You have a lot of patience. The "vagueness" principle apokrisis advances might work better. It would parallel the actual history of math.

The interval would be made so small that the length of the line was the same size as the width of the line - both being now effectively zero. — apokrisis

A line in turn would be arrived at as the constraint on the quality of “plane-ness”. Squish the 2D plane from either side and the limit of its compression becomes how a 1D line arises. — apokrisis

The sequence of contours of the form ${{Z}_{n}}(t)=t+i\tfrac{1}{\sqrt{n}}Sin\left( 2\pi tn\sqrt{n} \right)$ demonstrate convergence to both an infinitesimal straight line (on the x-axis) and the origin, 2D down to 1D down to a point. $t\to \tfrac{1}{\sqrt{n}}$ and $n\to \infty$ with ${{Z}_{n}}(t)\to 0$.

Oh oh, I almost forgot $\text{Length }{{Z}_{n}}(t)\to \infty$. You can ignore that I suppose :cool:

So the ontology is fundamentally complex. And hence not widely understood by folk....But good luck applying this kind of advanced systems logic to the simplicities of number theory. — apokrisis

"Behind it all is surely an idea so simple, so beautiful, that when we grasp it - in a decade, a century, or a millennium - we will all say to each other, how could it have been otherwise? How could we have been so stupid?"
John Archibald Wheeler

I'm just not feeling your theory. I don't see the point.

↪keystone

Some of your quote links are not going to the posts in which the quotes occur. — TonesInDeepFreeze

Oooh...I've been quoting wrong. Moving forward I'll do it the right way.

I'm just not feeling your theory. — keystone

It’s not “my” theory. And likewise, your feelings are irrelevant to the argument it makes. So whatever. :cool:

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

You answered the question. You're not serious. — TonesInDeepFreeze

Of course, I'm not saying that the natural numbers are actually equal. I'm saying that natural numbers as defined in an inconsistent system can be easily proven to be equal. IF ZFC were inconsistent and IF someone proved that to be the case then wouldn't this be the celebrated conclusion? But IF that were the case, that doesn't mean that math is wrong, it just means that ZFC is not a good foundation for mathematics.

I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist. — TonesInDeepFreeze

It's funny how you criticize me if I don't respond to some of your points but then you criticize me if you don't respond to some of my points.

And you egregiously obfuscate the terminology. 1/8 increments is not a continuum. You could at least give the consideration of not appropriating terminology in a blatantly incorrect way. — TonesInDeepFreeze

I'm saying that the wooden stick upon which tic marks are placed is the continuum.

The existence of the set of real numbers doesn't stop you from considering only a finite number of numbers for a given problem. — TonesInDeepFreeze

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them. The 1/8 tics on the ruler have purpose because in between them lies a space.

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

What on earth are you talking about? — TonesInDeepFreeze

I'm referring to the objects of set theory being beyond our grasp.

I've answered that and answered it and answered it already. The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it? — TonesInDeepFreeze

I get what you're saying, but I don't agree with it.

You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see? — TonesInDeepFreeze

Sometimes there's not enough room for two conflicting ideas.

Points are not "nothingness". — TonesInDeepFreeze

It occupies zero space.

You're adding things into what I wrote that are not there. — TonesInDeepFreeze

I'm confused why you embedded a geometric series into the definition.

----------

@TonesInDeepFreeze: Earlier today I was seeing that you responded to my posts and I was looking forward to our continued conversation on this thread but based on your responses it's clear that this conversation has run its course. As you've mentioned multiple times, we're going in circles. We both think the other is not listening or being reasonable. In any case, I do appreciate that you gave me a lot of your time at composing well written responses. I sincerely thank you.

It’s not “my” theory. And likewise, your feelings are irrelevant to the argument it makes. So whatever. :cool: — apokrisis

I do appreciate the discussion we've had though. Thanks!!!

Of course, I'm not saying that the natural numbers are actually equal. I'm saying that natural numbers as defined in an inconsistent system can be easily proven to be equal. IF ZFC were inconsistent and IF someone proved that to be the case then wouldn't this be the celebrated conclusion? — keystone

You don't know what you're saying.

ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty".

I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist.
— TonesInDeepFreeze

It's funny how you criticize me if I don't respond to some of your points but then you criticize me if you don't respond to some of my points. — keystone

What in the world? I don't criticize you if I don't respond to some of your points.

Anyway, my point stands: Your circle bit is yet another variation on your theme. I've responded over and over to such variations, even if not to the circle in particular. I've done enough.

the wooden stick upon which tic marks are placed is the continuum. — keystone

So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on.

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them. — keystone

Wrong. Look up the math sometime.

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

What on earth are you talking about?
— TonesInDeepFreeze

I'm referring to the objects of set theory being beyond our grasp. — keystone

You are very kind to your unresponsive dogmatism to just now omit the key point in my reply:

The mathematicians don't just assume the theory will work. Rather, they prove that it does, by deriving the existence of the real number system, then proving the theorems of mathematics used by the sciences. — TonesInDeepFreeze

I've answered that and answered it and answered it already. The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it?
— TonesInDeepFreeze

I get what you're saying, but I don't agree with it. — TonesInDeepFreeze

You don't get to disagree with it. It's a plain fact, no matter anyone's philosophy.

I'll explain it yet again for you: Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).

You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see?
— TonesInDeepFreeze

Sometimes there's not enough room for two conflicting ideas. — TonesInDeepFreeze

Don't worry about two. There's not enough room in your mind for even one coherent idea.

Anyway, my point stands that it is poor thinking to always try to jam the sense of one framework into another one incompatible with the first.

Points are not "nothingness".
— TonesInDeepFreeze

It occupies zero space. — TonesInDeepFreeze

Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness.

You're adding things into what I wrote that are not there.
— TonesInDeepFreeze

I'm confused why you embedded a geometric series into the definition. — TonesInDeepFreeze

Another doozy by you. I said that you add things into what I wrote, and you reply by adding it again!

There is no geometric series in my writeup about Thompson's lamp. Period. No geometric series. Look at it again, hopefully this time not hallucinating, and you will see that there is no geometric series there.

We both think the other is not listening or being reasonable. — keystone

We both think that, but you're wrong about it and I'm right about it. I have mulled over your remarks a pretty fair amount. I have turned them around in mind, including from what I understand to be your point of view, and I have responded on point to them as exhaustively as feasible for me. And I do have at least a little exposure to finitistic approaches and systems, and an interest in learning more about them.

You, on the other hand, just slide across what I say, not even taking a moment to understand, and instead getting it quite wrong, quite confused, and, at key points, essentially putting words in my mouth. And without a mote of intellectual curiosity to learn even the very first things about set theory.

And I don't insist on putting finitistic alternatives into the framework of set theory. But the core of your arguments, repeated over and over in same form or with variations, is to to insist on putting set theory into your finitistic frameworks.

I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse.

So nope, it's not a parity.

ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty". — TonesInDeepFreeze

In ZFC, is the equation 1+1=2 a definition, a theorem, or something else? My understanding is that if ZFC were inconsistent then one could prove both that the natural numbers are distinct and that they are equal.

So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on. — TonesInDeepFreeze

I dispute a continuum composed of points. I take it that you consider the points to equal the continuum.

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them.
— keystone

Wrong. Look up the math sometime. — TonesInDeepFreeze

I know the standard construction, starting with natural numbers then integers then rationals then reals, etc. And often we say that the naturals are defined as nested sets of sets. I am disturbed by this approach but I know in another thread you are already debating the definition of a set so let's leave it at that.

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity). — TonesInDeepFreeze

Then maybe ZFC is inadequate for analysis. Once this discussion has concluded I'm going to start a new post with my argument supporting this view. Perhaps we can discuss this further at that time.

Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness. — TonesInDeepFreeze

I see points as emergent from distance...but we've been here before...

I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse. — TonesInDeepFreeze

Care to try?

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" ... — TonesInDeepFreeze

Okay, I see. I forgot the details of the Thompson's Lamp paradox. f(n) corresponds to the incremental time of light switching, not the incremental distance Achilles travelled. To me that is a moot point, but let's hold off on distance for the moment and focus on your complaint that I'm trying to fit set theory into my finitist perspective.

Do you agree that your formal definition describes the informal notion that there exists a complete table (having no last term) as described below?

Step #[n], incremental time [f], current state of lamp [g]
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc.

Also, if you look at the Wikipedia page (https://en.wikipedia.org/wiki/Thomson%27s_lamp) you will see a table which is more closely aligned with the paradox:

Step #, cumulative time, current state of lamp
0, 1, on
1, 1+1/2, off
2, 1+1/2+1/4, on
3, 1+1/2+1/4+1/8, off
etc.

Do you think that the incremental time table and the cumulative time table convey the same information, just in a different format?

ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty".
— TonesInDeepFreeze

In ZFC, is the equation 1+1=2 a definition, a theorem, or something else? My understanding is that if ZFC were inconsistent then one could prove both that the natural numbers are distinct and that they are equal. — keystone

You tendered the notion that infinite sets are empty. I said that's a contradiction (more exactly, it's inconsistent). Then you replied that if set theory were inconsistent then set theory has that infinite sets are empty. And above you quoted me yourself instructing you that if set theory is inconsistent then still "infinite sets are empty" is inconsistent. (!!!)

So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on.
— TonesInDeepFreeze

I dispute a continuum composed of points. — keystone

Non responsive. You say there is no continuum, but in the imaginary world you describe, you have a ruler that you say is the continuum. Have cake or eat it. Choose one.

I take it that you consider the points to equal the continuum. — keystone

No, I never said that. I posted explicitly (in this thread or another that I think you were in) what the continuum is. The continuum is:

<R L> = {x | x in R} where L = the standard less than relation on R

In other context, it's okay to say that the continuum is:

<P M> where P = {<x 0> | x in R> and M = {<t s> | t in P & s in P & Exy(t = <x 0> & s =<y 0> & Lxy)}

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them.
— keystone

Wrong. Look up the math sometime.
— TonesInDeepFreeze

I know the standard construction, starting with natural numbers then integers then rationals then reals, etc. And often we say that the naturals are defined as nested sets of sets. I am disturbed by this approach but I know in another thread you are already debating the definition of a set so let's leave it at that — keystone

No, let's not leave it at that.

(1) I did not debate the definition of 'set'.

(2) That you are "disturbed" doesn't change the fact that in set theory, distinctness of natural numbers doesn't require consideration of a continuum. You are just plain flat out wrong.

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
— TonesInDeepFreeze

Then maybe ZFC is inadequate for analysis. — keystone

That is an idiotic non sequitur.

Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness.
— TonesInDeepFreeze

I see points as emergent from distance — keystone

Goody, your undefined 'nothingness' is justified by your undefined 'emergent from distance'.

I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse.
— TonesInDeepFreeze

Care to try? — keystone

At SAG-AFTRA rates. For that matter, I should already be charging you at least AFT rates for the instruction I'm giving you.

I just now recognized that it's 'Thomson' not 'Thompson'.

Okay, I see. I forgot the details of the Thompson's Lamp paradox. f(n) corresponds to the incremental time of light switching, not the incremental distance Achilles travelled. To me that is a moot point, — keystone

It's an important point, since it and the alternating states are what makes Thomson's lamp a different problem from Zeno's paradox.

Do you agree that your formal definition describes the informal notion that there exists a complete table (having no last term) as described below?

Step #[n], incremental time [f], current state of lamp [g]
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc. — keystone

No last entry. But keeping in mind that 'time' and 'current state' and 'lamp' are not in the mathematics itself.

Also, if you look at the Wikipedia page (https://en.wikipedia.org/wiki/Thomson%27s_lamp) you will see a table which is more closely aligned with the paradox:

Step #, cumulative time, current state of lamp
0, 1, on
1, 1+1/2, off
2, 1+1/2+1/4, on
3, 1+1/2+1/4+1/8, off
etc.

Do you think that the incremental time table and the cumulative time table convey the same information, just in a different format? — keystone

We can add whatever math you want to my writeup. Define:

s(0) = 1
s(n+1) = s(n)+(s(n)/2)

j(n) = <n s(n) g(n)>

And still my point about the writeup stands. We have an infinite sequence. There is no last entry in that sequence. (And we can also throw in the infinite series too, though it doesn't change the point). There is no contradiction there. Thomson's lamp is not a description of physical events. And it's not even model abstract set theory. Thomson's lamp does not show that set theory is inconsistent nor that set theory fails to provide mathematics for the sciences.

Ha! I hadn't read this until just now:

https://plato.stanford.edu/entries/spacetime-supertasks

"Benacerraf (1962) pointed out [that the] description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to “complete” the description of Thomson’s lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency."

And that is what my own writeup says too.

Here's what we have:

A putative description of an imaginary world (which is not a physical world).

The description is not coherent, since it posits that there is a last state for a process that does not have a last state.

Since the description is not coherent it does not specify even an imaginary world. Perforce, not a world that is a model of set theory.

Set theory is consistent*.

Set theory does provide a mathematical version of infinitely many steps. But not with a last step that is the successor to the previous step.

It's not the fault of set theory that it doesn't have a version of an impossible world. Indeed, it is a virtue of set theory that it doesn't have a version of an impossible word.

It is a fail to claim that Thomson's lamp impugns set theory. Indeed, if Thomson's lamp imgugns anything, it's the supertask that is described. Just as set theory does not assert that there exists such a supertask.

* Presumably it is consistent, (1) since no inconsistency has been found, and (2) by arguments regarding its hierarchical nature (see Boolos).

And one more detail:

And often we say that the naturals are defined as nested sets of sets. — keystone

The von Neumann characterization, which has been standard for a long time now is:

The set of natural numbers is the least successor inductive set.

df. Sn = n u {n}

thm. n is a natural number <-> (n = 0 or Ek(k is a natural number & n = Sk))

But now matter how we define the set of natural numbers, starting element, the successor operation and the starting element, as long as it is a Peano system*, then we get distinct natural numbers.

Distinctness does not depend on a particular characterization of the natural numbers (such as von Neumann's).

* And recall that all Peano systems are isomorphic with one another.

I just can't envision any computer holding even just the natural numbers without exploding. — keystone

Of course, one may adopt a thesis that mathematics should only mention what can happen with a computer (call it 'thesis C'). Then, go ahead and tell us your preferred rigrorous systemization for mathematics for the sciences that still abides by thesis C.

And one can reject thesis C. And there is a rigorous systemization of mathematics for the science that does not abide by thesis C.

I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane.

I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane — TonesInDeepFreeze

Any bird, bumblebee or dragonfly?

Perhaps maths, like logic and computation, is a view of nature that is accurate if nature were a machine. And perhaps nature is more than just that.

So no problem that you find maths useful. But nature seems larger in ways that still nag at the philosophical imagination.

Hence … systems science.

where we need to assume that the real line is composed of infinite points — keystone

You asked me about finitely many points, not about potentially infinitely many points. Be clear. — TonesInDeepFreeze

I'm not sure if this is what you're referring to but I don't mention finitely many points. As you know, I don't think the real line is composed of points. However, I can see the confusion resulting from my loose use of terminology. When I say real line or continuum you interpret that as the real number line because that's how it used. So that's my bad. Anyway, I'm talking about a continuous object, for example a line, a string, a surface, etc. These continuous objects can be used to do all the math that we typically do (e.g. graphing curves) but they're not composed of points.

I'm out of time, probably for a while.

But one more tidbit.

Some remarks here made my wonder how (using only first order PA (and theorems I already know proven in first order PA)) to prove:

There are no natural numbers m and n such that m < n < m+1.

I got stuck, so I looked it up. It's a bit trickier than one might think.

Toward a contradiction, suppose m < n < m+1.

Since ~ n < 0, let n = m+p, for some p, with ~ p = 0

Since n < m+1, let m+1 = n+q for some q, with ~ q = 0

So n+q+p = m+1+p = m+p+1 = n+1

So p+q = 1

Since ~ p = 0, for some j, let p = j+1

Since ~ q = 0, for some k, let q = k+1

So j+k+2 = 1

So j+k+1 = 0, which is impossible

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity). — TonesInDeepFreeze

Ok, I get what you're saying now.

Then you replied that if set theory were inconsistent then set theory has that infinite sets are empty. And above you quoted me yourself instructing you that if set theory is inconsistent then still "infinite sets are empty" is inconsistent. — TonesInDeepFreeze

Okay, maybe I need to refine my statement. How about, if ZFC is inconsistent then you can prove that infinite sets are empty and you can prove that infinite sets are infinite?

Non responsive. You say there is no continuum, but in the imaginary world you describe, you have a ruler that you say is the continuum. Have cake or eat it. Choose one. — TonesInDeepFreeze

If I stop using the word continuum and instead say that a ruler is a continuous object does that sit better with you?

I did not debate the definition of 'set'. — TonesInDeepFreeze

I'm referring to your discussion with MetaphysicsUndercover where he says that a set by definition cannot be the empty set. This I do not want to debate here.

That you are "disturbed" doesn't change the fact that in set theory, distinctness of natural numbers doesn't require consideration of a continuum. You are just plain flat out wrong. — TonesInDeepFreeze

This conversation is going in so many directions I'm willing to set this point aside for now. Suffice it to say that I see nested sets of sets containing no objects in a similar way as I see geometry constructed from points - they're both creating something from nothing. I believe I'd be rehashing the same sort of arguments that I've already provided which you surely don't want to hear again, nor will you be convinced by it. After all, I feel much more comfortable talking about geometry than set theory so let's please drop this for now and I will acknowledge that as far as we know today, in set theory distinctness of natural numbers doesn't require consideration of a continuum.

We can add whatever math you want to my writeup...And still my point about the writeup stands. We have an infinite sequence. — TonesInDeepFreeze

For the table involving cumulative time, there is no row corresponding to a cumulative time of exactly 2 min. Does that mean that the table (and your set theoretic description) only describes the state of the lamp as time approaches 2 min?

Thomson's lamp is not a description of physical events. And it's not even model abstract set theory. Thomson's lamp does not show that set theory is inconsistent nor that set theory fails to provide mathematics for the sciences. — TonesInDeepFreeze

I know it's not physically possible due to the physical limitations related to flicking a switch but I think we can set that detail aside. In this fictitious realm, can set theory can be used to describe the thought experiment of Thomson's Lamp all the way to 2 min? If Set Theory can't allow the cumulative time to reach 2 min, it seems that Set Theory fails to provide mathematics for the 'sciences' of this fictitious realm.

Benacerraf (1962) pointed out [that the] description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. — TonesInDeepFreeze

It sounds like by a similar reasoning we cannot say that 1+1/2+1/4+1/8+... = 2.

The description is not coherent, since it posits that there is a last state for a process that does not have a last state. — TonesInDeepFreeze

I agree, but as I just mentioned, by this logic we also cannot say that 1+1/2+1/4+1/8+... = 2 since this implies that there must be a last term to the summation. In other words, if we just look at the first two columns of the table...

Step #, cumulative time
0, 1
1, 1+1/2
2, 1+1/2+1/4
3, 1+1/2+1/4+1/8
etc.

...time never reaches 2 minutes.

Set theory does provide a mathematical version of infinitely many steps. But not with a last step that is the successor to the previous step. — TonesInDeepFreeze

Does the geometric series have the last step as a successor to the previous step?

It is a fail to claim that Thomson's lamp impugns set theory. Indeed, if Thomson's lamp imgugns anything, it's the supertask that is described. Just as set theory does not assert that there exists such a supertask. — TonesInDeepFreeze

It seems like the clock that counts to 2 minutes is performing a supertask. Can we say that set theory cannot describe clocks?

But now matter how we define the set of natural numbers, starting element, the successor operation and the starting element, as long as it is a Peano system*, then we get distinct natural numbers. — TonesInDeepFreeze

I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now.

if ZFC is inconsistent then you can prove that infinite sets are empty and you can prove that infinite sets are infinite? — keystone

Of course.

nested sets of sets containing no objects — keystone

There is only one set that has no members. That it is called a 'set' is extraneous to the formal theory. The formal theory doesn't even need to mention the word 'set'. We could just as well say "the object that has no members". And we don't even have to say "object". We could just say "There is unique x such that x has no members".

If I stop using the word continuum and instead say that a ruler is a continuous object does that sit better with you? — keystone

'continuous' is defined in mathematics. I don't know what you mean by it.

Does that mean that the table (and your set theoretic description) only describes the state of the lamp as time approaches 2? — keystone

That's close enough to what I already said. But, again, keep in mind, the mathematics does not describe the imaginary scenario in every respect - indeed, because the imaginary scenario is not coherent while the mathematics is consistent.

I know it's not physically possible due to the physical limitations related to flicking a switch but I think we can set that detail aside. — keystone

That's not even the issue. The problem is as I mentioned in my remarks.

Set Theory fails to provide mathematics for the 'sciences' of this fictitious realm. — keystone

There is only an incoherent description of something that can't even be a fictitious or abstract model of anything, because it can't be the case that there is a final state that is a successor state where, for each state, there is a successor state.

Especially a finitist would see that immediately. For a finitist there is no such realm, and for an infinitist too.

I'm not in a position to argue that Peano systems are inconsistent — keystone

In that context, I don't mean 'system' in the sense of axioms and a theory. I mean it in the sense of a tuple with a carrier set with a distinguished object and an operation, like an algebra. In that sense, 'consistency' or 'inconsistency' do not even apply.

Of course, one may adopt a thesis that mathematics should only mention what can happen with a computer (call it 'thesis C'). Then, go ahead and tell us your preferred rigrorous systemization for mathematics for the sciences that still abides by thesis C.

And one can reject thesis C. And there is a rigorous systemization of mathematics for the science that does not abide by thesis C.

I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane. — TonesInDeepFreeze

Every formal theory begins with an intuition. I don't have a formal theory. I also don't think you want to discuss the intuition further since it's not formal...but I believe it is a better airplane. Based on all of the infinity paradoxes, I can't help but think that the current "rigorous systemization of mathematics for the science that does not abide by thesis C" is inconsistent. I cannot prove that formally, but I can discuss the infinity paradoxes.

also cannot say that 1+1/2+1/4+1/8+... = 2 — keystone

WRONG. Yours is the typical claim of someone who knows not even the first week of Calculus 1.

An infinite summation is a LIMIT, not a final term in the sequence, as the sequence has no final term.

This is at the very heart of the paradox puzzles. The final state of the lamp is NOT a limit. There is no convergence between alternating "On" and "Off". So not only is there not a final state, but there's not even a limit of the sequence. The mathematics can't help, because the "realm" is impossible even on its own.

On the other hand, with Zeno's puzzle, the mathematics can offer that there is a limit to the sequence, thus infinite summation IS defiined.

Why don't you learn at least the first chapter in a Calculus 1 book?

There is only one set that has no members. That it is called a 'set' is extraneous to the formal theory. The formal theory doesn't even need to mention the word 'set'. We could just as well say "the object that has no members". — TonesInDeepFreeze

It seems that this fundamental particle of set theory needs to be defined then.

'continuous' is defined in mathematics. I don't know what you mean by it. — TonesInDeepFreeze

How about instead of continuous object, I say topological manifold?

There is only an incoherent description of something that can't even be a fictitious or abstract model of anything, because it can't be the case that there is a final state that is a successor state where, for each state, there is a successor state.

Especially a finitist would see that immediately. For a finitist there is no such realm, and for an infinitist too. — TonesInDeepFreeze

It's almost like you're saying that constructing the whole from the parts provides an incoherent description. I do see that immediately.

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity).
— TonesInDeepFreeze

Ok, I get what you're saying now. — keystone

Mark that as one of the very rare times a light goes on in a crank's mind. Alas, though, even when it happens, the crank will later double back to commit the error again.

In that context, I don't mean 'system' in the sense of axioms and theory. I mean it in the sense of a tuple of a carrier set with a distinguished object and an operation, like an algebra. In that sense, 'consistency' or 'inconsistency' do not even apply. — TonesInDeepFreeze

I can accept it as an algorithm for generating the set, but not as a completed set....but we've been here before....