Does L list itself in L?
Does L list itself in LL? — Philosopher19

I answered those questions exactly.

Let x in x. Let y not in x. So x not equal {x y}. x in {x y}.

So x in x, and x in a set other than x. — TonesInDeepFreeze

In terms of function and logic, there is no difference between "lists itself" and "is a member of itself". — Philosopher19

No, I explained the difference.

I'll say it again, a list is a sequence. A sequence is a function whose domain is an ordinal. So the members of a list are ordered pairs. The members of the range of a list are the items listed by the list.

The property P is instantiated based on what set the item x is in. — Philosopher19

I don't know what you mean by that. I don't know what you mean by a property being instantiated in this context. I referred to any property. As I said: If x has property P, then that is not qualified by "x has property P in some sets but not others".

In a theory that does not disallow self-membership, a set may be a member of itself and it is a member of other sets too. An example has been shown many times already:

Let x in x. Let y not in x. So x not equal {x y}. x in {x y}.

So x in x, and x in a set other than x.

This has been gone over already, but now we come back around full circle.

I mentioned part of this before, but it was skipped:

In ordinary mathematics: A list is a sequence. A sequence is a function whose domain is an ordinal. A function is a certain kind of set of ordered pairs. So, usually we're not looking to see whether a list is a member of a list or not. Rather, usually, we would look to see whether a list is a member of the range of a list.

So, suppose there is a set S whose members are all and only the lists. (No such set exists in ordinary mathematics that does not have unrestricted comprehension, but suppose we have unrestricted comprehension.)

Let L be a sequence whose range is S. (L is a "list of all lists")

Let K be a sequence whose range is {x | x in S & x in range(x)}. (K is a "list of all the lists that list themselves")

Is L in L? No.

Is L in range(L)? Yes.

Is L in K? No.

Is L in range(K)? Yes.

Is K in L? No.

Is K in range(L)? Yes.

Is K in K? No.

Is K in range(K)? Yes.

Next question:

So what?

/

In mathematics, it makes no sense to ask, "In which set does x have the property P?" Rather, we ask, "Does x have property P?" If x has property P, then that is not qualified by "x has property P in some sets but not others". So one can't give sensical answers to nonsensical questions such as 1) and 2) in the previous post.

That is another way of saying what Michael said in the post to which the previous post is in response.

That seems like a good synopsis to me.

You could make an argument from some basic results of model theory that mathematical formalism in most cases can`t be specific about the objects it`s supposed to speak about. When a set of axioms "uniquely" (up to the isomorphism) specifies a model we say that the theory is categorical. Hilbert and earlier Peano achieved a categorical axiomatization of Euclid`s geometry, Tarski proved this version of "Euclidean" geometry is consistent, complete and decidable. The "unique" model of it is the Cartesian plane. Beside Godel incompleteness features (undecidability and either consistency or completeness) any set theory pretending to be an axiomatization of mathematics can't hope to be categorical. There are weaker notions of the classes of models but I don't think it's possible to define a class of models zfc does specify. Isn't isomorphism weak enough to say a theory doesn't specify a mathematical object? Well an ignorant mathematical nominalist could make such an argument. There's nuance to it, you could step back and not even pretend that what mathematicians study are classes of models. — Johnnie

Agree on these points:

(1) A theory is categorical if and only if all its models are isomorphic with one another.

(2) First order Euclidean geometry is categorical.

But some points I would put differently:

(3) The incompleteness theorem implies that there is no recursively axiomatized, consistent, sufficiently arithmetical theory that is categorical. But that is endemic not just to set theory but even first order PA, Robinson arithmetic or many other theories for even just basic arithmetic. Roughly speaking, pretty much when you have successor, addition and multiplication, you don't have a categorical theory. But even more basic to the incompleteness theorem, from Lowenheim-Skolem we know that a theory with an infinite model has models of all infinite cardinalities, thus not categorical.

(4) We presume that ZFC is consistent, so there is the proper class of all and only the models of ZFC.

(5) It's not clear what is meant by [my paraphrase] "a theory specifying or not specifying an object". There are two notions of definition:

(a) In a theory, given an existence and uniqueness theorem, we may define a constant symbol. With a model for the language of the theory, that constant maps to a member of the universe, and if the model is a model of the theory, then that member of the universe is the one that satisfies the definition.

(b) Given a model, a member of the universe is definable in the language if and only if there is a formula with exactly one free variable such the formula is satisfied only by an assignment of the variables that assigns that free variable to said member of the universe. (This can be extended to relations too.)

But the class of all and only the models of a given theory is a proper class, so it cannot be a member of a universe. (I think the following is right:) On the other hand, in class theory, we can define the proper class {x | x is a model of ZFC}. Or in ZFC we can define a 1-place relation symbol M by: Mx <-> x is a model of ZFC. But in ZFC we can't prove that that is not the empty relation.

/

In any case, yes, usual formal theories for mathematics are such that each one has non-isomorphic interpretations. That is a mathematical fact. But I don't know that that blocks a realist from reasonably regarding mathematics to be referring to certain objects.

In what way do you regard 'Principia Mathematica' to be a fiasco?

First order predicate logic may be formalized in two ways:

(1) With logical axioms and rules of inference. (Known as 'Hilbert style'.)
or
(2) With only rules of inference. (E.g., natural deduction.)

In either case, we have the soundness theorem, since the logical axioms are true in every model and the rules of inference are truth preserving.

/

This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:

The law of identity is that a thing is identical with itself. Using '=' to stand for 'identical with' is not inconsistent with the law of identity. Indeed, the law of identity is stated:

Ax x=x

That is, for all x, x is identical with x. That is, for all x, x is x.

Not only is '=' as used in mathematics consistent with the law of identity, but the law of identity is itself an axiom of identity theory that is taken as part of the logic used for mathematics.

/

This point has been posted already, but, alas, the posts come back around full circle when the correct explanations are skipped again and again:

"the ordering of the set" is meaningful only when there is only one ordering of the set (in this context, by 'ordering' we mean a strict linear ordering). Any set with at least two members has more than one ordering.

The set whose elements are all and only the members of the Beatles in 1965 has 4 members, and there are 24 orderings of that set.

In other words, there is no "the" ordering of the 4 membered set {x | x was a member of the Beatles in 1965} since that set has 24 orderings.

/

"crackpot" is said by the crank calling the pot cracked.

Here's how I would put it:

2(x+5)= 2x+10
is understood to be implicitly universally quantified:
Ax 2(x+5) = 2x+10
and that is true

Then, by universal instantiation, we have:
2(x+5) = 2x+10
and that is true for every assignment of a value to the variable 'x'

2(x+5) = 3
is understood to be implicity existentially quantified:
Ex 2(x+5) = 3
and that is true

In high school algebra, we are asked to state the members of the "solution set", which is to say:
{x | 2(x+5) = 3} = {-7/2}
and that is true

There was developing an interesting discussion on the law of identity and (non-ordered) sets. — jgill

More a painfully needed, though unsuccessful, intervention than a discussion.

The points are simple:

* In mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc.

* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!"

{'The Maltese Falcon', 'Light In August', 'The Stranger'} = {'Light In August', 'The Stranger', 'The Maltese Falcon'}

{8, 5, 9} = {5, 9, 8}

And '=' reads fine whether 'equals', 'is identical with' or 'is'.

No law of identity is violated there.

/

Boss: Jake, tell me what is the set of items on our shipping clerk's desk?

Jake: It's the set whose members are a pen, a ruler, and a stapler.

Maria: But he also has another set on his desk! It's the set whose members are a ruler, a stapler, and a pen.

Boss and Jake: Wha?

Boss: Maria, take the rest of the day off. You're not quite with it lately.

/

{pen, ruler, stapler} = {ruler, stapler, pen}

Nobody says that the set of items on a desk is different depending on the order you list them.

On the other hand, mathematics does have ordered pairs and triples. For example:

<b c d> = <x y z> if and only if b=x, c=y and d=z.

With ordered tuples, yes, order does matter.

Within a different framework, say that of binary numeral system, 1 + 1 = 10 — RussellA

That's not an example of what I was talking about. I'm talking about general frameworks such as hold one's intuitions, perspective or philosophy, not matters such as variations in base numbering systems.

But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects.
— TonesInDeepFreeze

Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists? — RussellA

No.

What does "it, the knight on a chess board, refer to? — RussellA

When I wrote 'it', I was not referring to a particular piece of wood or particular array of pixels on a screen. I'm referring to the idea, the thing that players who are not even in each other's presence - thus not moving the same piece of wood or even seeing the same array of pixels - can still refer to as "the knight".

"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world. — RussellA

But it doesn't. Again, one may argue that leading up to the formation of the concept, there were particular pieces of wood or ivory or whatever that were carved to resemble a knight and that were moved around on a two-colored board. But soon enough, we have the abstraction that can be referred to. Indeed, 'chess' itself can be defined mathematically without even mentioning particular characters such as 'knight', but rather only a purely mathematical construct. We could say there are four objects, called 'WKL', 'WKR', 'BKL', 'BKR' ("white" and "black" "knights" on "left" and "right") and the other "pieces", then define a C-sequence (sequence of "chess moves") to be a certain sequence of matrices with those objects associated with cells in a matrix and successive matrices having a property that the "pieces" are in different cells only according to certain allowed ordered pairs of cells ("moves"), etc. So you see that when I say 'it', I'm talking about an abstract object, even though attaining that abstraction required previous concrete or ostensive understanding.

Innatism — RussellA

I am not opining whether or not the basic concepts 'is', 'exists', 'same', etc. are innate. Indeed, I have no ready argument that they are not first understood only ostensively. I'm only reporting that I don't know how I could arrive at successively more involved frameworks without them.

For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts.
— TonesInDeepFreeze

But how can there be wisdom in the absence of facts. — RussellA

I don't say that there can be.

It was commented "I wonder what "nicknamed" would imply in supposed rigorous logic."

The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas.

Moreover, the nicknaming (my word) I mentioned is not so much about the logic but rather about defined symbols in a theory such as set theory.

Set theory, as formalized, uses only formal symbols, not natural language words. A formal proof is not allowed to use connotations, associations or any of the suggested notions that natural language words have. However, for everyday communication of proofs among mathematicians and students it is unwieldy to recite exactly each formal symbol in the formulas that are sequenced for a proof. Moreover, it aids picturing the content of the theory to informally use words. I call that 'nicknaming'. For example, in set theory in all formality, there is no constant term 'the empty set'; instead there is a 1-place operation symbol, a pure symbol, with a purely formal definition. Moreover, as I've mentioned, the adjective 'is a set' or even a formal predicate for 'is a set' are not even required as in formal theories such as ZFC. So, as concerns the formal theory, it doesn't matter whether or not one's personal notion of sets allows that there is one special set that has no members that is called 'the empty set' but rather, the theory has a formal theorem, such as:

E!xAy ~yex
thus a definition
0 = x <-> Ay ~yex

So, in that particular regard, we could just as well use the nickname "zee zempty set". It would not change the "structure" of the mathematics, which is the relationships of the definitions and theorems.

Note that my remarks about this are not necessarily a commitment to extreme formalism expressed as "mathematics is just a formal game of symbols". Rather, in this context, we may note that, no matter what framework or philosophy one has for understanding mathematics, at least we have the formalization, even just the fact of that formalization, as a component in our understanding - whether a fully self-contained and isolated component (i.e. extreme formalism) or as merely a point of reference and a rigorous constraint against handwaving.

Setting aside whether it's good to move a thread from the main table of contents, the moderator in his role as moderator would have been better not to so subjectively, curtly and sweepingly over-reduce a thread.

If I'm not mistaken, the importance of Russell's role is that he noted that the paradox applies to Frege's system.

I have this in my notes:

Russell discovered the paradox while studying Cantor's argument that there is no function from a set onto its power set. Zermelo had discovered the paradox earlier in 1900 or 1901 (cf. "Zermelo's discovery of the "Russell Paradox"" by Rang & Thomas in Hist Math 8 pp. 15-22).

Frege proposed the fix by using the following axiom schema (here couched set theoretically) instead of unrestricted comprehension:

For any formula, P in which x does not occur free:

ExAy(yex <-> (y not=x & P))

But the above axiom schema, with identity theory, is inconsistent with Exy x not=y, as became known to Russell and to Frege, and Lesniewski provided a proof in 1938 (cf. Fixing Frege by Burgess - pg 32-34; "On Frege's Way Out" by Quine (in Mind 1955 and in Selected Logic Papers - pg 151))

The proof is pretty cool, and it's interesting that it's much much more complicated than the easy Russell proof.

Of course it was Zermelo who provided a robust fix with:

AzExAy(yex <-> (y in z & P))

accompanied with the rest of the existence axioms.

Math is a subset of logic that deals exclusively with relationships of quantity. — Kaiser Basileus

Whether or not mathematics is a subset of logic, it is decidedly not the case that mathematics is only about quantity.

It wouldn't be just the plain classic symbolic logics only in use. — Corvus

Right.

various types of non classic logics seem to have been implemented, and used such as Temporal Logic, Description Logic, Fuzzy Logic, Epistemic Logic, Many-Valued Logic, Probability Logic, Topological Logic, Assertion Logic, Deontic Logic — Corvus

Ordinarily, 'classical logic' refers to any of the equivalent formulations of predicate logic, in first or higher orders, with the ordinary features such as excluded middle, non-contradiction and explosion.

Advancements have been added to classical and non-classical logic, such as modal logic, temporal logic, etc. And some advancements may be incompatible with classical logic, such as multi-valued logic, relevance logic, paraconsistent logic, etc.

It is of course granted that predicate logic is not adequate for all forms of inference. However, that does not vitiate that predicate logic (and even just sentential logic) is useful.

Note also that I replied to a claim about formal, symbolic logics. And such things as temporal logic, deontic logic, fuzzy logic, etc. are given as formal, symbolic logics, even if they may also be studies informally. So their uses does not vitiate the point that formal, symbolic logic is useful, since they are themselves formal, symbolic logics. And usually an intellectual prerequisite for study of those advanced logics is an understanding of plain vanilla predicate logic. Moreover, as far as I've seen, set theory may be a meta-theory in which those advanced logic are studied, which is to say that in set theory we may formulate those other logics and prove theorems about them.

Classical logic is useful, even just in its sentential component, which is the Boolean logic used in ordinary computing, and further as classical logic is the logic for the ordinary mathematics for the sciences and for the study of recursive functions and the theory or computability that are at the very heart of the invention and development of the digital computer. And, while predicate logic cannot account for all forms of inference, predicate logic is usually prerequisite for study of the more advanced logics.

All topics are connected by finitely many degrees of separation.

Without comment on the specifics, I think that's a pretty good perspective.

Hilbert said, "Mathematics is the art of giving the same name to different things." — Ø implies everything

Seems you're conflating Hilbert with Poincare.

Formal logic and Symbolic logic are not able to deal with the real world phenomenon and states very well. — Corvus

They are at the very heart of the development of digital computers, such as the one you're reading right now.

The validity of the aleph-1 or non-countable infinity is adequately demonstrated by Cantor's Diagonal Argument — alan1000

Just to be clear: Cantor's showed that the set of real numbers is uncountable. He didn't prove that its cardinality is aleph_1. The assertion that the cardinality of the set of real numbers is aleph_1 is the continuum hypothesis, which Cantor did not prove, and which was later proven to be independent of ZFC.

it is a challenging exploration of ideas — jgill

To add to that, the practical benefits of abstract research are not always seen at first, but such investigations generate ideas that can lead to practical benefits, and lead to methods and ways of thinking that can be used in wider contexts. Mathematics is a beautiful and creative expression of human intellect and curiosity. I am always amazed when people want to put the kibosh on it.

The law of identity is:

Ax x=x

That is one of the axioms of identity theory.

I posted that earlier today, but of course you SKIPPED it.

Since the axiom of extensionality is consistent with identity theory, perforce it is consistent with the law of identity.

So I am not at all avoiding the issue or switching the issue.

/

You keep using the word 'consistent' while you show no inconsistency; we only learn from your bungled arguments that the axiom of extensionality does not accord with your confused and dogmatic views about mathematics. As I've said an uncountably infinite number of times if I've said it a countably infinite number of times, no one disallows you from positing your own framework of understanding, but the mere fact that mathematics does not adopt your framework (which is, meanwhile, confused) does not make mathematics incorrect. And you keep evading many of the other arguments and considerations about set theory, such as the application of classical mathematics to the sciences and computability, including the existence of the computer you're typing on right now. It is funny how not only is it the case that cranks can never answer that point, but they will never even recognize it.

Moreover, when you resort to dogmatically insisting that a set is determined by a particular ordering, you resort to silliness that was debunked years ago in posts to you in this forum. We went over it and over it back then, and you still don't get it. I think you don't get it because you have a mental block about such things. So I suggest that not only should you make an appointment with an ophthalmologist to find out why you can't see things right in front of your eyes, but you should see a cognitive psychologist to find out about the mental blocks you have that disallow you form understanding even such basic ideas that can be understood by a young child.

An axiom is a formula. It has a meaning upon interpretation of the language. But also, it has our ordinary reading of it in a natural language. To understand that ordinary reading of it, requires understanding the context, which includes what the axiom proves.

But, of course, you will resort to any argument you can to evade actually learning anything about the subject on which you dogmatically declare.

Let's refresh just recent matters alone:

You claimed that axiom of extensionality is inconsistent with identity theory. I proved it is not. You evade that, because you know virtually nothing about identity theory, the axiom of extensionality or consistency.

Most basically, you haven't a clue what the axiomatic method is about.

I explained for you how set theory does provide for the identity of indiscernibles and the indiscernibility of identicals.

You claim that a set and an ordering on the set determine the set. You were saying that years ago in this forum and it was debunked then. You still don't get it. Still don't get what even a young child can understand.

/

I don't care what you call yourself, but if you do change your name, at least I'll know that the denotation of 'Main Crank' is the denotation of 'Metaphysical Underground', which is the same foolish, arrogant, ignorant, illogical, irrational, dogmatic, confused, intellectually blocked, dishonest, lying poster he always was.

Now where were we?

Oh, yes, the main crank's insistence that set theory handle order in the way he thinks it should be handled, even though he is ignorant of how set theory does handle order.

Prediction: Corvus will reply yet again that he no longer wants to discuss the personal aspects of the postings, while he yet again renews his claims about the personal aspects of the postings.

As said, I am not interested in keeping talking with you on who has done what. — Corvus

And as I said, nobody's stopping you from not talking about it.

negative postings — Corvus

While you are free to not post in an insulting manner.

I wouldn't have replied to you at all. — Corvus

You were replying to me in an insulting manner (criticizing my arguments while not even addressing their key points and taking me for a fool with ignorant and false red herrings that study of mathematics is just a bunch of regurgitating what is in book, or however you actually phrased it) well before I said even a word about you personally. And as I observed you replying insultingly to another poster in this thread in even the earliest part of this thread.

Again, for the hundredth time, I don't remark on the ignorance, confusion and dishonesty of posters merely because they disagree with me.

And, of course, you read the first line, but not the context that justifies that first line.

Meanwhile, look at your own posts.

What you can "prove from the axioms" is irrelevant, when it is the acceptability of the axioms which is being questioned. — Metaphysician Undercover

You're making claims about the axiom vis-a-vis identity. So it is very relevant what the axiom proves regarding identity.

And as I said: There is a difference between what mathematics says and what one thinks mathematics should say. So anyone is welcome to say how they think mathematics should be formulated, and better yet, to provide an actual formulation. And anyone is welcome to say why they think the ordinary axioms are not acceptable. But to do that, one should at least understand what those axioms are and how such mathematics is formulated. And not to continually strawman about them. Also it always helps to understand context, which here includes why mathematics adopts axioms, what the axioms of classical mathematics are intended to provide, and the perspective of the use of classical mathematics for the sciences and computing.

I don't see a point in ad hominem posts. — Corvus

You don't see a point in them, but that doesn't stop you from posting insults.

And, again, it is very important to distinguish between an ad hominem ARGUMENT and, on the other hand, stating an non-ad hominem argument but in addition remarking that a poster is confused, ignorant and dishonest, especially when detailed explanation is given the poster as to what his ignorance, confusion and dishonesty are.

I have no time or inclination for getting involved in non-philosophical quibbles with you. — Corvus

You're free not to!

This is truth. — Corvus

First, it's not. Second, it is also truth that the main crank here is ignorant and confused about the subject.

So it seems you think that ""distortion [...] bias, prejudice and false judgement" is not an insult, because you think it is true, but "ignorant and confused" is an insult.

As to start of posts, there are different starting points: The start of a single post, the entry posts in a thread, and the first posts between posters in this forum. In the start of this thread, I did not make personal remarks. Over time, as the main cranks misrepresents, strawmans, posts in ignorance and confusion on the subject in this thread, then I remark on that. And this is in context of a MASSIVE amount of that kind of insulting dishonesty in many other threads in this forum.

It's not a question of replying to all of what poster writes, but it deserves remarking when you criticize posts while skipping their key points and misrepresent what they say.

The main crank continues to argue by mere assertion about the ordering, repeating over and over and over his dogma, without even taking a peek at the information provided him that explains his confusion.

/

Set theory is an axiomatic system with one non-logical primitive. From the axioms we prove there is a unique object such that there is no x such that x bears the relation denoted by 'e' to said object. If the nickname 'the empty set' does' comport with one's notions about set, then it can be called 'the zempty zet' or 'the-thing-that-has-no-things-on-the-left-of-it'. The nicknames would not alter the formal theory.

Generally as to what things don't have members other than the empty set: urelements.

As usual, you evaded the point. Again:

"distortion [...] bias, prejudice and false judgement."

That is not an insult but "ignorant and confused" is?

As to what is not pleasant, it is not pleasant to have one's posts misrepresented, strawmanned and outright lied about, as you regularly do, and to confront supposed arguments against them that skip their key points, as you regularly skip the key points.

You can look back in this thread to see that I posted back and forth with you with my not saying anything remotely personal, until I pointed out that you were skipping the points.

And copious evidence and argument have been given showing that the main crank in this thread is ignorant and confused about this subject - including right up to this very moment.

distortion [...] bias, prejudice and false judgement. — Corvus

That is not an insult but "ignorant and confused" is?

Did someone say that the axiom of extensionality "states identity"?

if you don't have any meaningful philosophy to write down — Corvus

See above post.

Of course, you can't deal with the plain fact of the record of posts, which document not only that you've been lying (which itself is insulting) but which also includes relatively detailed remarks by me about mathematics and philosophy.

Again for the crank:

There are different orderings on sets.

There is no such thing as "THE" ordering for sets with at least two members. There are often what we call 'standard orderings' but still there is not just "THE" ordering of a set with at least two members.

Again for the crank:

The axiom of extensionality pertains no matter what orderings are on a set.

{0 1 } = {1 0}

{<0 1>} is an ordering on {0 1}
and
{<1 0>} is an ordering on {0 1}

{<0 1>} not= {<1 0>}

{<0 0> <1 1>} is a sequence whose range is {0 1}
and
{<0 1> <1 0>} is a sequence whose range is {0 1}

{<0 0> <1 1>} not= {<0 1> <1 0>}

The treatment of orders and sequences is rigorous in set theory. And the axiom of extensionality is not inconsistent with the theorems about them.