keystone

I have developed a paradox which (I think) shows the problem of using limits to define equality. Let's shelve this discussion until then. If you're up...

Look at the history, I've tried to end the discussion on this point but you keep looping it back it!

I'm going to let you have the last word on this point for now since I'm trying to keep the discussion focused on Thomson's lamp and continuous objects...

So do you agree that the = in an infinite summation means something different than the = in a finite summation?

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

It seems that this fundamental particle of set theory needs to be defined then. How about instead of continuous object, I say topological manifold? It...

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 no...

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

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...

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

Ok, I get what you're saying now. Okay, maybe I need to refine my statement. How about, if ZFC is inconsistent then you can prove that infinite sets a...

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 poi...

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 dis...

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 ...

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 eas...

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

"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 ot...

Very cool. I'll keep it in my back pocket in case it will be useful to me in the future :D In my view 0D points are not objects so you don't glue them...

Agreed. But you've got to start somewhere and fortunately it's easy to understand what I'm talking about, even though I'm talking informally. No, I'm ...

I mean what a computer does, either to the virtual objects or the bits themselves, when it computes. Let's start with my main claim - Nothingness (i.e...

@"TonesInDeepFreeze": You complained that I skipped some of your arguments so I'm going back to them. I knew I'd make a mistake with those terms, but ...

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 actuall...

If it doesn't make sense to think that something comes from nothing, maybe we need to revisit the belief that something comes from nothing. If it does...

In Achilles' journey he arrives at the destination. He takes a final step. Does Set Theory model Achilles' journey or not? If mathematics does not all...

I don't think you're being reasonable here. You say that many of the paradoxes are resolved by Set Theory (essentially it is you who is making the lin...

Why do we need to order the curves?

You've repeated this a lot so it's clearly important. Maybe my view is in disagreement with Set Theory then. In any case, I'm not in a position to cha...

I agree that this is the biggest weakness of his view. He complains about the foundations of math but his foundation is set of tick marks on a white b...

You seem to be well read in math, philosophy, and science. Out of curiosity, what are you trained in? I disagree. While an algorithm can indeed tune a...

Can you explain what you mean by 'catalogue all continuous curves'?

Thanks. I have admired him for many years. I loved his videos on real numbers. I've reached out to him a couple of times but didn't get much of a resp...

I apologize if I'm not addressing the substance of your previous replies. It's not intentional, I thought I was. As I mentioned in my other post, it's...

I see your point. In reality there is one truth (e.g. God either exists or not). I have this romantic/naive notion that in Mathematics there is simila...

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...

LoL.

As this conversation progresses, I've warmed up to set theory (although I currently hold an unorthodox view that set theory might not actually be abou...

Ultimately I want a mathematics that is formally supported by axioms and my preference is to keep what we've got (ZFC) because I'm guessing that so mu...

If you're going to describe a continuum with numbers you will need infinite decimal places. But of course, it doesn't have to be decimal. Binary is mo...

Yes, the conventional way to think of a circle is as an (actually) infinite set of points. What I'm proposing is that we think of the circle as the de...

I concede that there may be no better way to formalize calculus than set theory. I don't want to challenge the efficacy/validity of set theory. I only...

Okay, I'll concede this point and agree that set theory is used to formalize many branches of mathematics...and I don't think it's going away. But as ...

Maybe we both agree that our universe is like a simulation by a finite computer and as such our universe has some limitations in measurement, e.g. as ...

Ok, if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number? Not an axiom, but a...

I believe that cuts made to a continuum are perfectly precise since I can draw it with no vagueness. For example, consider this drawing of y=0 and y=x...

A finite being cannot exhibit or work with an infinite set directly. To do so requires an infinite being. Since one cannot disprove the existence of a...

I think saying "there exists a set of all natural numbers" is equivalent to writing a program to print all natural numbers and running it through to c...

Let's drop a dimension for simplicity. I can think of x^2+y^2=1 without having to think of any points. I can also draw that function without drawing a...

I can't speak to the standard axiomatization of analysis, but the informal definitions that us engineers were taught didn't use sets. As written on Wi...