In the case of B = {A, 0}, is A a member of A/itself, or is A a member of B/non-itself? — Philosopher19

Both

When a set is a member of itself, it is not a member of another set — Philosopher19

And this is a fundamental misunderstanding of set theory.

If A = {A} and if B = {A, 0} then A is a member of A and a member of B.

But a set can either be a member of itself or a member of other than itself. — Philosopher19

A set can be a member of more than one set. You just don't understand the basics of set theory.

You should really take a few math lessons before you start telling mathematicians that they're wrong about maths.

What you say in response doesn't prove that Russell's paradox isn't a contradiction.

1. x is a member of R if and only if x is not a member of x
2. Let x = R
3. R is a member of R if and only if R is not a member of R

(3) is a very obvious contradiction. You don't even need to know maths to see that.

A is a member of both A and B.

I'll explain it to you in non-math terms:

I am a member of the football team and a member of the tennis team.

These are two different claims:

1. I am not a member of the football team
2. I am a member of a non-football team

(1) is false and (2) is true.

@Philosopher19

Regarding Russell's paradox, it is simply this:

1. $x$ is a member of $R$ if and only if $x$ is not a member of $x$.

Is $R$ a member of $R$?

Either answer entails a contradiction, and so (1) is a contradiction. Given that naive set theory entails (1), naive set theory is shown to be inconsistent.

@Philosopher19

These are two different claims:

1. A is not a member of itself
2. A is a member of some other set

Given this:

$A=\{A\}\\B=\{A,0\}$

(1) is false and (2) is true.

The Moving Spotlight Theory? Seems to be a hybrid view that allows for both eternalism and a dynamic time.

That would assume an eternalist view of time, in which time is treated much like a length, or as another spatial dimension. Whereas - prior to the untimely demise of this discussion - I was seeking to explore the limitations of eternalism, such as its logical omission of progress, happening or motion; characteristics that I consider to be absent from eternalism but logically aligned with the opposing view of presentism. However, many eternalists disagree. — Luke

I assume you're also against the growing block theory of time?

If you're arguing for presentism then this might be interesting:

Presentists Should Not Believe in Time Travel

The general gist being that the very concept of traveling to the past depends on the past existing in some sense as a location to travel to, and so requires either the growing block universe or eternalism.

If presentism is correct then any supposed time machine would work by rebuilding the universe into a facsimile of one of its past states, which isn't really time travel.

there exists a history — Luke

If there exists a history then presentism is false.

Does Trump think — Benkei

No

https://www.bbc.co.uk/news/world-us-canada-68266447

Donald Trump has said he would "encourage" Russia to attack any Nato member that fails to pay its bills as part of the Western military alliance.

At a rally on Saturday, he said he had once told a leader he would not protect a nation behind on its payments, and would "encourage" the aggressors to "do whatever the hell they want".

The ironic thing is that article 5 has only ever been invoked once; when the USA was attacked on 9/11.

I can’t speak on the more scientific aspects of that paper, but on that final section, although it’s the case that any randomly selected brain is most likely a batty brain, it’s also the case that any randomly selected non-batty brain is most likely a Boltzmann brain.

But, so far as this thread goes, if Boltzmann brains exist, that shows that there is a world. — Banno

Sure. Same with brains in a vat.

That does not render Boltzmann brains true. — Banno

I’m not saying they’re true, only that if our understanding of physics is correct then it’s most likely.

I am not a Boltzman Brain, nor am I the reincarnation of Cleopatra. I am Banno. — Banno

“I am Banno” and “I am a Boltzmann brain” are not in conflict.

You are Banno, and if our physics is correct then you are also most likely a Boltzmann brain.

If you are convinced by Boltzmann to believe you are a Boltzmann brain, then the universe is pretty much as physics describes it, since that description - physics - is what Boltzmann uses to reach the conclusion that you are a Boltzmann brain...

And yet somehow the argument is seen as reaching the conclusion that the world is not as it appears... — Banno

The conclusion is that there is an external world that behaves according to the laws of physics but that we are most likely brains floating in a vacuum rather than embodied humans living on Earth. Part skepticism, part external world realism.

Of course, the problem is when you want to accept the veracity of physics but reject the implication that we are most likely Boltzmann brains. How would you resolve that apparent contradiction without resorting to special pleading?

And how would you reply to each of these counterpoints, were you arguing my view? — Banno

I’m not sure if I would. I just accept the existence of a material world and that my everyday experiences are of that material world as a matter of faith, even though there might be good reasons to believe otherwise, such as Bostrom’s simulation argument or the implications of Boltzmann brains being physically possible (and even likely).

But I won’t pretend that this faith is more reasonable than the alternative view.

It seems to me that you must conclude that there is something more than just your thoughts. — Banno

Certainly more than my thoughts but possibly not more than my thoughts and experiences.

Novelty. We are sometimes surprised by things that are unexpected. How is this possible if all that there is, is already in one’s mind? — Banno

I can be surprised when I dream but it doesn’t follow that the things I dream about are “external” to my experience of them. So it’s not prima facie necessary that the same isn’t true of waking experience. It could be that dreams and waking experiences are two different modes of solipsistic existence.

Agreement . You and I sometimes agree as to what is the case. How is that possible unless there is something "external" to us both on which to agree? — Banno

It could be a shared hallucination. We’re both brains in a vat being fed the same misleading sensory inputs. Or it could be that you’re a figment of my imagination.

Error. We sometimes are wrong about how things are. How can this be possible if there is not a way that things are, independent of what we believe? — Banno

If solipsism is true but I believe that solipsism is false then my belief is in error. If solipsism and so mathematical antirealism are true then I still don’t know the square root of pi. Other minds and an external material world are not necessary to be wrong or ignorant.

Perhaps check out this article:

Epistemological Problems of Perception

The discussion in this thread, like all discussions, presupposes the existence of an "external" world in which the discussion is taking place... — Banno

Perhaps at the very least it presupposes that solipsism is false. It need not presuppose the existence of a material world (e.g. it allows for idealism), or that the world we experience is that material world (e.g. it allows for us being brains-in-a-vat).

But as philosophers we tend to want for something stronger than presuppositions, and so even solipsism is an open question.

There is no such a thing as "infinite" number. See this is an illusion, and source of the confusion.
Infinity is a property of motion or action, nothing to do with numbers. Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number until halted by break signal (as can be demonstrated in computer programming).

A set containing 3 numbers can be made infinite, when it is in the counting Loop 1, 2, 3, 1, 2, 3 .... ∞ Therefore a term "infinite number" is a misnomer. I bet my bottom dollar that you will never find a number which is infinite, because it doesn't exist. If it did exist, then it is not an infinite number. — Corvus

Extended real number line

In mathematics, the extended real number system is obtained from the real number system ${R}$ by adding two infinity elements: +∞ and −∞, where the infinities are treated as actual numbers.

BREAKING
Special counsel says there is evidence Biden 'willfully retained and disclosed classified materials' but will not be charged
https://www.nbcnews.com/politics/joe-biden/special-counsel-says-evidence-biden-willfully-retained-disclosed-class-rcna96666 — RogueAI

The full quote, in the body, continues with:

but the evidence "does not establish Mr. Biden's guilt beyond a reasonable doubt."

Biden needs to drop out. There's no way around it. — RogueAI

Sure, after Trump.

But urgently, how do you write matrices and footnotes and equations here? — Lionino

https://thephilosophyforum.com/discussion/5224/mathjax-tutorial-typeset-logic-neatly-so-that-people-read-your-posts/p1

A question for mathematicians: looking at what I've done above, can this be written as a matrix like this?

$\begin{bmatrix}0.1 & 0.01 & 0.001 & \cdots \\ 0.2 & 0.02 & 0.002 & \cdots \\ 0.3 & 0.03 & 0.003 & \cdots\\\vdots & \vdots & \vdots & \vdots \\ 0.11 & 0.011 & 0.0011 & \cdots \\ 0.12 & 0.012 & 0.0012 & \cdots \\ 0.13 & 0.013 & 0.0013 & \cdots\\\vdots & \vdots & \vdots & \ddots\end{bmatrix}$

We can then say that $a_{m,n} = {m\over10^{n+\lfloor\log_{10}(m)\rfloor}}$?

Yes, there are an "infinite number" of infinite sets:

$\{0.1,\text{ }0.01,\text{ }0.001,\text{ }...\}\\\{0.2,\text{ }0.02,\text{ }0.002,\text{ }...\}\\\{0.3,\text{ }0.03,\text{ }0.003,\text{ }...\}\\...\\\{0.11,\text{ }0.011,\text{ }0.0011,\text{ }...\}\\\{0.12,\text{ }0.012,\text{ }0.0012,\text{ }...\}\\\{0.13,\text{ }0.013,\text{ }0.0013,\text{ }...\}\\...$

visual experiences are biological facts that arise under certain conditions of satisfaction — jkop

The epistemological problem of perception seeks to understand the relationship between visual experiences and the external world objects that such experiences are putatively of. Roughly speaking indirect realists claim that the relationship is only causal, whereas direct realists claim that the relationship is more than that.

For example, we mostly agree that we experience a red colour when electromagnetic radiation of a certain wavelength stimulates the sense receptors in our eyes, and that the wavelength of this electromagnetic radiation is influenced by the atomic structure of an external world object's surface.

What we don't agree on is whether or not it is correct – or even sensible – to say that this red colour is a property of that external world object. Indirect realists say that it isn't, whereas direct realists (or at least naive colour realists) say that it is.

Whether it's direct or not might not be a fruitful debate, because the way the term is understood is so diverse as to be hopelessly confusing. — Jamal

On that I agree.

I've tidied up my comment. Perhaps you could explain where you think I've gone wrong?

We adduce a sentence G that is is true (to be more precise, it is true in the standard model for the language of arithmetic) if and only if G is not provable in T.

Then we prove that G is not provable in T. — TonesInDeepFreeze

If all it proves is that every T has the true and unprovable sentence "this sentence is true and unprovable" then it seems vacuous.

Or does it prove that every T has a "natural" example of a true and unprovable sentence, like the strengthened finite Ramsey theorem in Peano arithmetic?

Of course, the particular problem here is really just linguistic — Jamal

I agree. This is the issue with the very meaning of the terms "direct" and "indirect". Each often seems to be defined as the inverse of the other, which is no definition at all.

Traditionally direct and indirect realism provide two competing answers to the epistemological problem of perception. The epistemological problem asks if we can trust our experiences to provide us with accurate information about the nature of external world objects. Direct realists conclude that we can and indirect realists conclude that we can't.

The usual way direct realists phrase their position is just to say that we perceive external world objects. But it doesn't follow from this that we can trust our experiences to provide us with accurate information about the nature of external world objects – unless they mean something very specific by perceiving external world objects.

So what meaning of perceiving external world objects would entail the direct realist's conclusion that we can trust our experiences to provide us with accurate information about the nature of external world objects?

My own take is that the properties of the experience – the look, the sound, the feel, the taste, the smell – must be properties of the external world objects. If they are then we can trust our experiences to provide us with accurate information about the nature of external world objects and if they aren't then we can't.

But what if someone were to say that the properties of the experience are not properties of the external world objects but that we are nonetheless perceiving external world objects? At the very least it seems to show that the usual dichotomy is an overly simplistic division. There should in fact be four options to consider:

1. We perceive external world objects and the properties of the experience are properties of the external world objects
2. We perceive external world objects and the properties of the experience are not properties of the external world objects
3. We do not perceive external world objects and the properties of the experience are properties of the external world objects
4. We do not perceive external world objects and the properties of the experience are not properties of the external world objects

We can perhaps dismiss (3) as a viable option, (1) I understand as naive realism, (4) I understand as indirect realism, but what of (2)?

(2) disagrees with the indirect realist's claim that we do not perceive external world objects but entails the indirect realist's conclusion that we cannot trust our experiences to provide us with accurate information about the nature of external world objects.

Is (2) direct or indirect realism, or something else?

Although I suspect that indirect realists will claim that we perceive external world objects if and only if the properties of the experience are properties of the external world objects, and so that (2) is a contradiction.

No, it doesn't. It is kind of like asking what physics has to say about if the sun suddenly wasn't there. Would Earth continue to orbit for 8 minutes or would it immediately commence a straight trajectory?
Another question: Does an infinite sheet of material (a meter-thick slab of concrete say) result in a uniform gravitational field?

Physics has nothing to say about either case since there is no way to describe what any of the above even means.

Luke is exploring a philosophical question about the implications of various philosophical models on the concept of time travel. The current model seems to be a sort of growing block model, which is full of contradictions, most of which have been left unexplored due to the slow pace of working through even the trivial bits. — noAxioms

Well, I think any reasonable philosophy needs to take into account the facts as we best understand them. According to General Relativity time is the fourth dimension of spacetime. Talking about "overwriting" the "timeline" is like talking about "overwriting" the "heightline" or the "widthline" or the "lengthline". It seems pretty nonsense.

What does it mean to "overwrite" a direction in space(time)?

What does it refer to then? — RogueAI

Nothing really. "Tell me the truth" just means "don't lie".

Let's use Sherlock Holmes as an example. Does Sherlock Holmes exist as an idea? — RogueAI

What does "Sherlock Holmes exists as an idea" mean? Does it mean "the idea of Sherlock Holmes exists"? And does this mean "we (can) imagine Sherlock Holmes"? I agree with this. But this does not entail that the name "Sherlock Holmes" refers to something that exists.

Like with the word "fact" your question abuses the ambiguity of language.

You're talking about fictional things: ghosts, Dracula, Sherlock Holmes, God, etc. Fictional things exist as ideas, otherwise, we wouldn't be able to intelligently talk about them. — RogueAI

If "ghosts" referred to something that exists then ipso facto ghosts exist. Ghosts don't exist. Therefore, "ghosts" doesn't refer to something that exists.

Also, if a lawyer tells a jury, "You'll discover what the truth is when the trial is done" he's not talking about something like a ghost, is he? — RogueAI

Like the noun "ghost", the noun "the truth" doesn't refer to something that exists.

Ghost can refer to an idea, which is a physical thing. — RogueAI

Ghosts don't exist. Therefore the word "ghosts" in the sentence "ghosts don't exist" doesn't refer to something that exists.

But a noun is always a person, place, thing, or idea. Those are all physical things, in the materialist ontology. If a word is correctly being used as a noun, it has to refer to some physical thing. — RogueAI

Did you see the next sentence of my comment (I can't remember if I edited it in after)?

"Ghost" is a noun. The existence of the noun "ghost" doesn't disprove materialism. Ghosts don't exist.

Under materialism, don't all nouns have to be physical? — RogueAI

Just because a word satisfies the grammatical role of being a noun isn't that it corresponds to some object that exists in the universe.

"Ghost" is a noun. The existence of the noun "ghost" doesn't disprove materialism. Ghosts don't exist.

Doesn't truth have to be a physical thing? — RogueAI

No. We just use the word "true" to describe a sentence that we understand as describing some feature of the world. There's no reason to treat "truth" as being some object that exists.

So let's do away with the word "true", like we did away with the word "fact".

Either a sentence describes some feature of the world or it doesn't.

Then you get X := NOT X — Brendan Golledge

Yes, that's how the sentence "this sentence is not true" is translated into symbolic logic.

Is "belonging to" a physical thing? They have to be, right? — RogueAI

Do they? I think this is where you're over-interpreting physicalism. Physicalism, as I understand it, is the position that everything that exists is a physical thing. Balls exist and are a physical thing. I exist and am a physical thing.

Belonging to isn't something that exists, and so isn't something that needs to be physical for physicalism to be correct. If A belongs to B then A is a physical thing and B is a physical thing. There are just two things involved.

The notion that the belonging to relationship between A and B must be some third physical thing that exists seems spurious. And the notion that the belonging to relationship between A and B is some non-physical mental thing that exists also seems spurious.

Whereas before the issue was with ambiguous language, the issue now seems to be with reifying verbs. You're giving too much metaphysical import to language.

I actually deleted that comment because I recognise that mountain height isn’t the best example. I think my previous comment about property is more pertinent.

I think that's a big problem for materialism. — RogueAI

Why? Is the below a big problem for materialism?

At T₁ the ball is someone's property. At T₂ everybody dies. Nothing physical has changed about the ball but it is no longer someone's property.