As I understand, we agree. Godel gave an outline that leaves out needed details. But you said he was "slippery". I don't see what is slippery about it.

So it is sentence - proof sequence - axioms? — ssu

A proof is a sequence of sentences such that every sentence is either an axiom or follows by a rule of inference from previous sentences in the sequence.

What I meant that it itself is an indirect proof: first is assumed that all reals, lets say on the range, (0 to 1) can be listed — ssu

No, as I said, Cantor did not make that reductio assumption. Again:

Let g be an arbitrary list of denumerable binary sequences. (We do NOT need to ASSUME that this is a list of ALL the denumerable binary sequences). Then we show that g is not a list of all the denumerable binary sequences.

Turing constructed a quite important and remarkable proof for the uncomputability of the Entscheidungsproblem. But is that constructiveness a problem? — ssu

Church is the one who addressed the Entscheidungsproblem. Turing proved the unsolvability of the halting problem. My point was that Turing's proof is constructive.

this distinction wouldn't interest you since it relates to the ideas I'm proposing. — keystone

More childishness from you. From the fact that I'm not interested in going over all your stuff all over again, it is not entailed that I rule out being interested in any ideas you might mention. And again your childish penchant for turning blame around. You admit that you misused 'classical' but still manage to blame me anyway.

I don't believe any Cantor crank shares my perspective, if someone wants to label me a Cantor crank — keystone

For the record, I did not say you're a 'Cantor crank'.

I don't think we'll agree on terms. For example, in another message to you which you ignored I explained that I want to think of the infinite series 9/10 + 9/100 + 9/1000 + ... as a Turing computable algorithm, which can output arbitrarily precise partial sums but never output a 1. I get what you're saying, but in this sense, your function will never output intervals which will union to (0 1). — keystone

I didn't respond to your notion of an algorithm, since it doesn't vitiate that (0 1) is an infinite disjoint union of intervals. Again, an example of your modus operandi. You make a false claim, but then complain that it's not refuted because of some other red herring about some other notion you have.

What you just said is an utter disconnect. That no finite partial sum is 1 in no way contradicts that (0 1) is an infinite disjoint union of intervals. You argue on the basis of your kinda sorta associations about two different things rather than be responsible to make actual logical connections.

I feel like you dislike me or what I represent. I can’t debate classical mathematics to your level of formality, and you don't seem interested in my ideas — keystone

It doesn't matter whether I like you. For that matter, I can't have any fair opinion of you as a person aside from this extremely narrow context of posting. I have no reason to doubt that you are a decent and likable person away from posting. On the other hand, yes, I very much dislike your posting modus operandi.

It's not only a matter of discussing to a degree of formality. Rather, it's that you say a lot of things that are incorrect. But, yes, you handwave through just about everything.

As to my interest, I took a whole lot of time a while back to go through all the details of your proposal at that time. My participation was indeed generous. But even as I adapted to your many revisions, it ended up in a dead end where your proposal was still hopelessly vague and reliant on sophistical ambiguities. And even if I don't have the time and interest to engage yet again the broad handwaving scope and details of your musings, it is still eminently reasonable to point out particular clear falsehoods and misunderstandings you post. For example, rather than take a minute to understand my refutation of your false claim about (0 1) and to understand my proof, you fuss that I don't seem to like you.

if you could just define "identity theory" for me, and tell me what "=" means in that theory — fishfry

As I said much earlier in this thread, it is the first order theory axiomatized by:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For all formulas P(x):

Axy((P(x) & x = y) -> P(y)) (indiscernibility of identicals)

The meaning of '=' is given by semantics, and the standard semantics is that '=' maps to the identity relation on the universe. So, for any terms 't' and 's', 't = s' is true if and only if 't' stands for the same member of the universe that 's' stands for.

Still undefined but additional axiom. Sorry I don't follow. — fishfry

'=' is a primitive symbol. The axiom of extensionality is an additional axiom, not an axiom of identity theory.

In ZF, I define R={x∉x} — fishfry

Maybe you mean {x | ~ x e x} (you left out 'x |').

In Z we prove there is no set R such that Ax(x e R <-> ~ x e x). Therefore, the abstraction notation {x | ~ x e x} is not justified. How we handle that depends on our approach to abstraction notation. Personally, as a matter of style, I prefer the Fregean method, but reference-less abstraction notation is a whole other subject.

If extensionality is not an axiom, what is it? — fishfry

It is an axiom in ordinary set theory. I was describing a different approach, much less common, in which we don't have the axiom of extensionality.

Axioms and definitions are the same thing. You can take them as "assumed true," or you can take them as definitional classifiers, separating the universe into things that satisfy the definition and things that don't. — fishfry

There are two different senses:

(1) Syntactical definitions. These define symbols added to a language. In a theory, they are regarded as definitional axioms. But they are not like non-definitional axioms, in the sense that definitional axioms only provide for use of new symbols and don't add to the theory otherwise (criteria of eliminability and non-creativity).

This is the sense I'm using in my remarks about approaches to '=' in set theory.

(2) A set of axioms induces the class of models of the axioms. For example, we say first order group theory "defines" 'group'.

That is the sense that goes with the notion you mention

(2) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> z e y)

Axiom: x = y -> Az(x e z -> y e z)
— TonesInDeepFreeze

If two sets satisfy extensionality (the definition) then any set one of them is an element of, the other is also an element of? — fishfry

Yes.

I think that already follows from the definition. In fact I convinced myself I could prove it, but did not work out the details. So I could be wrong about this. — fishfry

Do the details. Remember that you don't have the identity axioms, so you can't use anything prior about '='. For example, you can't use substitutivity.

But what is the intent? — fishfry

To fulfill the other approach where we don't start with identity theory.

(3) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> y e z)

Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)

With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols.
— TonesInDeepFreeze

I didn't get all this, what's the intent of the axiom, what does it all mean? — fishfry

The intent is to arrive at the theorems of set theory but without adopting identity theory.

I did note one thing I disagreed with. You wrote:

"If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model')"

Perhaps we're using different terminology. When they do independence proofs, models are sets. So for example to prove that ZF is consistent, we are required to produce a set that satisfies the axioms. It's no good to just provide a proper class — fishfry

What you quoted by me agrees with that. A universe for a model is a set. But that doesn't entail that our meta-theory cannot be a class theory, as long as universes for models are sets.

I know universes that are not sets. For example:

The von Neumann universe and Gödel's constructible universe, both of which are proper classes (however you regard them) and are commonly called universes. — fishfry

That's a different sense from 'universe for a model'.

your definition of a universe. — fishfry

A model M for a language is a pair <U F> such that U is a non-empty set and F is an interpretation function from the set of non-logical symbols. U is referred to as 'the universe for M'.

Even informally, in ZF the universe is "all the sets there are." The axioms quantify over all the sets. And the universe of sets is not a set. — fishfry

I'm not talking about informal usage such as that.

If set theory has a model (which we believe it does), then the universe for that model is a set. That universe doesn't have to be "all the sets" (which is an informal notion anyway). It's a purely technical point: A model for a language has a universe that is a set.

why you think that the logical identity (whatever that is, I'm still a little unclear) is the same thing as set equality under extensionality. — fishfry

I think no such thing.

/

Rather than sorting out your questions in this disparate manner, it would be better - a lot easier - to share a common reference such as one of the widely used textbooks in mathematical logic. I think Enderton's 'A Mathematical Introduction To Logic' is as good as can be found. And for set theory, his 'Elements Of Set Theory'.

I understood that Tones was arguing that set equality is the law of identity — fishfry

I did not say that.

I said that classical mathematics has the law of identity as an axiom and that classical mathematics abides by the law of identity.

t seems to me that set equality is a defined symbol in a particular axiomatic system. — fishfry

I addressed that. Written up in another way:

Ordinarily, set theory is formulated with first order logic with identity (aka 'identity theory') in which '=' is primitive not defined, and the only other primitive is 'e' ("is a member of").

But we can take a different approach in which we don't assume identity theory but instead define '='. I don't see that approach taken often.

But both approaches are equivalent in the sense that they result in the exact same set of theorems written with '=' and'e'.

I believe you are trying to convince me that logical identity is the same thing as set equality as given by extensionality. — fishfry

No, I am not saying any such thing.

(1) I don't think I used the locution 'logical identity'.

But maybe 'logical identity' means the law of identity and Leibniz's two principles.

Classical mathematics adheres to the law of identity and Leibniz's two principles.

The identity relation on a universe U is {<x x> | x e U}. Put informally, it's {<x y> | x is y}, which is {<x y> | x is identical with y}.

Identity theory (first order) is axiomatized:

Axiom:

Ax x = x (law of identity)

Axiom schema (I'm leaving out some technical details):

For any formula P(x):

Axy((P(x) & x = y) -> P(y)) (Leibniz's indiscernibility of identicals)


But identity theory, merely syntactically, can't require that '=' be interpreted as standing for the identity relation on the universe as opposed to standing for some other equivalence relation on the universe.

And Leibniz's identity of indiscernibles cannot be captured in first order unless there are only finitely many predicate symbols.

So we make the standard semantics for idenity theory require that '=' does stand for the identity relation. And then (I think this is correct:) the identity of indiscernibles holds as follows: Suppose members of the universe x and y agree on all predicates. Then they agree on the predicate '=', but then they are identical.

(2) The axiom of extensionality is a non-logical axiom, as it is true in some models for the language and false in other models for the language.

As mentioned, suppose we have identity theory. Then we add the axiom of extensionality. Then we still have all the theorems of identity theory and the standard semantics that interprets '=' as standing for the identity relation, but with axiom of extensionality, we have more theorems. The axiom of extensionality does not contradict identity theory and identity theory is still adhered to. All the axiom of extensionality does is add that a sufficient condition for x being identical with y is that x and y have the same members. That is not a logical statement, since it is not true for all interepretations of the language. Most saliently, the axiom of extensionality is false when there are at least two urelements in the domain.

In sum: Set theory adopts identity theory and the standard semantics for identity theory, and also the axiom of extensionality. With that, we get the theorem:

Axy(x = y <-> Az((z e x <-> z e y) & (x e z <-> y e z)))

And semantically we get that '=' stands for the identity relation.

And if we didn't base on identity theory, then we would have the above not as a non-definitional theorem but as a definition (definitional axiom) for '='; and we would still stipulate that we have use the standard semantics for '='.

Think of it this way: No matter what theory we have, if it is is built on identity theory, then the law of identity holds for that theory, and that applies to set theory in particular. But set theory, with its axiom of extensionality, has an additional requirement so that set theory is true only in models where having the same members is a sufficient condition for identity.

Certain areas of mathematics, like combinatorics, are sufficiently distant from foundational issues and actual infinities. These areas transcend the label of 'classical' mathematics. — keystone

They are included in classical mathematics. They may be developed in set theory, without using the axiom of infinity, but they are still included in set theory. Moreover, with mathematical logic we have the formalizations of primitive recursive arithmetic that can be interpreted in set theory, and PA that can be interpreted in set theory, and set theory except with the negation of the axiom of infinity, which is inter-interpretable with PA.

That's why I ask if you're a Cantor crank. I just want to know who I'm dealing with. — fishfry

No Cantor crank would ever have the self-awareness to know that he or she is a crank.

Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. — keystone

What makes them cranks is not that they don't accept that there are infinite sets nor that they find the notion of infinite sets nonsense or fatally problematic. What makes them cranks is that their arguments about classical mathematics are ignorant, ill-informed, misrepresentational and astoundingly irrational. Sure, it's great that classical mathematics can be critiqued, but the ignorance, misinformation, misrepresentations and irrationality of the cranks is noxious

I think even constructivist and intuitionist set theories have a version of the axiom of infinity.
— TonesInDeepFreeze

But isn't it more like a potential infinity? — keystone

In classical set theory, we define 'is infinite'. I don't know whether there is a definition of 'is potentially infinite' in any theory. But, it seems reasonable to say that the notion of potential infinity is supposed to be upheld in constructivist and intuitionist set theories, which is the very reason I've brought them to your attention.

Note that Brouwer himself was not interested in formalization. But some constructivists and intuitionists have been.

You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it.
— TonesInDeepFreeze

To be honest it's because I didn't feel like continuing the dialogue with you because I find some of your comments offensive. — keystone

A litte earlier:

As always, I appreciate your comments and this dialogue. — keystone

But of course, people have the right to change their mind, even for the least of reasons.

Nicely done that at the end of a post in which you did continue to comment on my remarks, you say that you don't want to continue. And however you've taken other comments by me, the one I made about the link to a video is not very much offensive, if at all. On the other hand, it is at least a bit rude to link to a video without saying what point you want to make with the video or what part of the video you have in mind, so that I wouldn't have to have to watch all of a video without even context of your point about.

Anyway, I gave a proof that you are incorrect when you claim that the interval (0 1) is not an infinite union of disjoint intervals, whether or not you want to take a minute to understand the proof.

untested experimental vaccine shots — Tarskian

What untested vaccines? (Of course, they're untested for the people who are taking them in tests.)

Hilbert believed it so strongly that he insisted that all his colleagues should work on proving the above. — Tarskian

He proposed the project. But he insisted that all of them undertake it? Moreover, is there even one colleague to whom Hilbert insisted the colleague undertake it?

considering the first incompleteness theorem:

PA ⊢ ∃ A ( A ⇔ ¬Bew(ÍAÎ ) ) — Tarskian

Do you mean the diagonalization lemma applied to the negation of the provability predicate?

The diagonalization lemma doesn't have an existential quantifier like that. The diagonal lemma is:

If F(x) is a formula, then there exists a sentence S such that ('#' for 'the numeral for the Godel number of'):

PA |- S <-> F(#(S))

Applied to the negation of the provability predicate ('P' for the provability predicate):

There exists a sentence G such that:

PA |- G <-> ~P(#(G))

So the existence statement is in the meta-theory, not in PA.

For each element B in P(S0), form the sentence that is the coordinate conjunction of all the sentences in B. — The Vastness of Natural Language

Some of the elements of B are infinite. Those members of B that are infinite don't have a finite conjunction, but all natural language expressions are finite.

"In particular, the consistency of P is unprovable in P, assuming that P is consistent. (in the contrary case, of course, every statement is provable)." - Godel (His proof of this being very slippery.) — tim wood

The second completeness theorem is:

If S is a formal, consistent theory that adequate for certain arithmetic, then S does not prove the consistency of S.

Godel doesn't give a full proof for the second incompleteness theorem in his famous paper. But the details are supplied in subsequent articles and textbooks by other authors.

What the Turing Machine cannot compute is found exactly by using diagonalization (or negative self-reference) that we are talking in the first place. — ssu

I think the theorem you have in mind is that there is no algorithm that decides whether a program and input halt. The proof uses diagonalization. But, again, the proof is constructive. Given an algorithm, we construct a program and input such that the algorithm does not decide whether the program halts with that input.

With diagonalization, we get only an indirect proof. — ssu

No, diagonalization does not require indirect proof.

Is diagonalization a way to find mathematical statements that cannot be proven by a direct proof, but only can shown to be true by reductio ad absurdum? — ssu

No, diagonalization does not require indirect proof.

Proof of incompleteness is usually constructive. Given a system of a certain kind, we construct a sentence in the language for the system such that the sentence is true in the standard interpretation for the language (or, more informally, true of arithmetic) but not provable in the system.

Also, there are two kinds of proof by contradiction:

Assume P, derive a contradiction, infer not-P. That method is not generally controversial.

Assume not-P, derive a contradiction, infer P. That method is not accepted by intuitionists.

/

It seems people get false Internet memes in their head. I don't know where these memes originate from, but they ubiquitous and persistent in forums.

Provability, if I have understood it correctly, means that a truth of a statement/conjecture can be derived from some axiomatic system or logical rules. — ssu

Not in mathematical logic.

A sentence is provable from a set of axioms and set of inference rules if and only if there is a proof sequence (or tree, tableaux, etc.) resulting with the statement.

A sentence is true in an interpretation if and only if the sentence evaluates as true, per the inductive clauses, in the interpretation.

And we have the soundness theorem:

If the axioms used in a proof are all true in a given interpretation, then the proven sentence is true in that interpretation.

So, proving a sentence is not in and of itself proving the truth of a sentence. Rather, we have that if all the axioms used are true in a given interpretation, then the proven sentence is true in that interpretation.

Do [we] have uncountably many interpretations? — fishfry

Given a particular countable language and meta-theory with a countable alphabet:

This is correct:

Given a countable set of symbols, there are exactly denumerably many finite sequences of symbols, thus exactly denumerably many sentences.

There are uncountably many subsets of the set of sentences. And any set of sentences can be a set of axioms. Therefore, there are uncountably many theories. But there are only countably many ways to state a theory, so there are theories that are not stable.

I'm pretty sure this is correct:

There are exactly denumerably many algorithms. And for every formal theory and set of axioms for that theory, there is an algorithm for whether a sentence is an axiom. So there are only countably many formal theories.

Given a language for a theory, trivially, there are uncountably many interpretations for the language, since any non-empty set can be the universe for an interpretation, and there are not just countably many sets. But there are only countably many ways to state an interpretation, so there are interpretations that are not statable.

Given any theory, there are uncountably many models of the theory, since there are uncountably many isomorphic models of the theory. But there are only countably many ways to state a model, so there are models that are not statable.

You can't prove arithmetic from arithmetic because we created it. — Philosophim

Given a theory adequate for a certain amount of arithmetic, for example, PA, it's redundant to say that the theory proves all its theorems. But if the theory is formal and consistent, then there are truths of arithmetic that are not provable in the theory. This has nothing to do with who "created" the theory.

It is a false meme that the Cantor proofs mentioned here are by contradiction or indirect. Moreover, the proofs are constructive.

(1) The theorem known as 'Cantor's theorem' has the key part ('P' for 'the power set of'):

For all x, there is no function from x onto Px.

Proof:

Let g be a function from x to Px.
Let D be {y | y e x & ~ y e g(y)}.
So D is not in the range of g.
So g is not onto Px.

That's a direct proof. And it's constructive: Given any function g from x to Px, we construct a member of Px that is not in the range of g.

(2) Cantor's other famous proof in this regard ('w' for 'the set of natural numbers'):

There is no function from w onto the set of denumerable binary sequences

Proof:

Let g be a function from w to the set of denumerable binary sequences
Let d be the function from w to the set of denumerable binary sequences such that:
for all n in w, d(n) = 0 if g(n)(n) = 1 and d(n) = 1 if g(n)(n) = 0.
So d(n) is not in the range of g.
So g is not onto the set of denumerable binary sequences.

That's a direct proof. And it's constructive: Given any function g from w to the set of denumerable binary sequences, we construct a denumerable binary sequence that is not in the range of g.

/

Cantor did propose answers to the paradoxes (though his answers are not in the axiomatic method) but I don't know that Cantor's showing that there are always sets of larger infinite size was meant to evade the paradoxes. Indeed, it is the fact that there are always sets of larger infinite size that allows a paradox in Cantorian set theory. Cantor's answer to that paradox is another matter.

The sum is not the total addition of all the entries, but the limit of the total addition of all the entries. — Ludwig V

No, the infinite sum is the limit of the sequence of the finite sums.

The crank needs to follow the conversation but first he needs to learn some basic mathematics.

The matter under consideration is whether time can be divisible ad infinitum.

Density is a property of orderings. An ordering is dense if and only if between any two points there is another point. If time is divisible ad infinitum, then the ordering of points of time is dense.

I have not taken a position on the matter of whether time can be divisible ad infinitum. I only pointed out that, pertaining to the lamp puzzle, Michael's argument is not to refute continuousness but rather to refute density (to be clear, Michael is not the crank).

But, of course, the crank in his unfocused and ignorant way just sees some words on a screen, doesn't know what they mean and then lashes out wildly.

I possess concepts that would be found in an introductory calculus textbook — keystone

Such books don't axiomatize the principles used. And those books make use of infinite sets.

I think even constructivist and intuitionist set theories have a version of the axiom of infinity. But the logic of those systems is different from classical logic, so a statement in one system might not mean what it means in another system with a different logic.

I'd like to know whether a "no complete, only potential, infinity" concept has been axiomatized in a way that would be to the satisfaction of cranks if they were ever to actually learn about such things.

Also, keep in mind the amount of complication an alternative axiomatization might be. Already with intuitionistic logic, the semantics is much more complicated than with classical logic. Of course, that price might be worth paying.

That seems okay at face value. But since you've put the argument in a list, I'd make explicit all the premises.

I'd like to read Benacerraf's paper that disputes that there can't be a state at 12:00, and the papers mentioned in Thomson's paper for required context.

While you were posting, I revised my post to better explain about continuousness and density.

The lamp exists at 12:00 and as per the laws of excluded middle and noncontradiction is either on or off. — Michael

That is not at issue. Rather, as I've said twice now, at issue is whether its state at 12:00 depends on there being an immediate predecessor state. Thomson assumes that it does.

Given the way lamps work, or at least the lamp in this example, the lamp is on if and only if the lamp was off and the button was pushed to turn it on, and (after having been turned on at least once) the lamp is off if and only if the lamp was on and the button was pushed to turn it off. — Michael

Not just that it was off and then turned on, but rather that it was off at time t1 and on at time t2. That is, that it's not just a matter of the lamp having been off previously but rather that there is an off state that is an immediate predecessor of the on state and that that extends to 12:00 too so that for the lamp to be on at 12:00 there must be an immediate predecessor state in which the lamp was off, mutatis mutandis for the lamp being off at 12:00. Thomson mentions this. It's a premise that needs to be stated.

If you don't mean "Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" then it should be considered scratched.
— TonesInDeepFreeze

I do mean that. — Michael

(1) The first task is impossible to be performed. The second task is impossible to be performed. The third task is impossible to be performed ...

Quantified:

For all tasks, there is not a performance of any of them.

I think that is not what you mean.

(2) It is not possible for there to be a single performance of all the tasks.

Quantified:

There is not a performance that performs all the tasks.

I surmise that is what you mean.

I wouldn't write "it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" because it can be understood in sense (1).

It is not possible for the first dancer to do a flip today, for the second dancer to do a flip tomorrow, and so on.

I would take that to mean that none of the dancers can do a flip on their appointed day.

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]

P3. If (C) it is possible for time to be continuous then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]

C3. Therefore, it is necessary for time to be discrete [from C2] — Michael

The division of time mentioned in the thought experiment doesn't require continuousness of time; it only requires density time (via the density of the rationals).

Continuousness implies density, but density does not imply continuousness. So banning continuousness does not ban density. But you need to ban density of time.

You should have instead:

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens]

P3. If (C) it is possible for time to be dense then A
C2. Therefore, it is not the case that C [from C1 and P3 via modus tollens]
C3. Therefore, it is necessary for time to not be dense [from C2]

But:

(1) We may doubt P2.

(2) You have an unstated premise on which P2 is based, viz. that that state must be determined by an immediate predecessor.

(3) We may doubt P3.

As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it.
— TonesInDeepFreeze

This is the argument I am making:

P1. If (A) it is possible for a button to be pushed at 11:00, 11:30, 11:45, and so on, then (B) it is possible for a button to be pushed an infinite number of times between 11:00 and 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A [from P1 and P2 via modus tollens] — Michael

If you don't mean "Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" then it should be considered scratched.

if the button is pushed an infinite number of times between 11:00 and 12:00 then the lamp can neither be on nor off at 12:00. — Michael

As I mentioned, that is a premise that you don't include in your own argument. As I mentioned:

"his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state."

I can't imagine anyone denying that there is no immediate predecessor state, but some partisans who don't accept the argument deny that the state at 12:00 must be determined by an immediate predecessor state. So you must include the premise that the state at 12:00 must be determined by an immediate predecessor state.

In the quotation in that message, I made no statement. I just asked a question. — Ludwig V

My mistake. Just now I put an edit note in the post.

"for any finite number of tasks, there may be a completion of all the tasks" does not imply that there may be a completion of all of infinitely many tasks and does not imply that there may not be a completion of all of infinitely many tasks. — Ludwig V

Right.

we have defined each entry in the sequence — Ludwig V

I am using 'define' in the exact sense of making a mathematical definition. A definition is a single formula. It follows from the definition of the sequence that each entry in the sequence is "determined" (for lack of a better word), but that is not to say that each has been individually defined.

there is no finite upper bound to how many task may be completed in finite time." It occurred to me that that depended on how long each task takes. — Ludwig V

I was speaking in the context of the completion times halving.

Does it take less time or more to add 1,000,000 to a given number? — Ludwig V

Less time or more time than what? And what is meant by "time to add a number"? Does it mean number of steps in some given adding algorithm?

How long does it take to switch Thompson's lamp on or off? — Ludwig V

The problem presented states the increasing rate of alteration. Maybe you're asking about how long it takes to jab the button. I don't find that to be relevant, since whatever times it takes, we need only assume that it happens within the durations given in the problem. For that matter, I don't think the particulars about buttons, jabbing, or especially about human acts such as fingers reaching to touch a device are relevant, as the problem could be entirely abstract, as what is essential only is that the lamp goes on and off at the increasing rate mentioned, or, for that matter, it's not essential even that it's a lamp or any other particular device (could be clown klaxon going off an on for all it matters) as long as there are alternating states, whatever they may be.

the proof that sqrt(2) is irrational is also a proof that no rational number is sqrt(2)? — Ludwig V

For all practical purposes, yes.

Do I have to show separately and individually that each rational number is not sqrt(2)? I think not, but I have proved, of each rational number, that it is not sqrt(2). — Ludwig V

You proved a universal generalization:

For all x, if x is rational then it is not the case that x^2 = 2.

Then, with universal instantiation, for any given rational number, for example, 1.4 we prove that it is not the case that (1.4)^2 = 2.

What term do you use for a member of the sequence. — Ludwig V

Which sequence? There are different sequences involved in the puzzles here.

An infinite series that has a sum (some might say the series is the sum) requires first having an infinite sequence (each entry in the sequence is a finite sum) that converges, and the sum is the limit. The sequence whose entries are 0, 1, 0, 1 ... does not converge. However, whatever you mean by 'complete', there are infinite series that have a sum.

An infinite sequence is a function whose domain is an infinite ordinal. *

* In discussion about the task problems, we are modifying a bit by not having 0 in the domains of the sequences. That is, we index starting with 1 rather than with 0.

A function is a certain kind of set of ordered pairs.

The members of a function are ordered pairs.

The domain of the function is the set of first coordinates of the ordered pairs. We call members of the domain "arguments for the function".

The range of the function is the set of second coordinates of the ordered pairs. We call members of the range "values of the function". I also call them "entries".

We say that an argument maps to a value.

The infinite sequences in this context are:

(1) The function that maps every natural number n>0 to to 1/(2^n)

This is the function {<1 1/2> <2 1/4> <3 1/8> ...}

(1) The function that maps every natural number n>0 to either 0 or 1 ("off or on") depending on whether n is odd or even

That is the function {<1 0> <2 1> <3 0> ...}

what do we call the first "0" as distinct from the second "0"? — Ludwig V

0 is 0. There are not different 0s. But with the function just mentioned, and 0 occurs as values for different arguments of the function. 0 is the value at the arguments 1, 3, 5 etc.

This is where Thomson's lamp comes in. His argument is that if B is performed then a contradiction follows; the lamp can neither be on nor off at 12:00 but must be either on or off at 12:00. Therefore B is proven impossible. — Michael

But in that example, his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state.

And an earlier point:

C1. Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on. — Michael

As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it.

If the first task is performed at 11:00, the second at 11:30, the third at 11:45, and so on, then how many tasks are performed by 12:00? — Michael

It would be difficult not to say that denumerably (thus infinitely) many are performed. But I am reserving full committment to that, as there may be a finer analysis depending on complications in the notions of 'task' and 'performed' and perhaps in the inscrutabiliy of the overall hypothetical context.

P1. If (A) it is possible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on, then (B) it is possible for infinitely many tasks to be performed by 12:00
P2. It is not the case that B
C1. Therefore, it is not the case that A — Michael

That's okay (though, personally for me, modulo the residue of doubt that I mentioned about P1). But I wouldn't take P2 as a given without justification.

You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it.

When we define the series, we have defined each and every step in the series — Ludwig V

I like to keep the word 'series' for sums per convergences, and the word 'sequences' for sequences.

It is not the case that when we define an infinite sequence we must individually define each entry in the sequence. Example:

Definition of sequence S:

The domain of S is the set of natural numbers. For every natural number n, S(n) = n+1.

That's a finite definition (all definitions are finite) of an infinite sequence.

Again, even if there is no completion of all of infinitely many subtasks, it is not entailed that there is a finite upper bound to how many may be completed, so, a fortiori, it is not entailed that each of the subtasks is not completed.
— TonesInDeepFreeze
Are you suggesting that it might be the case that all of infinitely many tasks can be completed? — Ludwig V

I made no judgement on that. Again:

Suppose there can be no completion of all the tasks. That does not entail that there is a finite upper bound to how many task can be completed. That is, suppose for some finite n you say that no more than n tasks can be completed. But n+1 tasks can be completed without contradicting that there can be no completion of all of them. So there is no n that is a finite upper bound to how many tasks can be completed.And your statement (at least as you wrote it) was that none of them could be completed, which is even more wrong. [strikethrough in edit; the remark pertains to a different poster.]

To see that explained again, see Thomson's paper.

when Achilles catches the tortoise or finishes the race, he has completed all of infinitely many tasks. That might need some explaining, though, wouldn't it? — Ludwig V

Indeed. Hence 2500 years of philosophers, mathematicians and scientists talking about it.

After all if = is the identity relation on the universe, why does ZF need to redefine it then? — fishfry

There are three ways we could approach for set theory:

(1) Take '=' from identity theory, with the axioms of identity theory, and add the axiom of extensionality. In that case, '=' is still undefined but we happen to have an additional axiom about it. The axiom of extensionality is not a definition there. And, with the usual semantics, '=' stands for the identity relation. It seems to me that this is the most common approach.

(2) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> z e y)

Axiom: x = y -> Az(x e z -> y e z)

(3) Don't take '=' from identity theory.

Definition: x = y <-> Az(z e x <-> y e z)

Axiom: Az(z e x <-> z e y) -> Az(x e z -> y e z)

With (2) and (3), yes, '=' could stand for an equivalence relation on the universe that is not the identity relation. But it seems to me that even in this case, we'd stipulate a semantics that requires that '=' stands for the identity relation. And I think it's safe to say that usually mathematicians still regard '1+1 = 2' to mean that '1+1' stands for the same number that '2' stands for, and not merely that they stand for members in some equivalence relation, and especially not that it's just all uninterpreted symbols.

Suppose X and Y are objects in the universe, but they are not sets? — fishfry

Depends on what you mean by 'set' and what meta-theory is doing the models.

In set theory, contrary to a popular notion, we can define 'set':

x is a class <-> (x =0 or Ez z e x)

x is a set <-> (x is a class & Ez x e z)

x is a proper class <-> (x is a class & x is not a set)

x is an urelement <-> x is not a class

Then in ordinary set theory we have these theorems:

Ax x is a class

Ax x is set

Ax x is not a proper class

Ax x is not an urelement

If our meta-theory for doing models has only sets, then all members of universes are sets.

If our meta-theory for doing models has proper classes, still a universe is a set (proof is easy by the definition of 'model'). And no proper class is a member of a set.

If our meta-theory for doing models has urelements, and '=' stands for the identity relation, then the axiom of extensionality is false in any model that has two or more urelements in the universe or has the empty set and one or more urelements in the universe.

I see you've hijacked it to your hobby horse. — fishfry

That was said to Metaphysician Undercover.

Actually, I am the one who took up his misconception that sets have an inherent order. I don't consider that "hijacking", since his posts in this thread need to be taken in context of his basic confusions about mathematics, as mathematics has been discussed here.

nobody claims mathematical equality is identity — fishfry

In ordinary mathematics, '=' does stand for identity. It stands for the identity relation on the universe.

For terms 't' and 's', 't = s' is true if and only if what 't' stands for is identical with what 's' stands for.

When pressed, a mathematician would readily admit that mathematical equality is nothing more than a formal symbol defined within ZF set theory in the logical system of first order predicate logic — fishfry

An extreme formalist would say that. There is no evidence I know of that more than a very few mathematicians take such an extreme formalist view. Indeed, mathematicians and philosophers of mathematics often convey that they regard mathematics as not just formulas. Not even Hilbert, contrary to a false meme about him, said that mathematics is just a game of symbols.

And the semantics of first order logic with identity usually do require that '=' stands for the identity relation.

The poster wrote:

"you claim that the only relevant concept of "identity" is the mathematical one"

That is false. I've said very much the opposite in this forum. Of course identity is treated in philosophy aside from mathematics and as an everyday notion. And especially in philosophy and in certain alternative mathematics there are a great many differing views of the subject, all of which are we may benefit by study and comparison.

I wrote:

"Moreover, the context is the law of identity vis-a-vis mathematics."

That was in response to the poster's own claims about the law of identity vis-a-vis mathematics. That is, the context of this part of the discussion with the poster has been his attack that mathematics is incompatible with the law of identity. I have never at all claimed that mathematics has sole authority regard the subject of identity. Rather, I have shown (in posts in this forum) how the poster's attacks on mathematics vis-a-vis identity are ill-founded.

The poster disputes that mathematics upholds that law of identity. But the law of identity, both symbolically and as understood informally, is an axiom of classical mathematics. The poster has two prongs in reply:

One prong in the poster's reply is (as best I can summarize): Mathematicians may claim to state the law of identity, but those statements are incompatible with the actual law of identity, since mathematics regards numbers and such* as objects but those are not objects, and the axiom of identity pertains only to objects.

* If I say 'numbers and other things' or 'mathematical things', then that is tantamount to referring to objects since 'object' and 'thing' are synonyms. So, if we may not refer to mathematical objects then we should not even use the word 'thing' regarding, well, mathematical things.

The other prong in the poster's reply is: Mathematics regards sets as identical if and only they have the same members, but that ignores the orderings of the members of the sets, and the ordering of a set is crucial to the identity of a set, so sets are not identical merely on account of having the same members.

(1) OBJECTS.

If I'm not mistaken, objection to referring to numbers and such as objects is something like this: There are only physical, material or concrete objects, but mathematics regards numbers and such as abstractions.

But, without prejudice as to whether numbers and things are not physical, material or concrete, in mathematics, philosophy and in everyday life, people do refer to numbers and such as objects. It's built into the way we speak, as numbers and such are referred to by nouns and are the subject in sentences. If we weren't allowed to speak of numbers and such as objects then discourse about them would be unduly unwieldy.

But one might counter, "People talk like that, or think they need to talk like that, but that doesn't entail that it is correct that they do." To that we might say, "Fair enough. So when we say 'mathematical object' we may be regarded, at the very least, as using the word 'object' as "place holder" in sentences where it would be unduly unwieldy otherwise. The mathematical formulations themselves do not use the word 'object'. One could study formal mathematics for a lifetime without invoking the word 'object'. But to communicate informally about mathematics, it would be unwieldy to not be allowed to use the word 'object'. Moreover, when 'object' is used in that "place holder" way, one may stipulate that one does so without prejudice as to whether mathematical objects are to be regarded as more than abstractions, concepts, ideals, fictions, hypothetical "as if" things, platonic things, values of a variable, members of a domain of discourse, etc.

And it seems to me that the notion of 'object' itself may be regarded as primitive - basic itself to thought and communication. Any explication of the notion of 'object' would seem fated to eventually relying on the notion of 'object'.

In regards all of the above: Philosophy of mathematics enriches understanding and appreciation of mathematics, but one can study mathematics for a lifetime without committing to any particular philosophy about it. Moreover, one can study philosophy for a lifetime without committing to any particular philosophy. One may critically appreciate different philosophies without having to declare allegiance to a certain one. And one may use the word 'object' in a most general sense, even in a "place holder" role, without saying more about then that we may regard one's usage without prejudice as to how it should or should not be explicated beyond saying, "whatever sense of object that you may have about the "things" mathematics talks about".

The poster wrote:

"The law of identity in its historical form is ontological, not mathematical. Mathematics might have its own "law of identity", based in what you call "identity theory", but it's clearly inconsistent with the historical law of identity derived from Aristotle."

(2) LAW OF IDENTITY

To start, from the above quote, should the poster be charged here with argumentum ad antiquitatem? Even if not, there are more things to say.

Just to note, if I'm not mistaken, Aristotle's main comments about identity are in 'Metaphysics'. I don't have an opinion whether that's properly considered ontology.

The law of identity is usually stated as "A thing is identical with itself", or "A thing is itself" or similar.

Through history identity became an important subject in logic, philosophy and mathematics.

In logic, two central ideas emerged: The law of identity and Leibniz's identity of indiscernable and indiscernibility of identicals.

Eventually, mathematical logic provided a formal first order identity theory:

Axiom. The law of identity.

Axiom schema. The indiscernibility of identicals.

(The identity of indiscernbiles cannot be formulated in a first order language if there are infinitely many predicates, but it can be formulated in a first order language if there are only finitely many predicates.)

Along with the axioms, a semantics is given that requires that '=' stand for 'is identical with'. That is taken as 'is equal to', 'equals', 'is', 'is the same as' or any cognate of those.

So when we write:

x = y

we mean:

x is identical with y

x is equal to y

x equals y

x is y

x is the same as y

However, the poster, in all his crank glory, continues to not understand:

x = y

does NOT mean:

'x' is identical with 'y'

'x' is equal to 'y'

'x' equals 'y'

'x' is 'y'

'x' is the same as 'y'

but it DOES mean:

what 'x' stands for is identical with what 'y' stands for

what 'x' stands for is equal to what 'y' stands for

what 'x' stands for equals what 'y' stands for

what 'x' stands for is what 'y' stands for

what 'x' stands for is the same as what 'y' stands for


The law of identity is:

For all x, x = x

And by that we mean:

For all x, x is identical with x


And that does not depend on what kind of objects x ranges over. WHATEVER you regard a term 't of mathematics to refer to, the referent of 't' is identical with itself.

Classical mathematics does uphold the law of identity as it has been ordinarily understood in philosophy and as it came through Aristotle.

(3) EXTENSIONALITY

Still, the poster cannot say what THE ordering is of the set whose members are the bandmates in the Beatles. So, still, his claim (every set has and order that is THE order of the set) is not sustained, thus still unsustained is his second prong mentioned above.

If Hegel rejects the law of identity, but the poster endorses it, then it makes no sense for the poster to invoke Hegel as vindicating violations of the law of identity thus excusing the set theory that the poster abhors.

Moreover, the context is the law of identity vis-a-vis mathematics. As I said, it is not at all typical that philosophers of mathematics who are interested in set theory suppose the law of identity to be unacceptable. So it is not typical for mathematicians and philosophers of mathematics to vindicate set theory on the basis of denying the law of identity. Quite the contrary, it is typical for mathematicians and philosophers of mathematics to accept or endorse the law of identity, especially as the law of identity is one of the axioms used in set theory.