is What does 'real' mean when we are talking about (what seem to be) abstractions? What does 'exist' mean in the context of numbers existing?

Platonic thought has nothing to do with the visions of Saint Teresa. — David Mo

Both of them would say that there is an order beyond the physical image. As the hydrogen atom is an image of energy, the physical world is an image of a non physical order. In this way science and religion are based on a similar idea: that there is an order beyond physical particulars. Scientists call this order 'the laws of nature' religion/Platonism may call it other things, but it is 'the world beyond the world.'

That's not counting though. Anyone can make up a new definition of "counting", and use that definition to make whatever conclusion one wants to make about infinity. But that conclusion would be irrelevant to what "counting" really means to the rest of us. So if Cantor turned "counting" into some sort of abstract concept which has nothing do with the act of counting, as we know it, I don't see how that's relevant. You are just arguing through equivocation. — Metaphysician Undercover

The difference is really semantic. Counting is about associating a number with an object; 1 orange, 2 apples etc. But Cantor counts numbers with numbers by associating numbers with other numbers. In this way Cantor associates/counts the rational numbers with integers and comes to the conclusion that there are enough integers to count the rationals.

Infinity just means 'without end'. — A Seagull

In mathematics infinity is a set, such as Aleph Null, not a process. Infinity is not 'the biggest number' it is all numbers, together.

Sure, but we were talking about counting, not pure maths. The contested statement was:

Counting infinity has nothing to do with time. — Metaphysician Undercover

Cantor uses Aleph Null to count infinities. One can count an infinity conceptually, without time. How much time is there between the digits of pi? Likewise with the empty question 'What came before the beginning of time?' The real question is "What gives rise to time?" or "On what necessary condition is the world/universe contingent?" It is really an ontological question.

Positive integers can be generated by a process of iteration and partition:

Start with "/"
Iterate: //
And again: ///
And again: ////
So you get ///////////////////...

Partition each step: /, //, ///, ////,...

These partitions are sets

{/}, {//}, {///}, {////},...

and they are represented in Arabic numerals as

1, 2, 3, 4,...

The initial / need not be anything other than a concept of something or nothing.

In mathematics it can be the null set.

Counting infinity has nothing to do with time. An infinity of numbers does not require time to exist. They exist conceptually as a set.

Let me throw in another question: how does philosophy differ from "thinking" generally? Or does it? — Xtrix

What do you mean by 'thinking'? Abstract 'rational' thinking? Isn't simply being conscious thinking? If thought is energy 'flowing' through the mind then being is thinking. Thought is being. Being is thought.

Philosophy is not based on authority but on the exercise of personal reason. — David Mo

I'm not sure about this one. Early philosophy was closely aligned to mysticism (eg Plato's cave). Only in recent centuries did philosophy become heavily abstract and intellectual, 'reasonable'.

Philosophy is not religion
Philosophy is not sophistry
Philosophy is not science
Philosophy is not just ethics
Philosophy is not math
Philosophy is not just a form of literature — Pfhorrest

Can't surrealism be philosophy?

A tangent drawn to the curve on the leading edge of foreign policy is never parallel to a crow's beak at noon. And.

'cause there's all them other sorts of tautologies. — Banno

Some argue that there are mathematical or logical tautologies.

It is a linguistic tautology (Wittgenstein).

In the 'beginning' philosophy was more aligned with mysticism but since Descartes it has become more and more abstract and intellectual.

If messages could be sent between the twins at infinite speed that would show simultaneity:
"What are you doing now?"
- "I'm reading Dante"
But for all practical purposes simultaneity cannot be determined.

Evil cannot exist without good because good is being and evil needs being/God in order to exist. Evil is ultimately self destructive.

Good is life an being. Evil is the loss of being. It tends towards nothingness. Evil is not absolute, it must have some being in it if it is to have any potency; therefore God allows evil. Absolute evil is nothingness. Evil depends on being and therefore on good. It is inferior.

Marijuana can cause great psychic damage. https://www.youtube.com/watch?v=oZEivOBQ6nc

"what is the ontological status of institutions?" — Matias

An institution is an abstract concept made concrete, visible, by the people and objects that make it manifest. Communism is a concept. Capitalism is. They can be made manifest by imbuing them with energy.

If one assesses the evidence for Canada with intelligence the conclusion is that it must exist because its existence makes sense of the evidence. Likewise with God. God's existence is the most convincing answer to the available evidence.

Our apparatus is definitely classical, but it's a fairly radical direction to claim our apparatus imposes anything on the quantum realm ... as this seems to imply the apparatus exists first. — boethius

What I'm saying is that the geometry is imposed on the trace effects that register on photographic plates etc. It is not the particle that is being observed but the trace effect (eg a spot on a photographic plate). The point is that these trace effects are necessarily classical objects and any geometry that relates them is going to be a classical geometry.

Suppose you have a light source at A and a photographic plate at B with a spot made by a photon. Here are two trace effects with a straight line joining them. It is natural to assume the photon travelled in a straight line between A and B. But since photons exists in some exotic quantum geometry we cannot really say it travelled in a straight line, not least because it does not even live in our classical world.

Where then does the straight line come from? It is an artefact of the experiment itself. The experimental apparatus is a classical object in classical spacetime and likewise with the trace effects that are collected. Given this, the only geometry these trace effects can have is a classical geometry. But this tells us nothing about how the photon travelles from A to B since it is travelling in its own spacetime.

The crux of my idea is that there are two distinct spacetimes (quantum and classical) made manifest by ontological space. These spacetimes exist 'here' in our ontological space but because they are different geometries they are, from a geometric perspective, two different spacetimes.

The trace effects exist at the 'edge' between these two spacetimes, but on the classical side of it. The photon exists on the other side of it. So how can we measure quantum spacetime with classical rulers?

For example, understanding the complexities of Second Temple Judaism and the historical Jesus vs. the Jesus of what becomes the mythologized version of orthodox Christianity is quite lacking in most conversations. Same goes for the development of any religion really. None of them came out as perfectly christaline specimens — schopenhauer1

This is the point I am making re. mythology. People need to frame things in mythological terms and bare bones Christianity took on the mythological elements of the day. But that is all very well if it helped Christianity to take hold in the Roman Empire. Without the mythological packaging it might never have caught on. It is really the practical value of mythology that matters.

Uri Geller has been debunked on several occasions. — whollyrolling

Yes, that's true apparently. But the experiments were done under strict conditions and Geller is not the only one who could do these things. It comes to mind that Geller may have been able to do this but he lost his ability and started faking out of vanity. Otherwise we must call the author of the book a liar and I don't think he is.

There have been hundreds of experiments involving alleged psychics and alleged paranormal phenomena that have all come up completely empty. — whollyrolling

Uri Geller was tested in strict lab conditions and he bent strips of metal that were sealed inside glass tubes. It was also done by a number of British kids.

"The paranormal is a term that covers those weird phenomena that are seemingly beyond scientific explanation. For the past hundred years telepathy, extrasensory perception and psychokinesis have baffled researchers brave enough to fly in the face of the scientific establishment. Then came Uri Geller and in his wake others, some of them young children, to challenge orthodox science with their bewildering powers.... John Taylor, a distinguished and respected professor of mathematics, has shocked sceptics and scientists alike with his conclusions from this level-headed experimental investigation into the science of the paranormal; the open-minded will draw their own."

https://www.amazon.com/Superminds-Investigation-Paranormal-Picador-Books/dp/0330247050

Religion is hierarchical and only the top dog can be the master dog. — praxis

That would only apply to an evil religion. Hierarchies on the secular world also create great possibilities for evil.

Yes, but that would be true in a material sense that it is actually creating functions by harnessing natural processes and materials that are useful for survival, comfort, or entertainment and can be measured as to its development and effectiveness in solving the need or want. — schopenhauer1

True but that does not tell us anything about the veracity of belief. Personally I am very cautious about psychoanalytic views on religion. They are too vague and too easy to make up. It seems to me that humans are deeply attached to the language of myth. Myth may be older even than written language. You only have to look at tribes in far away places to see how mythological they are. Humans need to mythologize consciousness and that is why religion is so heavily mythologized. It is pointless to talk in terms of whether myth is 'true' or 'false'. Myth is only the 'packaging' for our spiritual reality (whatever you take that to mean). We build myth around these things because we are deeply mythological.

You can see myth evolving today in Hollywood movies. We have superheroes, fiends, angels and all manner of beings coming through our screens. These myths are 'archetypes' of realities deep in our psyche...

I'm not sure what you mean by everything happening simultaneous. If such were the case there would be no cause and effect? — boethius

Well, that was a throwaway comment. What I mean is anything could be the case for all we know.

Here is where my thoughts have led.

Space is two things. It is an ontological reality and a geometric reality. Ontologically space is there. It is not nothingness, it is a substance. But space as geometry seems to be more accessible to science.

Can it be that ontological space can simultaneously manifest more than one geometry or spacetime? It seems to me that quantum spacetime and classical spacetime (ie the macroscopic 4D world) exist simultaneously in the same ontological space. The spacetime that particles live in seems to be some kind of exotic N-dimensional geometry that is not classical.

I think it was Bohr that said it is meaningless to talk about where a particles is, outside measurement by a device in classical spacetime. In this respect 'where' means a position is classical spacetime. Apparently it is nowhere in classical spacetime at all, it is in quantum spacetime (geometrically speaking).

We don't see particles, we only see trace effects. A spot on a photographic plate is a trace effect, not a particle! What is important here is to see the both the detection apparatus and the trace effect are macroscopic, classical objects; they both exist in classical spacetime. This means that the trace effect is necessarily a classical object, obviously located in classical spacetime. But where is the particle before/after detection? Nowhere. Nowhere in classical spacetime that is. This is why Bohr says it is meaningless to say where it is. It is 'elsewhere'.

If there is a light source at A and a photographic plate at B and a photon is detected it is natural to assume that the photon travelled in a straight line from A to B. But, strictly speaking, all we can say is that the photon left a trace effect at A and a trace effect at B.
But these trace effects are classical objects and a straight line joining them is also a line drawn in classical spacetime.

If the photon is not really travelling in a straight line (because it is not even in classical spacetime) the straight line must be seen as an artefact of the experimental apparatus itself. This is because the whole experiment is taking place on 'this side' of the interface between these two spacetimes. Consequently any relationship between trace effects must be in terms of a classical 4-D geometry. That is, the positions of particles (in reality trace effects) is imposed on the situation because the experimental apparatus, being a classical object in classical spacetime, can do nothing else but force things into a classical geometry.