"undetermined (can be true or false)"

>>8009271
it's false (under normal circumstances like using LEM)
wolfram just doesn't know that

because x is not defined in what you gave to wolfram

>>8009303
Please elaborate by what you mean by "not defined". It's not any "more defined" in https://www.wolframalpha.com/input/?i=x+%3D+x%2B1 , yet the assessment is correct in that one.

Can you "define" any [math]x[/math] which would make [math]x \in \emptyset[/math] a true statement? Because Wolfram, while remaining vague and cryptic, seems to be accounting for such a possibility somehow.

>>8009271
[math] x = \emptyset [/math]

>>8009375
Well... you kinda can make it "true" if [math]x[/math] is replaced by something that is contradictory in itself, such as "a four-sided triangle". Such objects don't exist, thus the set of all such objects is empty, thus the set of all such objects is THE (as there's only one!) empty set, thus all such objects are elements of the empty set.

Also, a "vacuous truth" statement can ascribe any arbitrary property to the (nonexistent) elements of the empty set. This may also come into play here somehow.

https://en.wikipedia.org/wiki/Vacuous_truth

>>8009404
No, The empty set does NOT contain itself. All elements of the empty set don't exist, and the empty set itself certainly exists, thus it cannot be its own element.

>>8009271
Why are you typing LaTeX macros into wolfram?

>>8009415
Wolfram can parse a wide variety of expression formats with more or less success. This particular expression seems to have been parsed correctly (as evidenced by the "input interpretation" line, which shows what was intended), it's the evaluation that is puzzling/interesting.

>>8009404
Not true. If this was so then what would be difference between the empty set and the set that contains the empty set.

>>8009404
you are thinking about empty set being a subset of itself

>>8009409
no,

>there exists (object which can't exist) and x is in the empty set

is false. You don't know what you're talking about and it's not a vacuous truth

>>8009438
>Wolfram can parse a wide variety of expression formats with more or less success
I'd have never thought to try this. I wonder how far that can go.

Anyways, it's just a bug stemming from the way Mathematica handles unqualified expressions like "x." They certainly don't have code that says "always return false if someone asks if anything is in the empty set."

>>8009458
0

>>8009566
>>there exists (object which can't exist) and x is in the empty set
>is false

What's wrong about saying that a non-existent(!) object is an element of the empty set? It does not violate any property of the empty set. After all, the object in question doesn't exist, just as any element of the empty set doesn't exist (just as a shelf that contains only books that don't exist is empty - and you can claim of any book that doesn't exist that is sits on that very shelf, and such a statement is vacuously true).

Also, please try to be constructive and elaborate on your criticism and/or give counterexamples. "You don't know what you're talking about" is not conctructive criticism.

Also:

[math]x = x + 1 \iff x \in \emptyset[/math]

How about the above equivalence? Tautology? Contradiction? Or, can be true OR false depending on the choice of [math]x[/math]?

>>8009596
there exists which is in the empty set is false.
that's the end of it.
all you're saying has nothing to do with it, they're other things.

>>8009577
Wolfram parsing LaTeX is actually a very good idea, it would allow users to feed precise expressions into the system without having to rely on any Wolfram-specific syntax.

File: wolframalpha_x=x+1.png (22KB, 669x425px) Image search: [Google]

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>>8009577
>Anyways, it's just a bug stemming from the way Mathematica handles unqualified expressions like "x." They certainly don't have code that says "always return false if someone asks if anything is in the empty set."

But then why doesn't it have any problems with pic related, and (expectedly enough) just evaluates it correctly as an unconditionally false (i.e. contraditory) expression? After all, the [math]x[/math] in [math]x = x + 1[/math] isn't any more "qualified" than the [math]x[/math] in [math]x \in \emptyset[/math] is. The universal falseness of [math]x = x + 1[/math] stems from fundamental arithmetic laws, just as the universal falseness of [math]x \in \emptyset[/math] stems from fundamental set theory laws - Wolfram is expected to be perfectly aware of both, isn't it?

>>8009601
>there exists

Nobody insisted on that. When I say "all four-sided triangles are elements of the empty set", nowhere do I say that these supposed objects exist. What I do here is ascribing the (unsatifiable, as it is) property of being a four-sided triangle to certain elements of the empty set (which is vacuously true, as per
>In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.
stated by https://en.wikipedia.org/wiki/Vacuous_truth).

>>8009624
when you write
>x = x+1
wolfram assumes that it's an equation to solve over reals or complexes
when you type x \in \emptyset then x can be pretty much everything

>>8009605
I agree. The more places can type LaTeX the better.

>>8009624
I'm pretty sure anytime you feed Wolfram a polynomial it will attempt to solve it over [math]\mathbb C[/math].

>>8009601
EXISTENT objects cannot be elements of [math]\emptyset[/math], because it's elements DON'T exist. However, NONEXISTENT objects can and are elements of [math]\emptyset[/math], because it's elements ARE nonexistent. Furthermore, it is a vacuously true statement to assert any property to members of [math]\emptyset[/math] EXCEPT the property of existence. Therefore, "all elements of the empty set are green" is a (vacuously) true statement, while "all elements of the empty set exist" (or, "there exists an element of the empty set") is a false statement.

>>8009664
> it's elements
stopped reading there.
better try again next thread buddy!

>>8009664
Also, "all four-sided triangles are elements of [math]\emptyset[/math]" is a true statement, because said objects are non-existent (just as all elements of [math]\emptyset[/math] are).

>>8009684
Yes, it should be "its elements". Sorry for the redundant apostrophes.

>>8009660
>when you type x \in \emptyset then x can be pretty much everything

Ok, but so far we haven't seen a well-defined example of any x that could make it non-false, and there doesn't seem to be any (unless we consider "nonexistent" x, as >>8009664 says - that seems like the only option, i.e. saying "a certain x, which doesn't exist, belongs to the empty set").

>>8009596
>What's wrong about saying that a non-existent object is an element
an element is an object existing in a set, so what's wrong is saying a non-existent object is an object that exists - it's contradictory

>>8009710
>an element is an object existing in a set
>existing

>In mathematics and logic, a vacuous truth is a statement that asserts that all members of the empty set have a certain property.
https://en.wikipedia.org/wiki/Vacuous_truth

It says it right there, "all members of the empty set" ("members" and "elements" of a set obviously being the same notion). Nowhere does it say that they must exist - specifically, they don't (and can't) exist, if the set is [math]\emptyset[/math]. You seem to be trying to make an argument based on some implied semantic nuances which don't really appear to even be there. To say "there's some cats in my room, all of which are non-existent" is the exact equivalent of "there are no cats in my room". Nobody in their right mind would insist that mere mentioning of cats as well as a location necessarily means that they really must exist there.

I checked and Mathematica itself, while it has Boolean logic to play with, doesn't implement the empty set of the element relation in any way that you can check.

So I'd say they just didn't fix the meaning / interpretation of it.

Besides, what's true in logic is that for any predicate P, you have

[math] \forall (x \in \{\}). \, P(x) [/math]

i.e. the vacuous truth thing - for all x in the empty set, P applies to x.
E.g. for all x in the empty set, x^3=-4 and x+1=84 as well as x^2=0.

https://en.wikipedia.org/wiki/James_Anderson_%28computer_scientist%29#Transreal_arithmetic_and_other_arithmetics

If this guy is a crank, then his [math]\Phi[/math] ("nullity") [math]\in \emptyset[/math].

>>8009752
>To say "there's some cats in my room, all of which are non-existent" is the exact equivalent of "there are no cats in my room".

That's wrong. The first is a contradiction, the second one is a proposition.
Also fucking please stop linking wikipedia. It's embarrassing.

>>8009817
"I've got some money in my wallet, and the exact amount is $0."

"The amount of money in my wallet is $0".

"I don't have money in my wallet."

To me, all of these amount to the exact same thing. You will probably try to argue that the word "some" in the first sentence necessarily means "more than $0", but, surprise, it really doesn't (or, at least, doesn't have to, which makes an assumption that it does just as much invalid).

>>8009830
That's because 0 exists. 0 isn't an element in the empty set.

>You will probably (bullshit)
No, I'm not going to argue idiotic semantics.

>>8009832
>That's because 0 exists. 0 isn't an element in the empty set.

You are missing the point entirely. It's about comparing overall sentence structure and meaning, not the subjects contained inside. When I say "a/b = c/d", I'm not comparing a to c or b to d, but rather the ratio of a and b to the ratio of c and d. Specifically, if

"I've got some money in my wallet, and the exact amount is $0."

means exactly the same as

"I don't have money in my wallet."

then

"There's some cats in my room, all of which are non-existent" (which can also be rephrased as "There's some cats in my room, and there's precisely none of them", or "There's some cats in my room, and their exact amount is 0 cats")

means exactly the same as

"There are no cats in my room"

.

>>8009852
>it's about comparing overall structure and meaning
no.

the empty set is a very specific object and logic predicates have very specific rules. your interpretation on what it means relating it to language is useless

"All four-sided triangles are elements of set [math]A[/math]."

Four-sided triangles don't exist, therefore the set [math]A[/math] must be empty.

But there exists only one set that's empty (THE empty set), therefore [math]A = \emptyset[/math].

Therefore, "All four-sided triangles are elements of [math]\emptyset[/math]."

>>8009865
this is fine.
this doesn't show (x in empty) can be false.
stop making this argument.

>>8009858
>the empty set is a very specific object

Yes, it's a (the only one, specifically) set of cardinality [math]0[/math]. As such, the statements

"Any cats in this room don't exist"
"There's cats in this room, precisely [math]0[/math] of them"
"There are no cats in this room"
"All the cats in this room belong to [math]\emptyset[/math]"

mean the exact same thing.

>>8009890
when you introduce language it's feeble at best, keep it at logic
"There exist X, there exist no X" is a contradiction. And I think that's what you mean with you second sentence.
"All X in U are in empty" is a proposition that is true if and only if U is empty. It is not equivalent to the other one. I think this is what you meant with your fourth sentence.

>>8009794
>Besides, what's true in logic is that for any predicate P, you have
>∀(x∈{}).P(x)
>i.e. the vacuous truth thing - for all x in the empty set, P applies to x.
>E.g. for all x in the empty set, x^3=-4 and x+1=84 as well as x^2=0.

Yes, this was also brought up a few times in this thread. You can attribute any property (except the property of existence) to all elements of the empty set, because to disprove all elements of the empty set having that property, you would have to show at least one element of the empty set which doesn't have this property. But you cannot do that, because elements of the empty set don't exist. Therefore, you have proven by contradiction that all elements of the empty set have any property (again, except the property of existence, because then you would end up with the statement "all elements of the empty set exist", which implies "there exists at least one an element of the empty set", which is obviously false). The empty set could be described as "the set of all objects such that each of them has any property (except for existence)".

>>8009917
So were completely fine with

"I've got some money in my wallet, and the exact amount is $0."

but somehow you consider

"There's cats in this room, precisely 0 of them."

a contradiction? But they are exactly the same, just with different subjects ("money" vs "cats" and "wallet" vs "room").

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[math]x \in \emptyset[/math] is routinely used in Poland to denote that an equation in [math]x[/math] has no solution (or, more generally, to denote that an [math]x[/math] satisfying the conditions given doesn't exist).

In pic related (pages in a printed mathematical tables/formulas book) it is used when the given linear or quadratic equation in reals has no real roots. The non-existent roots are considered elements of the empty set.

Ok, so what is the answer to the question posed by OP thus far? Is this evaluation expected after all, or is this a likely error and Wolfram staff should be consulted to clarify on this?

>>8009941
language is ambiguous and contradictory. don't use language.

>>8009941
You're twisting words and hiding behind subtleties of the English language to make a mathematical point, and that is a very bad thing.

>>8010055
Raymond Smullyan has been doing exactly that in his multitude of logic puzzle books. He has been using natural language almost exclusively to phrase both problems and their solutions (most of the target audience probably wouldn't digest his popular books all too well if the subject was presented in formal language).

>>8010084
You need to know what metalanguage is and how to use a subset of language small enough that you avoid contradictions. He does that because he's a logician and a mathematician.

You, on the other hand, haven't stopped using arbitrary quantifiers and blurrying distinctions between them, intentionally running into the contradictions associated with language.

So don't use it.

>>8009624

Technically in the trivial ring, that can be true.

>>8010094
Alright, but returning to the thread's original problem: what is the true significance of the expression [math]x \in \emptyset[/math] and how should Wolfram's puzzling evaluation of it (i.e. "undetermined (can be true or false)") be approached/interpreted? Is there a point behind this cryptic and seemingly unlikely assessment, or is it likely the result of an error or ommission?

Also, is the use of that expression as presented in >>8009958 justified, or rather questionable?

>>8010098
Can you elaborate on this? It's not clear what you're precisely referring to (let alone, what exactly you mean by that).

>>8010119
It is clear for anyone who knows what the trivial ring is. It is the set {0} with trivial operations +, *. Here the element 1, the identity of *, is just 0 itself. So 0 + 1 = 0 + 0 = 0.

>>8010107
error definitely
the use of the expression is probably ok, something like:

Let's take P and an x such that P(x). But this implies x is in empty. (aka a contradiction, x doesn't exist)

>>8010130
>It is clear for anyone who knows what the trivial ring is. It is the set {0} with trivial operations +, *. Here the element 1, the identity of *, is just 0 itself. So 0 + 1 = 0 + 0 = 0.

Oh ok, so you were referring to the expression [math]x = x + 1[/math]. That's what was unclear.

>error definitely

Ok, so I guess it might make sense to bring this up in the Wolfram forums and see what they have to say about it.

>>8009752
>It says it right there in Wikipeedja
just stop fgt pls

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>>8009830
>I've got some money in my wallet, and the exact amount is $0
If you have some money in my wallet, then the amount cannot be $0, because Zero-Dollar bills do not exist, so both your premise and your conclusion are false, at one stroke. Good job.

>>8010279
>Some claims made in Wikipedia are doubtful, therefore none can be trusted

Why not adress the statement itself, instead of making reverse ad hominems (i.e. attacking the source in general)?

>>8010302
This kind argument was predicted in >>8009830 already:
>You will probably try to argue that the word "some" in the first sentence necessarily means "more than $0", but, surprise, it really doesn't (or, at least, doesn't have to, which makes an assumption that it does just as much invalid).

Also, there is no "premise and conclusion", because "I've got some money in my wallet, and the exact amount is $0" is not an implication.

>>8009271
Pick the real number x such that x^2 = -1. Then x belongs to the empty set.

>>8010823
No.

Shit son. By definition
[math]\forall x \neg(x \in\emptyset)[/math]

>>8010823
I know what you're trying to do, but it doesn't work this way.

>>8009271
It's just a base case that isn't explicitely tested for

>>8010837
>>8010906
Hm, according to the usage shown in >>8009958 it kinda does that.

After all, all propositions that are always false are trivially equivalent. So

[math]x = x + 1 \iff x \in \emptyset[/math]
[math]0 x = 1 \iff x \in \emptyset[/math]
[math]x^2 = -1 \iff x \in \emptyset[/math]

should all be true if we consider the reals as the domain for [math]x[/math].

Furthermore, to possibly satisfy the expression [math]x \in \emptyset[/math] (i.e. make it a true statement), we must pick an [math]x[/math] such that the fundamental property of the empty set (i.e. it being empty) is not violated.

Thus, such an [math]x[/math] would have to be

a) literally "nothing". Not sure how to express "literal nothing" mathematically though - obviously, neither the number 0, nor the empty set, are "literal nothing", because they are in themselves mathematical objects which by all means exist, hence aren't really "literally nothing" (that's why the expressions [math]0 \in \emptyset[/math] and [math]\emptyset \in \emptyset[/math] are both false), even though they are closely related to the intuitive concept of "nothing", and can represent such a notion to a certain degree;

b) an object which doesn't and cannot exist because its definition is (or leads to) a logical contradiction. Examples have been named in this thread already, such as

"a four sided triangle"
"a real or complex solution to [math]x = x + 1[/math]"
"a real or complex solution to [math]0 x = 1[/math]"
"a real solution to [math]x^2 = -1[/math]"
"a real or complex solution to [math]\sqrt{x} = -1[/math] (assuming that the radical symbol always represents the "principal root" exclusively)"

>>8009958 shows an example how [math]x \in \emptyset[/math] being is used in the sense shown in b) (i.e. to essentially state "no such [math]x[/math] exist" in the context of the given conditions). But even then, I don't think the expression would be considered to be true - it's merely equivalent to another false expression.

>>8010837
>>8010906

The whole concept of "vacuous truth" (mentioned by >>8009794) revolves around considering non-existent objects to be elements of the empty set (you can ascribe any property to all elements of the empty set and can't disprove it, as you can't provide a counter example, i.e. an element of the empty set that doesn't have that property, because there aren't any elements). With that in mind, >>8010823 is correct, as is the usage of [math]x \in \emptyset[/math] mentioned by >>8009958 (exact same thing really).

>>8011863
>Not sure how to express "literal nothing" mathematically though

Given the properties of the empty set, [math]x \in \emptyset[/math] actually seems like the perfect way to express exactly that.

Assume someone wants you to define a mathematical object which is "nothing". The number [math]0[/math] is not such an object - while it may represent "nothing" in the intuitive arithmetical sense, it's not "nothing" itself given that it's a number. The empty set [math]\emptyset[/math] is noth "nothing" either, because it is a set, i.e. a well-defined mathematical object. However, by definition it is a set which has only "nothing" (i.e. exactly what we're looking for) in it - thus, if you want to express an [math]x[/math] being "literally nothing" mathematically, [math]x \in \emptyset[/math] seems like the ideal way to do it. The expression is short and glaringly obvious - as the cardinality of [math]\emptyset[/math] is zero, it only has "nothing" inside it, so [math]x[/math] must be "nothing" (or "something that can't exist", which is the same thing really).

>>8012242
So [math]x \in \emptyset[/math] becomes a true statement when [math]x[/math] stands for "nothing" ("nothing is an element of the empty set" or "nothing is in the empty set" seem to be true enough)? Is this true?

If so, is it the solution to the question why Wolfram claims that [math]x \in \emptyset[/math] "can be true or false"?

Two questions arise:

1) can "nothing" be considered a mathematical object (and be defined by "[math]x[/math] such that [math]x \in \emptyset[/math]", as suggested by >>8012242)? Would acknowledging "nothing" as an object not cause it to become "something", causing a contradiction?

2) can "existence" be considered a property like any other (as suggested by >>8009924)? If yes, then could [math]\emptyset[/math] be defined as "the set of all objects which lack the property of existence", i.e. [math]\emptyset = \{x: \neg E(x)\}[/math], or [math]\forall_x (\neg E(x) \iff x \in \emptyset)[/math] (with [math]E(x)[/math] representing the "property of existence")?

So, nobody has any more ideas?

>>8010098
Technically, if x is an cardinal number and 1 is [math]1_{\Bbb R}[/math], it's correct too.

>>8016040
Large sites just indulge in censorship like that. Some new image-reading bot also curiously refuses to identify what's in certain pictures. >>8015211

>>8009404
no set contains itself as an element...

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>>8016519

Literally nobody in this board is capable of approaching and addressing these >>8014347 seemingly simple questions? Can "nothing" be regarded as an entity, and can "existence" be considered a property, in a sense strict enough to be of use in logic and set theory?

>>8016531
source please, kind sir

>>8017868
Source for what? These were questions, not assertions.

>>8017873
Uh ok, you meant the comic (rather than the reply above yours). It's a page from "Logicomix", a graphic novel loosely based on the life and work of Bertrand Russell.

Ok I wanna talk about antielements. Sets containing elements and their anti elements contain no elements. It's my replacement for the axiom of choice.

>>8017884
Define "anti-element".

>Sets containing elements and their anti elements contain no elements.
Then each of such sets would be equal to [math]\emptyset[/math], which is the only set containing no elements (i.e. having cardinality [math]0[/math].

>>8017895
exactly :D

Now we can have negative cardinality, talk about anti power sets, and then we can move on to complex valued or fractional cardinality. Sets that contain only half of an element!

>>8017961
Well... before fractal theory, hardly anyone thought that one could denote dimension by anything than a non-negative integer. After having sticked to understanding exponentiation by means of the intuitive definition, most students are bewildered to hear about negative or fractional (let alone irrational or imaginary) exponents. Also, I heard claims that non-natural number system bases are possible. So, could generalizing cardinality in a similar manner not at least be considered?

>>8017961
>>8017989
no. it couldn't. you don't know what cardinality is if you think it could

The empty set contains all triangles with four sides, square circles, etc

>>8018008
Yes, I guess we can assume that.

The question is, if the predicate [math]x \in \emptyset[/math] becomes a true statement if we substitute [math]x[/math] with either "nothing", or with what doesn't and can't exist (such as "a triangle with four sides", "a square circle", etc.).

According to what you said, and what appears to make sense at least intuitively, the answer seems to be "yes". But that really just leads us back to the question whether "nothing" can be considered an entity which placeholders such as [math]x[/math] can represent, and if "existence" can be treated like a property like any other (and one which "nothing" or "square circles" don't posess).

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>>8009415
Because it werks?

>>8009271
(for every x in emptyset)

>>8018008
Translate this thought into mathematical language and you'll run into problems, as >>8018023 notices.

The proposition [math]x \in \emptyset[/math] is equivalent to false. Things like [math]x\in\mathbb R \text{ and } x^2 + 1 = 0 \implies x \in \emptyset[/math] (replace the LHS by whatever you like: x is a triangle with four sides, etc.) are true precisely because the RHS of the implication is always false.

>>8018753
>replace the LHS by whatever you like
You can't exactly replace it by _anything_ for such an implication to stay true. As a matter of fact, as the conclusion is always false, the premise must be always false too. For instance, the implication [math]x = 1 \implies x \in \emptyset[/math] is not a tautology, because there exists a value of [math]x[/math] (namely [math]1[/math]) which makes the premise true, and obviously an implication with a true premise and a false conclusion is false.

If for as long as the premise is a contradiction, the implication holds - and because the right hand side is a contradiction too, it's actually and equivalence. Thus

[math]x[/math] is a triangle with four sides[math]\enspace \iff \enspace x \in \emptyset[/math]
[math]x = x+1 \enspace \iff \enspace x \in \emptyset[/math]
[math]0x = 1 \enspace \iff \enspace x \in \emptyset[/math]

are all true. So, if we try to define some [math]x[/math] by specifying contradicting conditions, then equivalently such an [math]x[/math] is a member of the empty set.

On the other hand, anything follows from a false premise, thus [math]x \in \emptyset[/math] as a premise implies anything, including true statements, or predicates which may be true. As such, [math]x \in \emptyset \enspace \implies x = 1[/math] is true (although, as noted above, the reverse implication is not true, because the number [math]1[/math] is not a member of the empty set.

>>8018846
>You can't exactly replace it by _anything_
Yes, I know, I was sloppy. I meant replace it by any of the objects that guy was claiming to be a "member of the empty set."

>>8009271
>What gives?
Nil.

>>8018023
>a square circle
God here. Can confirm square circles aren't in the empty set.

>>8018945
They must be though. See >>8018846
> if we try to define some x by specifying contradicting conditions, then equivalently such an x is a member of the empty set.

>>8018969
You're still missing the point.

>>8018969
>contradicting conditions
God here. Can confirm a square circle is in no way contradictory.

The statement "all elements of the empty set are square circles" is true. If you disagree, disprove it by pointing out a specific element of the empty set which _isn't_ a square circle.

>>8019047
Oh, it's not? Then can you show an example of one?