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.

It's not hot air that the poster's claim was (and is now back) that there is THE ordering of a set.

Evidence includes the poster's penultimate post: "I argued that the axiom of extensionality does not indicate identity in a way which is consistent with the law of identity, because the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements."

And revisionist: [strikethrough in edit, since 'revisionist' is not the right word there.]

"I do not argue that set theory is "wrong" on account of this violation, because some philosophers suppose the law of identity to be unacceptable."

It is not at all typical that philosophers of mathematics who are interested in set theory suppose the law of identity to be unacceptable.

What specific philosophers is the poster referring to?

And what the poster says makes no sense, again. He endorses the law of identity (no?). And even one post ago he falsely claimed that set theory is not consistent with the law of identity. But now, revisionistically, he says he doesn't fault set theory, and that the reason he doesn't fault set theory is that some philosophers deny the law of identity. But if he endorses the law of identity, then it would make no sense for him to let set theory off the hook on the basis that there are philosophers who disagree with his endorsement of the law of identity. The poster is brazenly illogical, again.

/

The law of identity is that a thing is identical with itself. It is an axiom of identity theory, which is presupposed by set theory. Not only is set theory consistent with the law of identity, but the law of identity is one of the pre-axioms of set theory.

/

And the poster is back to claiming that a set has an ordering that is THE ordering of the set. And still he does not address the natural rejoinder: If a set has an ordering that is THE ordering of the set, then which of the 24 orderings of the set whose members are the bandmates in the Beatles is THE ordering of that set?

/

As to the track here, the poster's position regarding mathematics in this context is fairly paraphrased as:

1/2 + 1/2 is not 1, because you can't cut pie without some crumbs falling around so that the resulting two pieces of pie are not precisely the same size.

No exaggeration. The poster's notion is that that 1/2 + 1/2 is not 1, since the '1/2 + 1/2' is not '1' (though the particular example was different). (And even the most utterly ridiculous irrelevancy that with '1 = 1', the first occurrence of '1' is not the second occurrence of '1'.)

And now, moreover that the identity assertion fails, because, for example, pie cutting is not exact.

Stern-Brocot is interesting. But a while back, your improvisations on it came to confused handwaving. I'm not inclined to start all over again about it with you.

Let me know when (or if) you have a system with formation rules, axioms and inference rules.

Otherwise, discourse with you is such that what obtains is just what you say obtains, without your interlocuters having access to checking your arguments by the objective reference of mathematical proof.

In practice, applied mathematicians do not actually use infinite sets. — keystone

Depends on what you mean by "applied". Ordinary mathematics, even as used for basic applied interests such as speed and acceleration use ordinary calculus, which is premised in infinite sets.

/

In context of ordinary mathematics, a line in 2-space is a certain kind of set of ordered pairs and a line in 3-space is a certain kind of set of ordered triples. If one wishes to propose an alternative, then they can set it up coherently with axioms and definitions. On the other hand, if you wish to limit yourself to ostensive, imagistic musings then I wouldn't begrudge you from entertaining yourself that way; only that count me out, as I have better avenues of intellectual engagement.

/

You ask where does length "come from" if not points. The mathematical definition of 'distance between points' is given by a well known formula in high school Algebra 1. Look it up.

Who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"? Specific quotes are called for. Otherwise the claim is a flagrant strawman.
— TonesInDeepFreeze

[...] I think it's better for me to just take that statement back. — keystone

Even better would be to figure out why you are prone to such things to begin with.

generate the infinite sequence (which is described with finite characters and, as per Turing, halts if executed.) — keystone

No, the execution does not halt. You have it completely wrong.

I see this informal forum conversation as the journey which may be slowly taking me toward that destination or may be leading me toward the junkyard to dump my ideas. Either way, this journey is useful to me. — keystone

That's nice. But, by magnitudes, less useful to me than actually reading mathematics and philosophy of mathematics. For that matter, even less useful to me than, say, watching my screen saver. But, meanwhile, I don't mind correcting falsehoods you post about mathematics itself and to comment on the irremediable handwaving in the description of your own musings.

It was claimed that Russell's paradox is "still there". In what specific post-Fregean systems is it claimed that the contradiction of Russell's paradox occurs?
— TonesInDeepFreeze

I think it's best for me to just take this claim back. — keystone

See earlier in this post.

Back to the poster who claims to offer an alternative to classical mathematics: The word 'isolate' keeps coming up. What is a rigorous mathematical definition of 'isolate'?
— TonesInDeepFreeze

In earlier posts I described 'potential' points which existed only in bundles that can be isolated by means of a cut. I have since scrapped that idea. Instead of proposing that cuts isolate points, I am now proposing that cuts create points. — keystone

You seem really not to recognize that it is a potentially infinite game with you, where you pretend to define or explain terminology or concepts in terms of yet more undefined terminology and concepts.

And who can keep up with you? So 'isolate' is now out? And 'create' is now in?

!⥱ — keystone

I guess I'm supposed to glean that that means something like "as goes to infinity" or something?

Anyway, what you said makes no sense and is not an answer to my correction of your false claim.

What I wrote:

Let f be the function whose domain is the set of natural numbers such that:

f(0) = (0 1/2)
for n>0, f(n) = [(2^n - 1 )/2^n (2^n+1 - 1)/2^n+1)

The range of f is an infinite partition of (0 1). — TonesInDeepFreeze

There is no "algorithm" or "going to infinity" there.

Purely that I proved that, contrary to your false claim, there is an infinite partition of (0 1). But you can't be bothered to actually looking at the proof to understand it (though it is very simple), as instead you're too preoccupied with handwaving and strawmaning that I've invoked an "algorithm" and a "going to infinity", which suggests of you a kind of a narcissism as you wish that other people indulge your undefined and confused musings while you ignore (worse, strawman) clear proofs and exact explanations from your interlocuter.

Defining a line as the union of smaller lines would be a circular definition — keystone

I didn't invoke any circularity. Read the proof.

'Dedekind cut' is not defined in terms of 'gaps', nor 'executions', nor algorithms. 'Dedekind cut' is mathematically defined and we see mathematical proofs that the set of Dedekind cuts is uncountable.
— TonesInDeepFreeze

In the context of computation, this distinction is moot since we can only ever speak of arbitrarily many cuts. — keystone

You wave "In the context of computation" here like it's a "Get Out Of Jail Free" card that you can use to evade that you're mischaracterizing what a Dedekind cut is.

I'm willing to withdraw my comment about uncountably many cuts made to countably many rationals. — keystone

Again, better not to be prone to making outlandishly false claims.

If we inspect the outcome of a real cut, we cannot determine what real algorithm was used to make that cut. — keystone

Again, Dedekind cuts do not invoke algorithms.

when executing real cuts we are forced to select a fractional interval at some finite depth in the Cauchy sequence of intervals, and whatever that interval may be could correspond to potentially aleph_0 different real cuts. — keystone

Learn to use mathematical terminology coherently. You suffer the misconception that throwing around jargon in inapposite imagistic clumps is mathematically meaningful.

Etc.

The conflation is just as I stated it.

But your new argument is quite different from the previous argument, and, as far as I can tell, it does not suffer the conflation.

But your new argument is not modus tollens.

Modus tollens is:

A -> B
~B
Therefore ~A

Your new argument is:

A -> B
~pB
Therefore ~pA

The antecedent and consequent of the conditional are A and B; they are not ~pA and ~pB.

I'm not sure whether the argument is modally valid (I'm very rusty in modal logic).

/

I'm not firmly opining as to whether A implies B nor as to whether B is possible.

First, though, what sense of 'possible' is meant? Thomson discusses physical possibility and logical possibility. If I'm not mistaken, he doesn't mention metaphysical possibility. Of course, discussion doen't have to be limited to Thomson's context, but 'metaphysical possibility' requires even more explication.

Anyway, I tend to favor that A is correct. But I tend to think Thomson may well be on the right track when he argues that it is not logically impossible to execute infinitely many steps in finite time.

The poster is very mixed up and adding lies to the ones he's already committed.

This is revisionist: [strikethrough in edit, since 'revisionist' is not the right word there.]

"the identity of a thing (by the law of identity) includes the order of the constituent elements, while the identity of a set (by the axiom of extensionality) does not include the order of the elements."

Yes, by the axiom of extensionality, sets do not have an inherent order. But in the past the poster argued that therefore the axiom of extensionality is wrong, because there IS the ordering of a set. That is why I asked: What is THE ordering of the set whose members are the bandmates in the Beatles.

Moreover:

A thing that has elements (or at least is made up only of its elements) is a set. So if things have an ordering that is "THE" [emphasis added] order of its elements, then, AGAIN, that is to say that, for sets, there is THE ordering of a set.

As I said:

Originally, his longstanding claim was that there is "THE" [emphasis added] ordering of a set . And that was the basis for him rejecting the axiom of extensionality. So I challenged by asking what does he claim is THE ordering of the set whose members are all and only bandmates in the Beatles (keeping in mind that there are 24 different orderings of that set, as, in general any set with n number of members has n! orderings).

But yesterday he said that sets do "NOT" [emphasis added] have an inherent ordering. So that was a reversal from his previous longstanding claim. But, even worse, yesterday he contradicted himself on the matter of whether there is "THE" ordering of a set.

Today he's back to:

There IS "THE" ordering of "things [with] constituent elements".

And that is what I said that he had said originally.

/

And contrary to the poster with his wildly ersatz, ignorant, stubborn, and mixed up ideas not just of mathematics but of everyday notions, I do recognize the sense of sets in everyday discourse. For example:

There is the set whose members are the bandmates in the Beatles. Suppose a painter asks for the Beatles to sit for a portrait - left to right in their order. Then they would say, "WHAT order?" There is no THE order of the Beatles. There are 24 of them.

Even if a person doesn't know to calculate that there are twenty-four orderings of a set with four members, a person does understand that there is not just one ordering of the set.

If a schoolmaster says, "Now, children, line up in your order." Even children would have the sense to say, "What order? Order by height? Order by age? Order by grade point average? Or what?".

But the poster still cannot grasp this fact of common understanding, though he presumes to take refuge in a sense of "common" language.

Sheesh!

As far as I can tell, you're committing the very conflation that Thompson warns about and as you quoted my paraphrase of it.

Think of the difference between "For all, there exists" and "There exists for all". Even if there does not exist a completion of all the subtasks, it does not follow that there there does not exist a completion of each of them. Just like that there is no completion of all of infinitely many additions but there is a completion of any finite number of them.

(1) We may question P2.

(2) C1 doesn't follow from P1 and P2. And it contradicts the point you quoted. 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.

(3) Thompson argues, "It is conceivable that each of an infinity of tasks be possible (practically possible) of performance [...] To deny [that] is to be committed to holding what is quite absurd, that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed."

(4) If I'm not mistaken, Thompson recognizes physical possibility and logical possibility, which are at least fairly well understood, but he doesn't mention metaphysical possibility. That's not to say that the notion of metaphysical possibility should be ruled out, but only that it requires explication.

Here's how it looks abstracted to a dialogue:

MU (many times a while back): The axiom of extensionality is wrong because sets have a certain order but the axiom allows that sets are equal without regard to their order.

TIDF (then and now): Sets of more than one member have more than one order. If every set has just one order, then, for example, what is the order of the set whose members are all and only the famous bandmates in the Beatles?

MU (many times): [silence]

TIDF (many times): Still interested in an answer.

MU (now): Sets do not have a certain order. And that is why sets are not identical with themselves.

TIDF (now): Many times you said that sets do have a certain order so that the axiom of extensionality is wrong. Now you say that sets do not have a certain order and that that is why they are not identical with themselves. But it's a violation of the law of identity that there are things that are not identical with themselves. So, sets, whatever they are, are identical with themselves. And from identity theory with the axiom of extensionality we have that sets are identical if and only if they have the same members, which denies that sets have only one certain order, and the law of identity is upheld not contravened. You are dishonest, self-contradictory and confused, and you don't know anything about this subject.

So put in terms the poster just used:

The set whose members are all and only the famous four bandmates in the Beatles.

That set is identical with itself (the law of identity applied to that set). But there is more than one ordering of that set. If a set being identical with itself required having one particular ordering, then no set with more than one member could be identical with itself. But the set that whose members are the bandmates in the Beatles is identical with itself.

So again the question for the poster: What is supposed to be THE one particular ordering of the Beatles that permits the set whose members are the bandmates in the Beatles to be identical with itself? Or, it seems that the poster's answer now is that sets are not identical with themselves. But that itself violates the law of identity, since the law of identity is that every thing is identical with itself - and whether material things or mathematical things.

The question is exact.

What is supposed to be THE order of the set whose members are all and only the famous four bandmates in the Beatles?

And woah! The poster is now saying the exact opposite of what he said for dozens of posts in this forum.

Over and over, the poster argued against the axiom of extensionality on the grounds that there is THE ordering of a set. Yes, I do remember.

And now he's denying he said that sets do have a certain ordering that is the ordering of the set.

The poster is flat out lying.

And completely reversing his previous argument. He says now that sets do not have an inherent order so that the law of identity does not apply to them. But he had been saying that sets do have an inherent order so that that order must be regarded when we evaluate self-identity. But it is the axiom of extensionality, which the poster denies, that provides that sets do not have an inherent order. And the poster denied the axiom of extensionality on the basis that sets do have an inherent order.

Then the poster mentions "things", though he denies that mathematical abstractions are things. But now he talks about sets as things. And he says their identity depends on their ordering but that a set can't be identical with itself. (What? A set can't be identical with itself?) Why can't a set be identical with itself? Now he says it's because a set does not have a certain ordering. Not only does it make no sense that a set is not identical with itself, it makes no sense that the reason a set is not identical with itself is that it does not have a certain ordering, especially when the poster's argument was that the reason sets fail the law of identity is that they do have a certain ordering.

The poster is not only lying, but he's very confused.

For any set that has more than one thing as a member (whether mathematical things or material things), there is more than one ordering of that set. That does not contradict the law of identity.

The principle of division does not indicate that a material object can be perfectly divided in two. Whether a pie can be perfectly divided so that the masses of the resulting pieces are exactly equal is not a mathematical question.

I predict that the crank will just say his incorrect argument over again.

And notice no response from the crank to the point that "there is no finite bound to the number of calls to division" does not require that each division be by the same divisor, as the paradoxes discussed don't require that the divisors all be the same.

And still interested in what is supposed to be the inherent ordering of a set such as the set of bandmates in the Beatles. If that question can't even be addressed by the crank then his claim that sets have inherent order (thus that the mathematical notion, and even the everyday notion, of sets is wrong) is not sustainable.

Thank you for the quote. His statement of his view that the notion of super-tasks is nonsense is wedged in a chain of reasoning, and somewhat hedged by saying 'if'. It's odd the way he slipped it in rather than stating it more centrally, since it would seem to be the most important conclusion in the paper. But for practical purposes, yes, we would take him as holding that notion of super-tasks is nonsense. Though it is not clear to me how he reached that conclusion (but I admit not really following certain parts of the paper).

But two points he does stress are the faulty inference from "no finite upper bound to the number of subtasks that can be completed" to "there is a completion of an infinite number of subtasks" and (2) that there are good grounds for doubting the premises of the paradox and that if there is misunderstanding in in those grounds than they are found in the "mathematical solutions" too (if I'm not mistaken, he has in mind that those faults are an incorrect understanding of the notion of an infinite sum).

And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case.
— TonesInDeepFreeze

It is taking this hypothetical premise – that there is no smallest unit of space and time – that gives rise to such things as Zeno's Paradox, Bernadete's Paradox of the Gods, and Thomson's lamp.

So if these arguments prove that contradictions follow if we assume that there is no smallest unit of space and time then as a refutation by contradiction it is proven that there is some smallest unit of space or time. — Michael

My reaction to that is the same as I wrote above to Ludvig V.

Sticking to the supposition of this puzzle creates confusion. The only possible solution is to look at it differently, not being hypnotized by 1/2, 1/4, ..... But I accept that it is your choice. — Ludwig V

Combined with another premise, it yields a paradox.

Thomson outlines the paradox:

Premise 1. To compete the main task requires completing infinitely many subtasks.

Premise 2. Infinitely many subtasks cannot be completed.

Conclusion. The main task cannot be completed.

But the main task can be completed. So one of the premises must be false.

One may say that Premise 1 is false. But that is a kind of "jumping the gun" when we would want to examine whether Premise 2 is false and Premise 1 is true. To say that Premise 1 is false on the grounds that it yields a falsehood skips that it's not Premise 1 alone that yields a falsehood, but rather it's the conjunction of Premise 1 and Premise 2 that yields a falsehood.

So Thomson does not right away say that it is Premise 1 that is false but rather he examines arguments for and against both premises.

Also Thomson argues:

"It is conceivable that each of an infinity of tasks be possible (practically possible) of performance [...] To deny [that] is to be committed to holding what is quite absurd, that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed."

That is, if there is a finite upper bound to how many times division can be executed, then there is some finite number k such that division can be executed k number of times, but division cannot be executed k+1 number of times. But why? Again, it's a leap to say that the answer is that otherwise there would be the false conclusion of the paradox, as the false conclusion might stem from the other premise of the paradox. Also, keeping in mind that Thomson is not arguing about the fact of the matter as to divisions of whatever, but rather about conceivability.

And I would consider that it's reasonable that the problem need not be contained to what happens to be true or false per some theory of physics, but rather that the problem of tasks may be abstract so that we may wish to resolve the paradox while granting the logical possibility or logical impossibility of the premises.

And Thomson's first point is one of logic, whatever the truth of falsity of the premises:

It is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.

The difficulty is, I think, the assumption that "divide" means exactly the same thing in all contexts — Ludwig V

Who makes that assumption?

not understanding that infinity means endless (but not necessarily limited) — Ludwig V

There are different philosophical notions of infinity. But I don't know what specific approach to the paradoxes you think depend on a certain notion of infinity that, on its own terms, carries a misunderstanding. At least in context of mathematics that is mentioned in the puzzles, we have definitions:

S is finite if and only if there is a 1-1 correspondence between S and some natural number.

S is infinite if and only if S is not finite.

S has an upper bound in S if and only if there is a member of S that is greater than or equal to all members of S.

S has an upper bound if and only if there is an x such that x is greater than or equal to every member of S.

S has a lower bound in S if and only if there is a member of S that is less than or equal to every member of S.

S has a lower bound if and only if there is an x such that x is less than or equal to every member of S.

g converges to L if and only if for any d, there is an n such that for all k greater than or equal to n, |g(k) - L| is less than d.

If g converges to L, the L is the limit of g.

So, regarding the infinite sequences in Thomson's lamp problem:

The range of the sequence 1, 2, 3 ... has no upper bound in the range, but it has an upper bound as far as ordinals go (the least upper bound is wu{w}).

The sequence 1, 2, 3 ... does not converge.

The range of the sequence 1, 1/2, 1/4 ... has no lower bound in the range, but it has a lower bound (the greatest lower bound is 0).

The sequence 1, 1/2, 1/4 ... converges to 0.

The range of the sequence 1, 0, 1 ... has an upper bound and a lower bound in the range.

The sequence 1, 0, 1 ... does not converge.

Those are just the ordinary mathematical definitions, which (in some sense) we don't evaluate as true or false but rather they're just stipulative definitions, and as such, they don't carry any "misunderstandings".

By the way, the poster, in his earlier remark about mathematics, has it exactly backwards, as usual with him, just as he has it backwards with me. Mathematics doesn't itself make claims about whether material objects can be divided any number of finite times without bound on the finite number ("infinitely divisible") but rather that there is no finite upper bound on reiterating division of numbers. That is a distinction that I hold myself; it's not a distinction that, contrary to the poster's confusion, I equivocate about.

I didn't argue that dividing a pie in half proves anything about the thought experiments.

And the poster again states his big fat stupid red herring about approximation, thus deserving a restatement by me:

"Of course it is understood that crumbs fall off and that we can't ensure that a knife cut will result in two pieces that are very much more than quite roughly the same weight. But that has no bearing on the principle of division. Moreover, the notion of dividing ad infinitum, such as in the paradoxes does not even depend on always dividing by halving or any precise number at all. The different divisions could be by different divisors, while still we get smaller and smaller distances or times." The poster argues like a child.

Just two of many now:

(1) The thought experiments don't depend on exactitude of division. Only a person who hasn't thought about the matter would overlook that one divisor might be 2 and the next divisor 2.1 and the next divisor 10, and we'd still have diminishing lengths.

(2) After several iterations of the challenge in threads, the poster still won't say what he supposes is the inherent ordering of, for example, the members of the Beatles, as lack of facing that challenge illustrates that the poster cannot sustain his attack on the notion of the extensionality of sets, either in everyday life or in mathematics, as the principle of extensionality is basic to the mathematics he presumes, without any education in the subject, to refute.

And then the poster cites that halving material object is merely an approximation, as if that has any bearing here.

Oh please! Of course it is understood that crumbs fall off and that we can't ensure that a knife cut will result in two pieces that are very much more than quite roughly the same weight. But that has no bearing on the principle of division. A person who drags in the fact that material objects are only measured approximately as if that refutes anything in a discussion such as this one is a person who has the mentality of a juvenile.

Moreover, the notion of dividing ad infinitum, such as in the paradoxes does not even depend on always dividing by halving or any precise number at all. The different divisions could be by different divisors, while still we get smaller and smaller distances or times. The poster brought in a big fat stupid red herring with his pointless point that measurements of material objects are not exact.

he only argued that "talk of super-tasks is senseless." — Michael

Where in the paper does Thompson say that?

Mathematics doesn't say there is no limit to the ways objects may be divided.
— TonesInDeepFreeze

That really depends on what you mean by "object". If you mean "physical object" then mathematics doesn't say anything about them at all, and whether or not some physical object is infinitely divisible is a matter for empirical investigation. — Michael

I had in mind not 'object' but that 'ways of dividing' is vague. Or maybe the poster just refers to the fact that there are many different divisors. And, yes, among the reals there are uncountably many divisors ("ways to divide").

This is less directed to you than it is directed to the poster who has ridiculous, ignorant ideas about mathematics. These are points that are too terribly obvious to anyone of even barely adequate intelligence and education, but the poster drags discussion down to the level that these things need to be made explicit:

Regarding numbers, when we say 'divide x by y to get z', that is an instruction for a procedure. That procedure merely upholds that x/y is the unique number z such that y*z = x. That does not imply, for example, that from the number x we to create two other numbers - one x/2 and another x/2. Rather, x/2 is itself a number. There are not two x/2, each one a separate object made by dividing x.

However, when mathematics is applied, then a material object may have x number of units, such as x number of square inches, or x number of pounds, or x number of grains of salt. Then, when we say, for example, "divide the object in half", we mean that we will have two different objects, each with x/2 square inches, or x/2 pounds, or x/2 grains of salt, as the case may be.

Those two different senses of "divide" are not in contradiction when a reasonable person considers the two different contexts.

The poster claimed that I equivocate about this. On the contrary, I am clear of quite clear of the distinction and none of my comments employ any equivocation regarding it.

As it stands one cannot have half a photon (or any point particle) and if spacetime is quantized then there is a limit to how far one can divide some section of it into two. — Michael

The paradoxes discussed don't require splitting such things as photons. And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case.

I maintain that the issue is about how you choose to represent my step, and representing my step as composed of infinitely many segments is only one of many representations. — Ludwig V

The puzzle supposes infinitely many segments. Of course, if you deny the supposition of the puzzle, then it may be easy to dispense the puzzle. But one may wish not to take the easy way out but instead grapple with the puzzle under the suppositions it makes. Of course, one can hold that there are empirical or even physics theoretical bases to hold that there is a finite upper limit to how many times distance and duration may be divided, but one might still wish to drive a harder bargain, which is that the question is not that of physical but rather of logical possibility.

Previously I wrote:

"Thomson says [...] there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time."

I may have erred there. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.

"Complete" does not apply to infinite series, by definition. — Ludwig V

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.

The infinite sequences in this context are:

(1) The function that maps n to to 1/(2^n)

(1) The function that maps n to either 0 or 1 (off or on) depending on whether n is odd or even

Mathematics doesn't say there is no limit to the ways objects may be divided.

Where does such a claim about mathematics even come from? What actual piece of written mathematics is claimed to say such an unfocused thing?

Rather, division of real numbers (which is the subject here) is merely by definition:

for y not= 0, x/y = the unique z such that z*y = x. And that is based on having previously proven that, for y not= 0, and for any x, there is a unique z such that z*y = x.

I know of no mathematician who wrote gibberish saying that "there is no limit to the ways objects may be divided".

the mistake is in the application of transfinite numbers — Michael

The only infinite number in the puzzle is the domain of the sequence.

It's not that complicated. Imaginary numbers have a use – even in electrical engineering – but I cannot have an imaginary number of apples in my fridge.

There's nothing wrong with maths, just sometimes an improper use of it. There is no smallest number, but if paradoxes like Zeno's and Thomson's are valid then it would suggest that there is a smallest unit of space and/or time – and that this isn't just a contingent fact about the physics of our world but something far more necessary. — Michael

So it seems your analogy is between misuse of imaginary numbers and misuse of infinite numbers.

I might not put it that way. But I do understand Thomson's point that there cannot be infinitely many task steps executed in a finite duration.* And I understand the different argument that the puzzle may dissolve if we allow that there is a shortest distance and shortest duration.

* EDIT: As I mentioned in an edit a few posts ago, I may have erred. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.

I'm sure it could count as a human right. — Ludwig V

A deep reading of American history reveals that the right to the arithmetic operations was to be enshrined in the Bill of Rights. But it failed to pass because the mid-Atlantic states feared that too much public exercise of arithmetic would allow citizens to become too number savvy and that would hamper the sports betting industry that was legal back then, especially in New Jersey.

The trouble with Thompson's lamp is that no switch can function in an infinitely small time. — Ludwig V

The lamp puzzle doesn't require anything to occur in an infinitely small amount of time.

And I don't think the discussions about a switch being moved or any aspect of the agency by which the light changes are relevant. It is missing the point to quibble about the mechanics of how the light changes. We need only take it for granted that it does change at the rate stated in the puzzle.

Infinite means without limit. — Ludwig V

Not in mathematics.

Everyone knows that tea is taken at at the tea time hour and that one is not to dawdle still drinking it, not even hypothetically, not even gedankenishly, past the tea time hour.

If space and/or time being infinitely divisible entails that supertasks are possible, and if supertasks being possible entails a contradiction, then it is proven that space and/or time are not infinitely divisible. — Michael

Did Thomson make that argument? Was that part of his answer to the paradox?

Meanwhile, still interested in knowing what the poster would claim to be the inherent ordering of the set whose members are the bandmates in the Beatles.

If the poster can't answer that question, then he lacks basis for his dogma that every set has an inherent ordering, which goes to the heart of his bizarre imaginings about mathematics.

Underphysician Metacover as the Baker:

Customer: I'd like a cherry pie, divided in two. I'm going to give one half to my niece and that other half to my yoga teacher.

Baker: That makes no sense. I would have to cut the pie, and then there would be two different things, not two halves of the same pie.

Gina: Excuse me?

Baker: You heard me.

Gina: I heard you. But I don't understand.

Baker: What don't you understand about the fact that when you slice something apart, there are never halves of anything, only two new things? If you still don't get it, then I suggest you read my posts at the 'Phil's Ossify For 'Em' website. It's a philosophy place where I write my posts showing that all of mathematics is wrong.

Gina: Well, I've studied philosophy and mathematics, and have not read anything like what you're saying.

Baker: Exactly. If you want to know what's really up, you have to come to me for it.

Gina: I just want you to cut the pie in half and put the separate halves in separate boxes. If necessary, I'll pay extra for you to do that.

Baker: There is no money in the world that would permit me to cut a pie in half. It's not a matter of money. It's that it is metaphysically, ontologically and mathematically impossible to do. Now I can slice a pie. But I can't call the pieces halves or sell them as halves. I can only sell you two new things that are not to be referred to as halves of anything. So would you like one new thing in its box and another new thing in its box?

Gina: Yes please.

Baker: Fine. But not if you call them halves!

Gina: Okay, I promise not to do that.

Later that day Gina arrives home and talks to her husband Ralph and son Timmy:

Gina: I had the most peculiar conversation with the new owner of the bakery downtown. He insists that he can't sell me two pie halves but only two different things that are made of pie.

Ralph: Yes, I know. He is a bit odd. Last week I asked for a baker's dozen of bagels and he gave me only twelve even though I reminded him that a baker's dozen is thirteen. He said that is a contradiction in terms and instead I need to ask for thirteen at the same price as for twelve and he'd do that. So I said, okay, just give me one more bagel to add to the twelve, since 12+1 is 13. Then he went into thing about how 12+1 is not 13, that numbers aren't even things of any kind, and that people are all wrong about the law of identity when they say things like "1+1 is 2". He even said that '1 is 1' is false because the first symbol '1' is not the second symbol '1'. Very strange fellow.

Timmy: I talked with him too. He has all kinds of very strong opinions about math, but he doesn't know anything at all about.

If there is a greatest divisor, then there is a greatest natural number, call it 'g'. So then what is g+1? If one says addition is not allowed with g as a summand, then one needs to come up with a different definition of addition, which becomes very very complicated if we wish to still have addition in a formal theory.

Also quite inelegant. If, for arbitrary example, we say that g = 66589080980923842343287098023450390811321445645098760011287390453735490233999934393 is the largest natural number, then naturally a person would want to say, "My, that's awfully specific for mathematics that we would like to be most general."

Moreover, why should numbers be limited to only how many particles there are? Such a limitation would preclude the natural human inclination to ruminate on such things as, "Suppose we made a mistake and there are actually twice that number of particles" or even with g as all the possible finite combinatorial arrangements of the particles. Well, that entitles us to talk about numbers larger than g.

Ultrafinitists are welcome to try to convince me, but I am a tough customer when it comes to giving up my natural prerogative to add 1 to any number.

I'd need to check Thomson's paper again to ascertain whether that properly describes the particulars of his view.

But, yes: The infinite sequence of durations converges to 0. And the marked time converges to 2 minutes. But it seems that Thomson is saying that it is in the nature of tasks that there is not an infinite sequence of them such that they are all completed in finite time.

Thomson says that there's a false premise, which is that infinitely many tasks cannot be completed in finite time. He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time.

[EDIT: I may have erred in the paragraph above. At least in the first part of Thomson's paper, he does not say that infinitely many tasks may not be completed in finite time. Rather, he says only that it is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks.]

I think "valid paradox" is, at best, ambiguous and confusing.

Ordinarily, an argument is valid if and only if it is not possible for the premises to be true and the conclusion false. But a paradox has seemingly true premises, seemingly correct logic, but a false (indeed, often a contradictory) conclusion. So it is not clear what you mean by a "valid paradox", though perhaps you mean that the premises indeed entail a falsehood (or even a contradiction) or that the premises are true but entail a falsehood?

I understand the view that there is no smallest number but that there are smallest distances and durations. But I am asking whether some people here do believe there is a smallest number.

I'm not claiming that the implications are or are not complicated.

The analogy with imaginary numbers and apples is amiss in this regard: Yes, apples are counted by integers, not imaginary numbers, so indeed imaginary numbers are not the correct kind of number to count with. But distances and durations are measured by real numbers, so smaller and smaller real numbers are not a difference in the kind of number.

An argument has been given that physics is impaired by an improper application of mathematics. So the question arises whether that argument extends to a claim that mathematics must be revised. The context is that paradoxes about motion involve application of certain mathematics to distance and time. So, if someone claims that the mathematics is to blame, then we would ask whether the mathematics itself (which holds that there is no smallest number) needs to be rejected, or whether the way in which the mathematics is applied needs to be rejected, or both.

My question was about mathematics not physics. Suppose there is a smallest number usable for a given application of mathematics. Then, must mathematics not allow smaller numbers? If mathematics must not allow smaller numbers, then how would that be rigorously enforced in a mathematical theory? Suppose p is the smallest number that is to be allowed. The ordinary operations are defined by:

For all x and y, x-y = the unique z such that y+z = x.

For all x and y, x/y = the unique z such that y*z = x.

What would be the definitions when there is a smallest number p?

Note that in informal contexts, we use a notion of 'undefined'. But in a fully formalized theory, we don't allow 'undefined' as it would violate the definitional criteria of eliminability, which is crucial for the requirement that the syntax be recursive.

If there is a maximum number of divisions, then what is that maximum number?

That is, for what natural number n is it the case that 1/(2^n) is not a number?

/

"non-dimensional points [...] dimensional separation"

Maybe start by defining 'non-dimensional point' and 'dimensional separation'.

/

If the discussion is about points in ordinary real 2-space or real 3-space then points are distinguished by being a different ordered tuple.

In 2-space, the point <x y> differs from the point <z v> iff (x not= z or y not= v).

In 3-space, the point <x y t> differs from the point <z v s) iff (x not= z or y not= v or t not= s).

If a particular line, say the ordinary horizontal axis, then <0 x> differs from <0 z> iff x not=z.

This is not the least bit baffling.