Runner's Paradox: Finishing the Race in d Metres

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The discussion revolves around Zeno's paradox, specifically the scenario of a runner attempting to cover a distance of d meters by completing an infinite series of tasks. The paradox suggests that since the runner must complete an infinite number of steps, he can never actually reach the distance d. However, it is clarified that while the series of tasks is infinite, the sum of these tasks can converge to a finite distance, allowing the runner to complete the race in finite time. The key point is that the completion of each task leads to a continuous motion that ultimately results in reaching the destination at time d. Thus, the paradox highlights the complexities of infinite sequences and their implications in motion and distance.
  • #91
Your argument assumes that if each task on a list is completable, then the list as a whole is completable.

I did misspeak; I was saying "can complete" when I was thinking "completes", which makes a huge difference.
 
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  • #92
Originally posted by Hurkyl
I did misspeak; I was saying "can complete" when I was thinking "completes", which makes a huge difference.

Yes it does. Now you've brought tense into play. As a result your assumption is either incorrect or begs the question depending on where you set the present moment.
 
  • #93
"can" connotes something vastly different from temporal concerns. (at least depending on your presumptions!)

But that's a little misleading. The current hypothesis with "can complete" gets formalized to:

P(n) := the set of the first n tasks can be completed. (whatever "can be completed" means)

while the "completes" version gets formalized to

P(n) := Achilles completes the n-th task (meaning he is at the specified destination at some point in time)


The present moment has nothing to do with it.
 
  • #94
Originally posted by Hurkyl
"can" connotes something vastly different from temporal concerns. (at least depending on your presumptions!)

But that's a little misleading. The current hypothesis with "can complete" gets formalized to:

P(n) := the set of the first n tasks can be completed. (whatever "can be completed" means)

while the "completes" version gets formalized to

P(n) := Achilles completes the n-th task (meaning he is at the specified destination at some point in time)


The present moment has nothing to do with it.

"completes" introduces temporal concerns. "can" is just an alethic modality.

The present moment has a lot to do with it. If you take "completes" to be present tense, then your assumption that if Achilles completes task n, then he completes task n+1 is simply wrong. If you are using "completes" atemporally or with a specious present, then the assumption begs the question as it tacitly assumes the completion of the list just as the use of a future tense would.

In any case you still would not have addressed the issue of composition.
 
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  • #95
And we go back to why I was so insistent on getting precise definitions of terms. The english language is not only horribly imprecise, but highly subjective.
 
  • #96
Unfortunately you're stuck with the messiness of natural language any time you try to make mathematics connect to the real world. Take something as simple as 1+1=2. Is it true? Of course it is mathematically speaking. But once you try to tell me that if I put one apple in a bag and then another that I will have two apples and you give as your reason the fact that 1+1=2, then you've gone and botched it up.

Since the original question deals with the physical rather than the mathematical world, we have to show that the math correctly depicts the real world. Doing that invariably requires some messines. In particular, our original problem is inherently temporal. The question isn't whether the sequence of ever decreasing segments has a finite sum. The question is whether it can be completed when done one task at a time.

So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.
 
  • #97
drnihili wrote:
Not all numeric properties are composable. For example, for any natural number n, if there is there is a natural number bigger than n, then there is a natural number bigger than n+1. However, there is no natural number bigger than all natural numbers.

Actually if you are happy with this condition then you should have no problem accepting that even an imaginary space can only be divided up into finite many distances.

My whole basis for the argument that a finite line can only be said to contain an infinite number of points is precisely the same logic used here. Just because the set of natural numbers itself is infinite doesn't force that property of infinity onto any of its elements. In other words, no natural number itself is infinite.

Imagine starting with a line defined only by its two endpoints. (That line can represent a distance or space) We can now place a point between the two endpoints. We now have 3 points. We can place points halfway between those existing points. We now have 5 points. Toss in more points between those, we have 9 points. Do it again we get a line with 17 points and so on.

We are building a set of the finite elements, {2, 3, 5, 9, 17,…}

There is no end to how many times we can do this. In other words we can add more elements to this set endlessly by simply placing more points between existing points. However, it is quite clear (to me anyway) that just because there is no end to the number of times we can do this, it does not follow that we can make the line contain an infinite number of points. On the contrary this example shows that no matter how hard we try we can never force the property of infinity onto any of the elements within the set. So while the number of points that we can add to a line appears to be unbounded, it must also always retain the property of being finite.

I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

drnihili wrote:
Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."
Now for any n, the initial segment whose last member is S_n is consistent. However, the entire seequence of sentences is inconsistent. (This, btw, is Yablo's paradox. Perhaps it's inclusion here is a hint that we may really be dealing with omega-inconsistency. I'll have a think on that.)

I don’t see the problem of a finite number of points in a finite line, or the finite divisibility of a finite distance, being a problem of self-reference.

However you might have a point concerning the completion of an infinite number of tasks. If so, all this would do is move you from Zeno's Paradox into Yablo's Paradox which may very well be a similar situation. (i.e. Something that isn't solvable: a Paradox)

If that’s the case, then you will have just added yet another way of saying that it isn't possible to infinitely divide up a finite space. :smile:
 
  • #98
Hurkyl wrote:
First off, only the energy is quantized. The times at which QM allows the atom to change states is not.

Alright, I confess I jumped the gun on that one. :frown:

I'm starting to just post my first thoughts now. That's not good on a forum board.

This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I'm still convinced that they are, and I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

I do like that way of thinking, and this does imply that time and space are related to material objects rather than to some imaginary idea of an absolute space.

So while I confess to being a complete idiot here, I still claim to be an idiot who's on the right track!

And this was still good food for thought too! :smile:
 
  • #99
Originally posted by NeutronStar
Actually if you are happy with this condition then you should have no problem accepting that even an imaginary space can only be divided up into finite many distances.

That depends on what you mean by "be divided". If you mean that any activity of dividing yields only finitely many divisions, then I agree. If you mean that there is some inconsistency in supposing that the line is constituted by infinitely many points, then I disagree.


I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

Simlarly here. I'm with you until you switch to set theory. Set theory is not about tasks or actions. When one talks of constructing the hierarchy of sets, there is no pretense that one actually builds the entire hierarchy. Rather one is setting out rules which define the hierarchy. Those definitions cover an infinite number of cases. However no one can actually carry out the construction for an infinite number of cases.



I don’t see the problem of a finite number of points in a finite line, or the finite divisibility of a finite distance, being a problem of self-reference.

Yablo's paradox does not contain any self reference, so I'm not sure where you're headed with this one.
 
  • #100
drnihili wrote:[/]
Simlarly here. I'm with you until you switch to set theory. Set theory is not about tasks or actions. When one talks of constructing the hierarchy of sets, there is no pretense that one actually builds the entire hierarchy. Rather one is setting out rules which define the hierarchy.[/color] Those definitions cover an infinite number of cases. However no one can actually carry out the construction for an infinite number of cases.


It was in fact the idea of the rules[/color] that I was referring to with the set theory example.

There is no rule (or axiom) in set theory that states that the quantitative property of a set must necessarily be transferred onto any of it's elements. There mere fact that we can continue to constructs elements without bound does not in itself imply that any of the elements within the set must necessarily become infinite.

All I'm saying here is that there is no rule[/color] in set theory that allows us to do that, or make that conclusion in any way. If we jump to any such conclusion then we are doing that intuitively on our own, and we really have no logical reason to do it.

On the contrary the rules of addition tell us just the opposite. We have started out with a finite number of point (the two end points). We then begin to add points half way between these two end points, etc. Well each time we do this we are adding a finite number of points to a finite number of points, and we have an axiom somewhere that says that the result of any such operations on sets must also produce a finite set. So from the operation of addition we have proven that all individual elements in our infinite set must indeed be finite.

It seems to me that putting all of this together set theory has indeed shown that any finite line must contain a finite number of points. Even though there is no bound on the finite number that we may choose to express this quantity.

The Points Salemans:

There is a points salesman who has cards that each contain a finite number of points on them. He has an infinite number of cards. Before you begin the race you much purchase a card to describe the number of points in your line. Then you may begin your race. No matter which card you chose from his infinite inventory you will end up with a card that contains a finite number of points. Then when you run the race, you will only need to complete a finite number of tasks.

Zeno's Paradox:

Zeno comes along and says, "Hey wait! Try doing this instead,…". And then he tells you to try stepping half way to the next point each time instead of using the points on a card from the set that defines the possible number of points in a line.

All that Zeno has done was to force you to use the last card in an infinite set of cards. But that is impossible because an infinite set has no last card. Zeno has set you up to fail. He has transferred the infinite property of the main set onto the elements.

How did he do that? It should be obvious. Imagine that you begin with a line that contains only two points. To step half way to the end point you must trade in your card with the 2 on it and request a card with a 3 on it. (in other words, you must dynamically redefine your line and move to the next card in the set). But after you make that step you again run out of points and must trade in your 3 card for a 5 card. Only then can you make the next step. This will continue forever because you are not actually completing the number of points of any particular card, instead what you are being forced to do is to continually choose a new card from the infinite deck with each step. In other words, you are being forced to redefine your line with every step you take!''

To me Zeno's Paradox simply proves that we can't construct an axiom that would allow us to force the quantitative property of a set onto its elements, because to do so would cause a logical contradiction. :smile:

Anyhow, this is how I think of the problem from the point of view of set theory. I'm sure that there are other ways to think of it as well.

drnihili wrote:[/]
Yablo's paradox does not contain any self reference, so I'm not sure where you're headed with this one.


I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!
 
  • #101
This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I would too like to see a discrete model of space-time (marginally different than "quantized", but I think you mean discrete anyways)... but I don't go promoting the idea because there's no proof.


I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

Right. In fact, in my casual reading of both GR and QM, the issue of what constitutes a "rod" and "clock" is one of the more interesting and still active research topics. One interesting fact is that, according to QM, any physical clock has a finite probability of giving the same reading twice!


So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.

I am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)


I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

Logic does not tell us this is an impossible task; logic merely tells us that induction is not sufficient to prove it possible.


Incidentally, the question about lines is not whether we can place a set of points on a line, but a question about the set of points that are on the line. So while induction cannot (by itself) prove that we can place infinite points on a line, it can prove that there are an infinite number of points on a line.

(given the reponse written while I wrote this, I think this warrants proof, so I'm writing it now)
 
  • #102
(the context is Euclidean geometry)

Theorem: Any line has an infinite number of points on it.

Proof:

Let L be any line.
Choose P on L.


Lemma: For n > 1, there is a set of n points on L, Sn, such that there is a point Qn in Sn such that for any other point R in Sn, R*Qn*P. (That is, Qn is between R and P)

Base case:
An incidence axiom guarantees that any line has at least two distinct points. One of those points has to be different than P; call it A.

A betweenness axiom guarantees a point Q such that A*Q*P. Define S2 := {A, Q}. This satisfies the statement of the lemma for n = 2.

Inductive step:
Let Sn satisfy the statement of the lemma. The same betweenness axiom guarantees the existence of a point Qn+1 such that Qn*Qn+1*P.

For every point R in Sn different from Qn, it is true that R*Qn*P. We also have Qn*Qn+1*P, thus we can conclude R*Qn+1*P. (I don't know wehre my text is, I can't remember if this is an axiom or a theorem) This allows us to conclude that R is distinct from Qn+1

Define Sn+1 := Sn U {Qn+1}. This set satisfies the statement f the lemma.

Thus, the lemma has been proven for all natural numbers n > 1.


Let T be the union of all of the Sn for natural numbers n > 1

Every point in T lies on L (because it is an element of one of the Sn's)

|T| >= |S| = n for all natural numbers n > 1, so T cannot be finite.

Therefore T is an infinite set of points on L, and this concludes the theorem.
 
  • #103
Originally posted by NeutronStar
I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!

Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

But whatever, this isn't a thread about self-reference. I was merely pointing out that there are lots of cases where we can find things such that if they are true of n, they are true of n+1 and yet they are not true of the natural numbers taken as a set.
 
  • #104
Originally posted by Hurkyl
II am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)

That's fine. In that case you have to agree that calculus (nor any branch of mathematics) cannot by itself resolve Zeno's paradox. Any resolution would have to consist not only of a set of axioms and cosequences but also an interpretation of the axioms in terms of the real world and also an argument that the axioms correctly model the world. The first of these tasks might, perhaps, be mathematical, but the second clearly isn't.

So from your revised formalist perspective, you still owe an argument that you have an accurate model of what's going on. At this stage it's entirely unclear how such an argument might go.
 
  • #105
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.
 
  • #106
Originally posted by Hurkyl
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.

Ah, no answer. Not surprising I suppose.
 
  • #107
drnihili wrote:
Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

I was referring to logical self-reference not just any self-reference. Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

Now if the sentence were to be rewritten as ""The first sentence of this post begins with the letter 'S' only if this statement is false." Then we would have a logically self-referenced situation. It's true that the self-reference would be with respect to the entire post, and not merely with respect to the sentence itself. But it is still a form of logical self-reference none the less.

drnihili wrote:
Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."

In this situation each element of the proposed set is making a demand that a certain truth value exist with respect to other specific elements within the same set to which this element belongs. I clearly see the logical self-reference here with respect to the set as a whole. I don't understand how you can seriously deny this. [?]
 
  • #108
I have no idea what you mean by "logical self-reference". However I can assure you that it's pretty uncontroversial that there's no self reference there. Try looking up the original article (by Yablo in Analysis. The whole point of the paradox is to avoid self reference. If you can find a critic who has claimed that Yablo failed in that, I'd be interested.

I grant you that there is a certain oddity to the sentences, but the oddity has absolutely nothing to do with self reference. You also said

Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

which strikes me as exceedingly odd. Of course the sentence's truth depends on the condition stated. How could it be otherwise?

**Edit**
Are you prehaps worried because the sentence refer to the truth or falsity of other sentences rather than making direct claims?
 
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  • #109


drnhilili wrote:

NeutronStar wrote:
Your example, "The first sentence of this post begins with the letter 'S'." is merely as statement. It may be true or false. It doesn't demand that its truth value be dependent on the condition stated.

which strikes me as exceedingly odd. Of course the sentence's truth depends on the condition stated. How could it be otherwise?

My point is that your example here would not be an example of logical self-reference. It would merely be an independent form of self-reference.

The only time that statements get involved in logical self-reference is when they start making statements about the state of their own truth value, or the truth value of other statements that are directly related to them (like other members of the same set that they themselves belong to).

If expert logicians believe that Yablo's Paradox is not a form of logical self-reference then I have lost complete faith in expert logicians. I can see the logical self-reference clear as day. Not as a self-reference of any particular element onto itself, but as a self-reference relative to the set as a whole.

All of the elements in the set make up the entire set. Therefore if any specific element within the set demands that the truth value of a statement contained in any other element in the same set is dependent on its position relative to the original specific element,… Well, it's clear to me that any set composed on such a rule is definitely referencing its own composition to decide a particular truth state about its own composition.

If that's not logical self-referencing then I don't know what is. :smile:
 
  • #110
drnhili wrote:
quote:
--------------------------------------------------------------------------------
Originally posted by Hurkyl
Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.
--------------------------------------------------------------------------------



Ah, no answer. Not surprising I suppose.

You mean- no answer that you understood.
 
  • #111
Originally posted by HallsofIvy
You mean- no answer that you understood.

No, I mean no answer. I understood what was said. If you'll take the time to read my post just prior to the one you quoted from Hurkyl, you'll see that I point out that a certain kind of explanation is required based upon Hurkyl stated position. Instead of providing such an explanation Hurkyl merely retreated to his "psuedoparadox" refrain. Then he goes on to say that the question of modeling is a separate one, which is of course absurd given his earlier post about the demarcation.

So, he provided no answer to my post. He has yet to provide any justification for his position that Zeno's is a pseudoparadox. His demarcation post does finally admit that calculus is not sufficient to resolve it. But faced with that admission he simply retreats to a domatic position and avoids the question at hand.

Please understand, I am not saying that calculus is somehow flawed or wrong. It's just not the right sort of theory. Neither is arithmetic, but that doesn't mean arithmetic is wrong.

But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it.
 
  • #112
If you'll take the time to read my post just prior to the one you quoted from Hurkyl, you'll see that I point out that a certain kind of explanation is required based upon Hurkyl stated position.

I'd like to see, in your own words, what you think my position is.
 
  • #113
Originally posted by Hurkyl
I'd like to see, in your own words, what you think my position is.

Your position in this regard is that science and mathematics have different purposes. Mathematics is supposed to tell us what follows from what. Science is supposed to tell us what axioms correctly model the real world (or some relevant portion thereof). Further, you believe that science and mathematics have some overlap, at least in practice, but probably also in theory.

So, how close is that?
 
  • #114
Well, yes.

But I was asking about my position with regard to Zeno's paradox; the position to which you make an accusation like

"But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it."
 
  • #115
Originally posted by Hurkyl
Well, yes.

But I was asking about my position with regard to Zeno's paradox; the position to which you make an accusation like

"But Hurkyl somehow knows he's right. 2000 years of study and effort have all been misguided, he's got the one true answer. And it's so obvious he doesn't even need to provide justification for it."

Ah, since you quoted a place where I was talking about your position on science and math, I assumed that was the postion you were asking about.

I'm on my way out the door now, so I can't say much more than that you think there is no contradiction involved. When I get back I'll try to give a fuller account.
 
  • #116
Ok, as I understand it, your position is roughly the following.

1. The consistency of calculus demonstrates the consistency of an inifinite divisibility model.

2. Induction is sufficient to show that all of the ever decreasing segments lie on Achilles' path from 0 to d.

3. Since d is the limit of the series of segments, completing the series is sufficent for reaching d.

(I agree with you on 1, 2, and 3 by the way.)

4. The above points suffice to show that Achilles can reach d by covering each of the ever decreasing segments in order. (At least barring a fully rigorous derviation to the contrary.)

(Point 4 is where I see the disagreement)

This is all from memory of posts, so I may be mistaken at one or two spots, but that's where I see your position.

Oh, you also seem to have a general background belief that all paradox is the result of a lack of clarity.
 
  • #117
4. The above points suffice to show that Achilles can reach d by covering each of the ever decreasing segments in order. (At least barring a fully rigorous derviation to the contrary.)

You sure you stated this right? If Achilles does cover all of the ever decreasing segments (in order), then we just have point 3.

(actually, this is contingent on the assumption that paths are continuous, and that there aren't any silly tricks like Achilles ceasing to exist at the very instant he would be at d)


Oh, you also seem to have a general background belief that all paradox is the result of a lack of clarity.

I think this warrants further explanation.

Mathematicians have known about the various paradoxes for a while, and to the best of my knowledge they have sufficiently tweaked the formalism so that all known paradoxes are either impossible to state (such as the liar's paradox) or have been reduced to pseudoparadoxes (such as Russel's paradox).

The prototypical example of "lack of clarity" is that of the Twin paradox; the applied formulas require that the reference frame be inertial, but this condition is overlooked and the formula is applied in a noninertial frame. A number of paradoxes arise in this manner merely by making logical mistakes.
 
  • #118
Originally posted by Hurkyl
You sure you stated this right? If Achilles does cover all of the ever decreasing segments (in order), then we just have point 3.

(actually, this is contingent on the assumption that paths are continuous, and that there aren't any silly tricks like Achilles ceasing to exist at the very instant he would be at d)

Yes, I stated it correctly, but there is sufficient ambiguiity in the English to warrant further explanation. A lot depends on emphasis that isn't recoverable from text.

I think you hold that 1-3 demonstrate that the series is completable, that by running one segment at a time Achilles can run all of the segments.

I think we're all in agreement that if Achilles completes the series, then he will have arrived at d. The question is whether Achilles can complete the series. I think you hold that 1-3 provide sufficient argument that he can.
 
  • #119
Originally posted by Hurkyl
I think this warrants further explanation.

Mathematicians have known about the various paradoxes for a while, and to the best of my knowledge they have sufficiently tweaked the formalism so that all known paradoxes are either impossible to state (such as the liar's paradox) or have been reduced to pseudoparadoxes (such as Russel's paradox).

The prototypical example of "lack of clarity" is that of the Twin paradox; the applied formulas require that the reference frame be inertial, but this condition is overlooked and the formula is applied in a noninertial frame. A number of paradoxes arise in this manner merely by making logical mistakes.

All I can say is that the best of your knowledge isn't very good on this point. The Liar paradox continues to be an active area of research and there is no generally accepted solution to it. Tarski's hierarchical approach and Kripke's fixed point theories are perhaps the most commonly cited, but not far behind are defalationsist and various context dependent approaches including at least one based on set theory with an anti foundational axiom. Not all solutions make the paradox impossible to state, revision theory and paraconsistency are good examples of theories that aim to preserve the paradox while removing it's sting.

There are wide variety of ways of tweaking the formalism to avoid the repurcussions of the Liar in formal systems. Even so, it remains a very open question which, if any, of them correctly models the Liar in natural languages. It would be a gross misrepresentation to say that Liar paradox has been shown to be the result of a lack of clarity. Three are some theories that take that stance, but by no means all of them, nor even the most important of them.

If by Russell's paradox, you mean the Barber paradox, then I could perhaps agree. But that was never really meant as a full blown paradox anyway. It was merely an example of the problems with unrestricted comprehension. Set theory was modified to have only restricted comprehension and thus avoided inconsistency. However the root problem remains, and set theory's response just isn't very illuminating. It is by no means clear that restricted comprehension is ultimately the right answer. Though admittedly it does pretty well for most things. I suspect that Neutron Star would have similar misgiving about resorting to restricted comprehension as he does about resorting to the empty set.

In general, initial mathematical responses to paradox are primarily frantic attempts to avoid inconsistency. They generally succeed in doing so, but it doesn't follow from that that the paradox has been laid to rest or shown to be just the result of foggy thinking.
 
  • #120
The question is whether Achilles can complete the series. I think you hold that 1-3 provide sufficient argument that he can.

Ok, this is where you have me wrong.

When we went over Zeno's paradoxes in my Greek philosophy class, they were presented as being an actual contradiction in Greek times; geometery permitted space to be infinitely divisible, but Greeks held infinite sequences to be impossible, and Zeno could derive from that a contradiction... we know Achilles can get from here to there, but we know Achilles cannot get from here to there because of the infinite sequence of tasks in the way.

Most of the time, when the paradox arises on forums, the poster tends to be of that same persuasion, so I typically start off by addressing the typical reasons why one might think an infinite sequence of tasks is impossible... thus the discussion about calculus and infinite series.


You, however, are asking a completely different question. The root of your question is why should one accept a given model of reality. Dressing it up in the guise of Zeno's paradox only serves to obfuscate the issue. And bundling your question up with Zeno's paradox connotes that there's some aspect about the paradox that is relevant to your question, causing me to respond attempting to show that your question + Zeno's paradox is no deeper than your question by itself. (I did comment on this earlier, but it seems to have been missed)

IOW, if you're serious about pressing the "Why should we use model X to describe reality", you really should ask it in its own context, at which point I make the boring statement that mathematics doesn't care about your question, science gives only empirical evidence that model X is practical, and that there has not yet been given any good reason to accept any other particular model. Certainly not logical proof of anything, but I think you're savvy enough to realize you're asking an unanswerable question. (though I do think it was bad form to complain that the unanswerable question was, in fact, not answered)


All I can say is that the best of your knowledge isn't very good on this point. The Liar paradox continues to be an active area of research and there is no generally accepted solution to it.

I beg to differ; mathematics has accepted a solution to it. P(Q) is simply not in the language of mathematical logic. I'm not trying to imply that this is the final word on the issue, just that in typical mathematical fashion we have taken a minimalist structure that avoids all the contradictions yet permits us to do all the logical steps we like to do in mathematics. (though possibly not metamathematics... I don't know all of the gritty details I would like to know about that subject)


However the root problem remains, and set theory's response just isn't very illuminating.

But it is practical. As with logic, the idea of ZFC was to take a minimalist approach that permits us to do everything we like to do, but not have enough power to derive the contradictions. I don't see anything wrong with that, and unless you really want to work with pathologically large sets, nothing is lacking in the solution either.


In summary, one can answer all the paradoxes. It might not be the best answer, but it is an answer, and it is probably contained in any better answer to boot.
 

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