we never actually manipulate the infinite sets directly. — keystone
I currently hold an unorthodox view that set theory might not actually be about sets — keystone
the mainstream view is that it exists in the Platonic Realm — keystone
While [Thompson's lamp] cannot exist in our world, it should have no problems existing in the Platonic Realm which is infinite. — keystone
I apologize if I'm not addressing the substance of your previous replies. It's not intentional, I thought I was. — keystone
In the Platonic Realm, can infinite objects exist but never be constructed? — keystone
My view of the Platonic Realm is that it's a world where infinite processes can be completed — keystone
In Set Theory we say 'There exists a set...'. What do we mean by this? — keystone
What I'm proposing is that there is no "contest" involving infinitely many "contestants". For example, I'm proposing that to do calculus we don't need to assume that a continuum is built from the assembly of infinitely many points. Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points? — keystone
All bijections are injections. So you're confused to begin with
The reason old-fashioned terminology is not so bad. One-to-one onto, etc. Bourbaki may be to blame? — jgill
You seem to be well read in math, philosophy, and science. Out of curiosity, what are you trained in? — keystone
No computer processing can make the image higher resolution than reality. — keystone
For a fully accurate depiction, the line width should be exactly 0. — keystone
I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers. — TonesInDeepFreeze
'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you. — TonesInDeepFreeze
Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points?
— keystone
Day one of high school Algebra 1:
"Students, we start with the set of real numbers and the real number line." — TonesInDeepFreeze
When math starts with points that are then assembled into curves, there is a way of describing those points on the real line, identifying a point with .5 for example. If you are starting out with curves or geometric figures you need to be classify them, order them somehow, for you then wish to create points by intersections I suppose. You need rigorous definitions for curves, then an axiomatic structure. All of which seems far-fetched. But who knows? — jgill
Basically, you have your concepts of how the statements of set theory should be understood, so you posit a realm to embody those concepts. But your concepts are not analogues of any actual statements in set theory. So it's a cheat. — TonesInDeepFreeze
By the way, are you capitalizing 'Platonic Realm' as a proper noun to shade discussion, even if just by hair, about mathematical realism? — TonesInDeepFreeze
What about the "final state"? There is no final state. — TonesInDeepFreeze
It is the very point that mathematics is not capable of such nonsense. — TonesInDeepFreeze
I'm referring to the fact that set theory proves there exists a complete ordered field and a total ordering of its carrier set. And then I highlighted that the carrier set is the set of real number and the total ordering is the standard less-than relation on the set of real numbers.
— TonesInDeepFreeze
You've repeated this a lot so it's clearly important. — keystone
'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you.
— TonesInDeepFreeze
How is that a contradiction? — keystone
[bold original]where we need to assume that the real line is composed of infinite points — keystone
Your proposal is finitism. It's a cool proposal and an interesting philosophical topic on its own right, but entirely unrelated to the paradox you were trying to solve (i.e. irrelevant). — Kuro
Basically, you have your concepts of how the statements of set theory should be understood, so you posit a realm to embody those concepts. But your concepts are not analogues of any actual statements in set theory. So it's a cheat.
— TonesInDeepFreeze
I don't think you're being reasonable here. — keystone
You say that many of the paradoxes are resolved by Set Theory — keystone
essentially it is you who is making the link between the two — keystone
but then you say that criticisms of the paradox can't touch Set Theory. You can't have it both ways. — keystone
Zeno's Paradox — keystone
Thompson's lamp.
It's a non-converging sequence.
Set theory doesn't have a "final state" with that.
But here's what set theory does have:
Let N = the set of natural numbers.
Let f be a function.
Let dom(f) = N
Let for all n in dom(f), f(n) = 1/(2^n)
So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...
0 is not in ran(f).
Let g be a function.
Let dom(g) = ran(f)
Let ran(g) = {"off", "on"}
Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)
So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ...
So I've mathematically "translated" Thompson's lamp.
What about the "final state"? There is no final state. The mathematics doesn't have a "final state" here, exactly because the mathematics doesn't have nonsense like "an infinite process with a final state". Of course you get a paradox from having an infinite process with a final state. It is the very point that mathematics is not capable of such nonsense.
But this:
Let (h) = g u {<0 "off">}
So the "final state" of h is "off". No contradiction.
Let j = g u {<0 "on">}
So the "final state" for j is "on". No contradiction.
Choose whichever "final state" you like - the "final state" with h or the "final state" with j. But neither is determined by g. — TonesInDeepFreeze
If mathematics does not allow for such nonsense then it cannot claim to resolve Zeno's paradox. — keystone
Does Set Theory model Achilles' journey or not? — keystone
Why do we need to order the curves? — keystone
'Infinite sets are empty' is a contradiction....
— TonesInDeepFreeze
How is that a contradiction?
— keystone
I'm sorry, but are you serious? — TonesInDeepFreeze
Here's another answer: Because if it had only finitely many points, we wouldn't be able talk about points greater than any of the positive points you chose or less than any of the negative points you chose. And we wouldn't be able to talk about those that are in between any two adjacent points. — TonesInDeepFreeze
All bijections are injections. So you're confused to begin with. — TonesInDeepFreeze
You're wasting our time. We already know that in set theory, infinite sets differ in this salient way from finite sets. — TonesInDeepFreeze
There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position. — TonesInDeepFreeze
so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory. — TonesInDeepFreeze
You require that sets can be "built" only in finite "processes". — TonesInDeepFreeze
In Set Theory we say 'There exists a set...'. What do we mean by this?
— keystone — TonesInDeepFreeze
Depends on what you mean by 'manipulate'. — TonesInDeepFreeze
Indeed, if no one mentioned a connection between the paradoxes and set theory, I'd have no objection at all! — TonesInDeepFreeze
You are completely missing the point. And now your lily pad jumping from Thompson's lamp to Zeno.
I gave you a rigorous answer regarding Thompson's lamp, but instead of comprehending that, you skip it as if it doesn't exist and frog hop to another pad.
You're not in good faith. — TonesInDeepFreeze
mathematics shows how to calculate that Achilles did finish the race. — TonesInDeepFreeze
With the mathematics, we have that certain functions are continuous. This allows that mathematics is modelled by such things as Achilles crossing the finish line. And we know that Achilles does cross the finish line. So the mathematics works. The paradox though is a model in which Achilles does not finish the race. So we're not so attracted to that model, and indeed thankful that it is not a model of the mathematics. — TonesInDeepFreeze
If you want to start with curves (or continuous objects) in order to derive points you need a systematic way of talking about them. — jgill
If you plan to assemble contours or curves by gluing together tiny straight lines, then you are doing nothing more than is done when one calculates the length of a contour in complex analysis. — jgill
So my comment is what difference does it make how you deal with that? — jgill
We've had posters here who spend years working up what they consider astounding revelatory articles, only to be more or less ignored. They become so enraptured with their ideas they get caught in that spiral in which the more effort you exert the more you think your product is of value, losing their objectivity. — jgill
My focus is systems science. Which means all of the above really. But neuroscience in particular. — apokrisis
But if you just want to resist the concept of vagueness, that’s your lookout. I can only say it was about the single most paradigm shifting thing I ever learnt. — apokrisis
How do you glue actually 0D points together to make a continuous line? How do you glue 0D width lines together to make a plane? — apokrisis
The issue to be resolved is how divisibility can co-exist with the continuum that it divides. — apokrisis
You don't build the whole from the parts. In my view you go the other way around. You start with the whole and cut it up. — keystone
'Infinite sets are empty' is a contradiction....
— TonesInDeepFreeze
How is that a contradiction?
— keystone
I'm sorry, but are you serious?
— TonesInDeepFreeze
If I write N = {1, 2, 3, ...} it seems that N has infinite elements. But appearances can be decieving. If someone proved that 1=2=3=... then N actually only contains one element. — keystone
Here's another answer: Because if it had only finitely many points, we wouldn't be able talk about points greater than any of the positive points you chose or less than any of the negative points you chose. And we wouldn't be able to talk about those that are in between any two adjacent points.
— TonesInDeepFreeze — keystone
I'm not proposing that the real line is composed of an unchanging, finite set of points. I'm proposing that the real line has infinite potential to 'give birth' to points as they are needed. — keystone
You never did respond to my post where I drew a circle for you. That post depicts what I mean, but in any case, I think the following paragraph will as well. — keystone
If you're a teacher and you tell the students to measure a table, you wouldn't hand them a bunch of nothing (points) to make the measurements. You would hand them a ruler (continuum) which has perhaps tic marks every 1/8 inch. If the table lines up halfway in between the tic marks, then and only then do you add a new tic mark based on the average of the adjacent tic marks. And if matter weren't discrete, you would have the potential to endlessly add tic marks to the ruler. It's just that you never would complete the job. — keystone
And in fact, if you somehow were able to completely populate the ruler with tic marks the ruler becomes useless. It's just one big black tic mark. In this example, The distinction between numbers is lost and the set of real numbers is no more useful than a set containing only one element. — keystone
Here's my airplane analogy. — keystone
Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly. — keystone
I'm acknowledging that they're good engineers and they've built a good plane but is that ritual really necessary? Is an actually infinity of points really necessary? — keystone
You complained that I skipped some of your arguments — keystone
in set theory, infinite sets differ in this salient way from finite sets.
— TonesInDeepFreeze
The way I would phrase it is that we already know that in set theory, infinite sets don't conform at all to the intuitions we've developed from all sets that we've actually worked with direction. — keystone
There's no law of the universe that one can't just chug along proving theorems without declaring a philosophical position.
— TonesInDeepFreeze
[...] philosophy of mathematics we can refine our intuitions and apply them in our quest to understand our universe. — keystone
paradoxes are the most important guidepost in our quest to see truth. — keystone
so you get a contradiction but outside the original terms you set: an imaginary world corresponding to set theory.
— TonesInDeepFreeze
I believe all worlds (physical and abstract) are simulated by computers. Not necessarily a computer made of silicon, just an entity that computes. One might say that 'God is a mathematician'. I'm not trying to squeeze set theory into a specific world, I'm just trying to imagine a computer with enough capacity to hold it. If you say that set theory is a finite set of rules, axioms, properties, etc. from which theorems could be derived then Set Theory could exist on a finite computer and all is good.
However, if you say that set theory describes the behavior and asserts the existence of another computer which directly works with the sets themselves then such a computer would necessarily be infinite in capacity (and infinite in speed as well to complete infinite operations in finite time). I'm willing to begin by assuming that such a computer exists, and then explore whether it would explode. I do not have the skill to prove that the computer that holds set theory will explode. However, I think I have the skill to at least discuss whether the computer that holds the infinity paradoxes will explode. — keystone
You require that sets can be "built" only in finite "processes".
— TonesInDeepFreeze
I'm willing to explore computers that perform supertasks, like one that simulates Zeno's Paradox where he performs infinite steps in finite time. And IF that computer explodes and IF I can form a 1-to-1 correspondence between the processes on that computer and the processes on the computer that holds the infinite sets then it's reasonable to say that that computer also explodes. If you think I haven't proven this yet...well that is true! — keystone
In Set Theory we say 'There exists a set...'. What do we mean by this?
— keystone
When I say something exists I mean there is a computer where it is in memory. [...] I just can't envision any computer holding even just the natural numbers without exploding.
So when you write "For a given model M, ExPx is true in M iff there is at least one member of the domain such that that member of the domain is also in the subset of the domain that is the denotation of 'P'", are you assuming the existence of an infinite computer? — keystone
Let's start with my main claim - Nothingness (i.e. points) cannot be assembled to form a something (i.e. a continuum), no matter how much of it you have. Even infinite nothingness is still nothingess. — keystone
I honestly thought you were covering both Thompson's Lamp and Zeno's Paradox simultaneously since you included the geometric series in your description — keystone
Step #, incremental distance, current state of lamp, dist. travelled
0, 1, on, 1
1, 1/2, off, 1+1/2
2, 1/4, on, 1+1/2+1/4
3, 1/8, off, 1+1/2+1/4+1/8
etc. — keystone
I argue that any valid demonstration uses a parts-from-whole (points-from-continuum) construction. — keystone
With the mathematics, we have that certain functions are continuous. This allows that mathematics is modelled by such things as Achilles crossing the finish line. And we know that Achilles does cross the finish line. So the mathematics works. The paradox though is a model in which Achilles does not finish the race. So we're not so attracted to that model, and indeed thankful that it is not a model of the mathematics.
— TonesInDeepFreeze
Aha! Your answer rests upon continua, not points! — keystone
The mathematics works because you're considering the journey as a whole. The complete journey exists first and only then can we choose to talk about what happens at t=0.5 or t=0.95 or any other instant. But we cannot talk about the journey starting from t=0 proceeding to the adjacent instant.....because there is no adjacent instant. Atalanta cannot even begin her journey. — keystone
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.